The Principle of Linearised Stability for Non-Linear

Universität Ulm

The Principle of Linearised Stability forNon-Linear Parabolic Volterra Equations

Vorgelegt von

Marie-Luise Hein

aus Halle (Saale) im Jahr 2018

Dissertation zur Erlangung des Doktorgrades Dr. rer. nat. der Fakultät fürMathematik und Wirtschaftswissenschaften der Universität Ulm

Amtierender Dekan:Prof. Dr. Alexander Lindner

Gutacher:Prof. Dr. Rico ZacherProf. Dr. Wolfgang Arendt

Tag der Promotion:18. Mai 2018

Contents

Introduction iii

I Preliminaries 1

1 The Laplace Transform 31.1 Definition and General Properties . . . . . . . . . . . . . . . . 31.2 About the Laplace Transform of Holomorphic Functions . . . 5

1.2.1 An Analytic Representation Theorem . . . . . . . . . . 51.2.2 Asymptotic Behaviour of Vector-Valued Holomorphic

Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Scalar Volterra Equations and Kernels 192.1 Scalar Volterra Equations . . . . . . . . . . . . . . . . . . . . . 192.2 Completely Positive Kernels . . . . . . . . . . . . . . . . . . . 212.3 Completely Monotonic Functions and Kernels . . . . . . . . . 23

2.3.1 Properties of the Laplace Transform of CompletelyMonotonic Kernels . . . . . . . . . . . . . . . . . . . . 24

2.3.2 Permanence Properties of Completely Monotonic Func-tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.3.3 Examples for Completely Monotonic Kernels . . . . . 33

3 Linear Volterra Equations 373.1 Resolvents and Integral Resolvents . . . . . . . . . . . . . . . 373.2 Parabolic Equations . . . . . . . . . . . . . . . . . . . . . . . . 423.3 Existence and Estimates for Resolvents and Integral Resolvents 44

3.3.1 Existence and Boundedness of Resolvents . . . . . . . 443.3.2 Existence and Integrability of Integral Resolvents . . . 47

3.4 Time-Weights for Linear Volterra Equations . . . . . . . . . . 523.5 Commutator Term . . . . . . . . . . . . . . . . . . . . . . . . . 59

II Semilinear Parabolic Volterra Equations 61

4 Stability for Semilinear Parabolic Volterra Equations 634.1 Local Existence and Uniqueness of Mild Solutions and Maxi-

mal Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.1.1 Local Existence and Uniqueness . . . . . . . . . . . . . 644.1.2 Maximal Solutions . . . . . . . . . . . . . . . . . . . . 65

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Contents

4.2 Stability Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 69

5 Instability for Semilinear Parabolic Volterra Equations 755.1 Spectral Projections associated with Spectral Sets and Volterra

Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.1.1 Spectral Sets and Projections . . . . . . . . . . . . . . . 755.1.2 Projections of Resolvents and Integral Resolvents . . . 76

5.2 Exponentially Weighted Scalar Resolvents and Integral Resol-vents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.2.1 Long-Time Behaviour . . . . . . . . . . . . . . . . . . . 835.2.2 Convergence Rate . . . . . . . . . . . . . . . . . . . . . 87

5.3 Instability Theorem . . . . . . . . . . . . . . . . . . . . . . . . 96

III Quasilinear Parabolic Fractional Evolution Equations 105

6 Maximal Lp-Regularity 1076.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076.2 Maximal Lp-Regularity for Fractional Evolution Equations . . 110

7 Local Well-Posedness and Maximal Solutions of Quasilinear Prob-lems 1157.1 Local Well-Posedness . . . . . . . . . . . . . . . . . . . . . . . 1157.2 Maximal Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 122

8 Stability for Quasilinear Fractional Evolution Equations 1278.1 Commutator Term Estimates . . . . . . . . . . . . . . . . . . . 127

8.1.1 Product Rule . . . . . . . . . . . . . . . . . . . . . . . . 1278.1.2 Lp-Estimates for the Commutator Term . . . . . . . . 128

8.2 Stability Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 1358.3 Example – Time-Fractional Diffusion Equation . . . . . . . . 154

Appendices 159

A Operators and their Spectrum 161A.1 Closed Operators . . . . . . . . . . . . . . . . . . . . . . . . . 161A.2 Spectrum and Resolvent Operator . . . . . . . . . . . . . . . . 162

Bibliography 163

Zusammenfassung in deutscher Sprache 173

ii

IntroductionThis thesis is concerned with the long-time behaviour of solutions to non-linear abstract parabolic Volterra equations. The focus is both on a wideclass of semilinear parabolic Volterra equations and on quasilinear fractionalevolution equations. We aim at results which generalise the well-knownprinciple of linearised stability for classical evolution equations to theseclasses of non-local in time equations. On the basis of stability properties ofthe linearised problem we derive conclusions about the stability behaviour ofthe non-linear problem. The key idea is the same for both problem classes andconsists in a perturbation argument that allows to control small (understoodin an appropriate sense) non-linear terms. However, the required analyticaltools and methods to make this work differ significantly for the consideredclass of problems. In particular, in the quasilinear setting we use the methodof maximal Lp-regularity.

The Classical Principle of Linearised Stability for ODEs. The princi-ple of linearised stability is a well-known result in the theory of ordinarydifferential equations (ODEs). To describe it, let g : U ⊂ R

n → Rn be a con-

tinuously differentiable map, where U ⊂Rn is an open set. We consider the

differential equation ∂tv = g(v). Assuming that v∗ ∈ U is an equilibrium ofthis equation, i. e. v = v∗ is a stationary solution with g(v∗) = 0, we can rewritethe differential equation as follows. Setting u = v − v∗, A = g ′(v∗) ∈Rn×n, anddefining f : U − v∗ ⊂ R

n→ Rn by f (x) = g(x + v∗)− g ′(v∗)x, for x ∈ V = U − v∗,

the differential equation ∂tv = g(v) is equivalent to the transformed equation

∂tu +Au = f (u), (1)

with f (0) = 0 and f ′(0) = 0, for which u∗ = 0 is an equilibrium. The linearsystem ∂tw+Aw = 0 is the corresponding homogeneous linearised problem.

The stability behaviour of linear autonomous systems of ODEs is wellunderstood. By the principle of linearised stability it is possible to drawconclusions for the non-linear equation by referring to the linearised problem.Only the eigenvalues of the matrix A and their position in the complex planeare necessary to decide whether the equilibrium u∗ = 0 is stable or not. Denot-ing as usual by σ (A) the spectrum of the matrix A the principle of linearisedstability reads as follows.

Theorem. Let A ∈Rn×n, f ∈ C1(V ;Rn) with f (0) = 0, f ′(0) = 0.If σ (A) ⊂ λ ∈ C : Reλ > 0, then the equilibrium u∗ = 0 is asymptotically

stable for the equation (1). If there is an eigenvalue λ0 ∈ σ (A) with Reλ0 < 0, thenthe equilibrium u∗ = 0 is unstable.

iii

Introduction

A proof of this result can be found in many textbooks on the qualitativetheory of ordinary differential equations, for example see Prüss and Wilke[PW10], Amann [Ama90b] as well as Chicone [Chi06]. We point out that inthe case of asymptotic stability one gets an exponential convergence rate forthe decay to the equilibrium.

Semilinear Evolution Equations. The equation ∂tu +Au = f (u) can begeneralised to abstract semilinear evolution equations in an arbitrary Banachspace X where A is an unbounded linear operator. It can be treated in theframework of linear semigroup theory. We refer to the monographs of Pazy[Paz92], Lunardi [Lun95], Cazenave and Haraux [CH98], as well as Engeland Nagel [EN00] for more details about the semigroup approach. Thereare several works which transfer the stability result for ODEs to the abstractsemilinear setting. A principle of linearised stability for abstract evolutionequations can be found in Kielhöfer [Kie75], he considered the Hilbert spacecase. For results in Banach spaces we refer for example to Henry [Hen81],N. Kato [Kat95b] and Ruess [Rue03]. It is an important fact, that also inthese generalisations we have the exponential decay estimate in the case ofasymptotic stability.

Fractional Differential Equations. In the last decades fractional differ-ential equations have attracted a lot of interest for the modelling of processesin science and engineering. For example they can be used to model transmis-sion processes for media with memory, see [Prü12], and diffusion of fluids inporous media with memory, see [Cap99]. Moreover, they play an importantrole in describing anomalous diffusion processes, for more details we refer tothe survey articles [MK00] and [MK04], as well as [KLM12] with further ap-plications of fractional dynamics. Furthermore, we mention the monographsabout fractional differential equations of Podlubny [Pod99], Kilbas, Srivastavaand Trujillo [KST06] and Diethelm [Die10], which provide a good account ofthe theory at least in the finite dimensional case.

Before turning to non-linear fractional differential equations we first con-sider the linear scalar fractional differential equation

∂αt (u −u0) +µu = 0, t ∈R+, u(0) = u0, (2)

where α ∈ (0,1) and µ ∈C.Here, ∂αt (u −u0) denotes the Riemann-Liouville derivative of u −u0 given by

∂αt (u −u0) = ∂t [g1−α ∗ (u −u0)], with g1−α(t) = t−α/Γ (1−α), t ∈ (0,∞), and f ∗ gdenotes the convolution on the positive half-line of the functions f and g.1

1Note that we have to distinguish between the Riemann-Liouville derivative of u, given by∂αt u = ∂t [g1−α ∗u], and the so-called Caputo derivative of u which given by C∂αt u = ∂αt (u −u0).

iv

The solution of the equation (2) is given by u(t;u0) = Eα(−µtα)u0, t ∈R+;where Eα denotes the so-called Mittag-Leffler function – a generalisation ofthe exponential function. The decay behaviour of the Mittag-Leffler functionEα(−µtα) depends on the parameter µ. Indeed, for all µ ∈ z ∈C \ 0 : |argz| <π −απ2 there is some constant C > 0 such that

∣∣∣Eα(−µtα)∣∣∣ ≤ Ct−α, t ∈ (0,∞).

This means the Mittag-Leffler function decays to zero algebraically like t−α andit can be shown that this rate is optimal. For µ ∈ z ∈ C \ 0 : |argz| > π −απ2 the Mittag-Leffler function is unbounded. Note that for a fixed α ∈ (0,1) thebehaviour of the Mittag-Leffler function is substantially different from theexponential function, in particular the decay as t→∞ is much slower.

A similiar behaviour is also known for linear fractional differential equa-tions ∂αt (u − u0) +Au = 0, t ∈ R+, u(0) = u0, with A ∈ Rn×n. Depending onthe location of the eigenvalues of the matrix A in the complex plane one getsasymptotic stability if σ (A) ⊂ λ ∈ C \ 0 : |argλ| < π − απ2 or instability ifthere is one eigenvalue λ0 ∈ σ (A) with |argλ0| > π − απ2 . For this result werefer to the work of Matignon [Mat96], which also provides the decay rate t−α

in case of asymptotic stability.Now, we come to non-linear fractional differential equations. There is an

analogous version of the principle of linearised stability.

Theorem. Let α ∈ (0,1), A ∈ Rn×n, and f ∈ C1(V ;Rn) with f (0) = 0, f ′(0) = 0,where V ⊂R

n is open with 0 ∈ V . Then the following statements concerning thestability of the equilibrium u∗ = 0 of the equation

∂αt (u −u(0)) +Au = f (u), (3)

hold true.If σ (A) ⊂ λ ∈C : |argλ| < π −απ2 , then the equilibrium u∗ = 0 is asymptoti-

cally stable for (3). If there is an eigenvalue λ0 ∈ σ (A) with |argλ0| > π−απ2 , thenthe equilibrium u∗ = 0 is unstable.

This result of linearised stability had been an open problem for a long time.Only in 2016, Cong, Doan, Siegmund and Tuan [CDST16] gave a correct proofof the stability part of this result. The instability result has been establishedin [CDST17]. Their proofs rely essentially on properties of the matrix-valuedMittag-Leffler function and the Jordan normal form of the linearisation. Thereis no explicit statement about the decay rate in case of asymptotic stability.

Volterra Integral Equations. For the so-called standard kernel gα(t) =tα−1/Γ (α), t ∈ (0,∞), we have the property that gα ∗ g1−α ≡ 1 on (0,∞). Con-volving the fractional differential equation (3) with gα (and assuming that u

v

Introduction

is sufficiently smooth with u(0) = u0) leads to the Volterra integral equation

u + gα ∗Au = u0 + gα ∗ f (u).

In general, Volterra equations can be considered with an arbitrary locallyintegrable kernel instead of the standard kernel gα. There is a vast literatureabout such Volterra equations. Moreover, these equations can be also treatedin the abstract setting of Banach spaces with unbounded linear operators A.Important monographs in this context are the monograph of Gripenberg, Lon-den and Staffans [GLS90] for the finite-dimensional case and the monographof Prüss [Prü12] for arbitrary Banach spaces. Further important contributionsare the works of Londen [Lon78], Clément et al. [CLN78], [CN79] and [CN81],Gripenberg [Gri79] and [Gri85], as well as Arendt and Prüss [AP92], aboutthe existence and uniqueness, regularity and long-time behaviour of solutionsof abstract linear and non-linear Volterra equations.

For the class of semilinear Volterra equations a principle of linearisedstability is obtained by N. Kato [Kat95a] and [Kat97] as well as by Londen andRuess [LR17]. Here A is an accretive operator and the kernel of the Volterraequation is completely positive, but bounded at zero. In particular, theseresults do not include the important case of the (singular) standard kernel.

Abstract Semilinear Volterra Equations. One goal of this thesis is toestablish a principle of linearised stability for a wide class of abstract semi-linear Volterra equations which can be also reformulated as abstract integro-differential equations and include in particular the case of fractional evolutionequations. Let us describe the class of problems to be studied. We supposethat a ∈ L1,loc(R+) is an unbounded, completely monotonic kernel and denoteby b ∈ L1,loc(R+) the corresponding completely monotonic kernel with a∗b ≡ 1on (0,∞); it is known that there is always a unique kernel b with this property.Furthermore, let A be a linear, closed and densely defined operator on the Ba-nach space X, f ∈ C1(X;X) with f (0) = 0 and f ′(0) = 0, as well as u0 ∈ X. Forthe unknown function u : J → X we consider the semilinear Volterra equation

u + a ∗Au = u0 + a ∗ f (u), t ∈ J ; (4)

where J is either a compact interval [0,T ], T > 0, or R+. Due to the kernelproperty a ∗b ≡ 1 we are able to rewrite the equation (4) as integro-differentialequation given by

∂t [b ∗ (u −u0)] +Au = f (u), t ∈ J, u(0) = u0. (5)

An equilibrium of the Volterra equation (4) and the integro-differential equa-tion (5), respectively, is given by u∗ = 0, which is a steady solution for the

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problem. Note that the stationary solutions of the Volterra equation (4) andintegro-differential equation (5), respectively, are the same as in the case ofthe corresponding classical semilinear evolution equation ∂tu +Au = f (u),t ∈ J , u(0) = u0, but the dynamical behaviour is completely different.

In contrast to classical evolution equations there are no semigroups forVolterra equations. In fact one has to generalise the concept of semigroups,leading to the notions of so-called resolvent families S(t)t∈R+

⊂ B(X) andintegral resolvent families R(t)t∈R+

⊂ B(X), see [Prü12]. With these twoobjects one can represent the solution of the Volterra equation in a waysimilar to the well-known variation of constants formula in the classical case,provided that these objects exist. By means of these operator families onecan introduce a mild solution u ∈ C(J ;X) of the semilinear problem (4) assolution of the fixed point equation u(t) = S(t)u0 + (R ∗ f (u))(t), t ∈ J . We provethe existence of a unique local mild solution in case of a locally Lipschitzcontinuous function f . Subsequently, we define the continuation of such asolution. Unlike the classical case one cannot restart the solution at a latertime with a new initial value, since the memory effect of the convolution termmust be taken into account. Note that all these considerations assume thatthere are a resolvent family and an integral resolvent family for the problem(4), see Chapter 3 for definitions.

The main result about the stability of semilinear Volterra equations readsas follows.

Theorem. Let X be a complex Banach space and A be a linear closed and denselydefined operator in X. Furthermore, let a ∈ L1,loc(R+) be an unbounded, completelymonotonic kernel and b ∈ L1,loc(R+) be the corresponding kernel such that a ∗ b ≡ 1on (0,∞). Moreover, let f ∈ C1(X;X) with f (0) = 0 and f ′(0) = 0. We assume, thatthe Volterra equation (4) admits a bounded resolvent S(t)t∈R+

and an integrableintegral resolvent R(t)t∈R+

.Then the equilibrium u∗ = 0 is stable for the Volterra equation (4). Moreover,

if a < L1(R+) the equilibrium u∗ = 0 is asymptotically stable with rate sε(t) forsome sufficiently small ε > 0, where sε is the unique solution of the scalar resolventequation sε + ε(a ∗ sε) = 1 on R+.

Note that in case of the standard kernel a = gα we have sε(t) = Eα(−εtα)and so the decay rate is algebraic. It is important to note that exactly the samedecay rate as for the scalar resolvent is obtained for the abstract semilinearVolterra equation.

The proof of the exact decay rate is one of the hardest and for our con-siderations most important parts of this thesis. In order to succeed we useappropriate time-weights and prove the boundedness of the time-weightedsolution, which in turn solves a more complicated Volterra equation in the

vii

Introduction

mild sense. The difficulty comes from an additional commutator term, whichhas to be studied in detail. Actually, this is the same approach like in the caseof classical evolution equations. However, in the classical case the situation ismuch simpler since an exponential time-weight merely leads to a shift of theoperator A, which is no longer the case in our situation.

A large part of the theory about Volterra equations heavily relies on thestudy of the behaviour of the Laplace transformed equation. It is necessaryto develop further the known tools relying on the Laplace transform to getmore precise results on the asymptotic behaviour. For this reason we prove anextended version of the known Analytic Representation Theorem from Arendtet al. [ABHN01, Theorem 2.6.1] and the Tauberian Theorem [ABHN01, Theo-rem 2.6.4] for the Laplace transform of holomorphic functions. In contrastto the known result we allow for a more general decay behaviour such as t−α,α ∈ (0,1]. Note that these generalisations also are interesting in its own.

In case of parabolic Volterra equations the existence of a bounded resolventS(t)t∈R+

is well-understood, see the monograph of Prüss [Prü12]. The exis-tence of an integral resolvent family has not been finally clarified. Althoughthere is a generation theorem for the integral resolvent R(t)t∈R+

, sufficientand easily verifiable conditions for the existence of the integral resolvent andtheir integrability seem to be not known. Some comments concerning thisquestion can be found in [Prü12, Comment 2.7 d)] and [Prü96b, p. 297], butthe proofs of the sufficiency of the given conditions are missing. In this thesis,we prove criteria for the existence and integrability of an integral resolvent incase of parabolic Volterra equations which are easy to check. These criteriacover a large class of kernels, in particular the standard kernel, and lead tothe following version of the stability result.

Theorem. Let a ∈ L1,loc(R+) be an unbounded, completely monotonic kernel whichis θa-sectorial with some θa ∈ (0, π2 ]. We denote by b ∈ L1,loc(R+) the corresponding

kernel with a ∗ b ≡ 1 on (0,∞). Assume that∫ 1

0 a(1/t)dtt < ∞,

∫ 10 b(1/t)dtt < ∞

and[t 7→mina(1/t)/t, b(1/t)/t2

]∈ L1(R+). Moreover, let A be an invertible and

sectorial operator with spectral angle ϕA < π − θa and let f ∈ C1(X;X) withf (0) = 0 and f ′(0) = 0.

Then the equilibrium u∗ = 0 is stable for the Volterra equation (4). If a < L1(R+)then u∗ = 0 is asymptotically stable with rate sε(t) for some ε > 0.

Observe that, apart from the decay behaviour of the kernels, we onlyrequire that the invertible sectorial operator A is compatible to the kernel a inthe sense of parabolicity.

Moreover, we are also able to give sufficient conditions for the instabilityof the solution u∗ = 0. With the spectral projection along a compact spectralset we show that our Volterra equation as well as the resolvent and integral

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resolvent family decomposes in the expected sense. In case of the standardkernel we get instability if the spectrum of the operator A has a compact partin the complement of the stability sector. In the general case the instabilitycondition depends on the mapping properties of the Laplace transform ofthe kernel a. The used estimates for the proof require a deep insight into thegrowth behaviour of the scalar resolvent and integral resolvent.

In case of the standard kernel the instability result reads as follows.

Theorem. Let X be a complex Banach space, α ∈ (0,1) and A be a closed linearoperator in X with dense domain such that σ (−A)∩∂Σα π

2= ∅ and −σ+ = σ (−A)∩

Σα π2

is a non-empty compact set and σ− = σ (A) \ σ+ is a closed subset of C.We denote by P+ the spectral projection of the operator A associated with the

compact spectral set σ+. Let A− be the part of A in Rg(Id−P+) with σ (A−) = σ−. Weassume that A− is an invertible sectorial operator with spectral angle ϕA− < π−α

π2 .

Moreover, suppose that f ∈ C1(X;X) with f (0) = 0 and f ′(0) = 0.Then, the equilibrium u∗ = 0 is unstable for the Volterra equation (4) with the

standard kernel a = gα.

It should be noted that the admissible pairs of kernels and operators in ourtheorems cover a large class of Volterra equations. Observe that in particularthe finite dimensional case for fractional differential equations from [CDST16]and [CDST17] is included.

Quasilinear Equations. A more general and particularly with regardto applications interesting class of equations are quasilinear evolution equa-tions. A good example is provided by diffusion equations where the diffusioncoefficients depend on the concentration and/or its gradient. Important contri-butions to the theory of abstract quasilinear parabolic evolution equations arethe monographs of Lunardi [Lun95] and the very recent monograph of Prüssand Simonett [PS16] which give a very good overview about the current stateof research in the field of abstract evolution equations. Furthermore, we referto the publications of Amann [Ama86, Ama88, Ama90b, Ama90a, Ama93].

For abstract evolution equations it is known that linearisation techniquesand maximal regularity, together with a fixed point argument, yield an elegantapproach to quasilinear problems, in particular local well-posedness can beshown. Here, we refer to the papers of Clément and Li [CL94], Clément andPrüss [CP92], Prüss [Prü02], Prüss and Simonett [PS04] and Amann [Ama05].Especially with regard to the application of the abstract theory to partialdifferential equations the work [DHP03] and [DHP07] of Denk, Hieber andPrüss have to be mentioned here.

Maximal Lp-regularity and the linearisation method can be also used forthe stability analysis of equilibria of quasilinear parabolic evolution equations,

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Introduction

see the work of Prüss et al. [PSZ09], [PWS13] and [PS16]. Their considerationseven go beyond the case of discrete equilibria. Here, smooth manifolds ofequilibria are studied, for this reason the concept is called generalised principleof linearised stability.

Beside the approach based on maximal Lp-regularity one can also usethe framework of continuous maximal regularity. As to evolution equationswe refer to the work of Simonett [Sim94] and Clément and Simonett [CS01].Fractional quasilinear parabolic evolution equations are considered in thissetting by Clément, Londen and Simonett in [CLS04]. Via continuous max-imal regularity results for linear equations they obtain result on existence,uniqueness and continuation of solutions for the quasilinear equation. Notethat in this context there is no linearised stability result known.

The second focus of this thesis is on the stability of equilibria of quasilinearfractional evolution equations in the setting of maximal Lp-regularity.

Firstly, we want to mention the work of Prüss [Prü91], where quasilinearVolterra equations in spaces of integrable functions are investigated, andhis monograph [Prü12, Chapter 8] about parabolic Volterra equations in Lp-spaces. Next, we refer to Zacher, who considered abstract Volterra equationsin the setting of maximal Lp-regularity in [Zac03] and [Zac05], as well as[Zac06]. Moreover, in this framework it is important to refer to the work[Zac09], [PVZ10], [Zac10] and [Zac12], as well as to the work about long-timebehaviour of a special class of Volterra equations [VZ15], [KSVZ16], [VZ17]and [KSZ17]. Additionally, we mention the thesis from Bajlekova [Baj01], herealso abstract fractional evolution equation are considered, but in particular thewell-posedness result for quasilinear equation is formulated for the Riemann-Liouville derivative of the unknown function and not the Caputo one.

Let us now describe the maximal Lp-regularity setting for the quasilin-ear problem, for details see Part III. For Banach spaces X0,X1 of class HTwith dense embedding X1 → X0, p ∈ (1,∞) and α ∈ (1/p,1) we consider thequasilinear fractional evolution equation

∂αt (u −u0) +A(u)u = F(u), u(0) = u0 ∈ V , (6)

where V is an open subset of the real interpolation space (X0,X1)1− 1αp ,p

, and

(A,F) : V → B(X1;X0) × X0. We are interested in solutions of maximal Lp-regularity, i. e. u ∈Hα

p ([0,T ];X0)∩Lp([0,T ];X1).We want to use the maximal Lp-regularity results for the linear equation

to get results on existence, uniqueness and continuation for the quasilinearequation. The local well-posedness of the quasilinear problem for initialvalues in the neighbourhood of u0 is proved under the assumption that themaps A and F are locally Lipschitz continuous and the operator A(u0) has

x

the property of maximal Lp-regularity on [0,T ]. The idea of the proof isexactly the same as in the classical case of quasilinear evolution equations.Subsequently, we show that any local solution can be continued to a maximalinterval of existence. Observe that in contrast to the classical case, it isnot possible to restart straightforwardly a local solution of the quasilinearfractional evolution at a later time with a new initial value, due to the non-local nature of the equation.

Our main result about the stability of equilibria of quasilinear fractionalevolution equations reads as follows. Here, RS(X0) denotes the class of R-sectorial operators in X0, see Chapter 6 for the definition.

Theorem. Let p ∈ (1,∞), α ∈ (1/p,1) and X0, X1 as described above. Supposethat (A,F) ∈ C1(V ;B(X1;X0)×X0) and assume that u∗ ∈ V ∩X1 is an equilibriumof the fractional evolution equation (6), i. e. A(u∗)u∗ = F(u∗).

For the linearisation of (6) given by A0v = A(u∗)v + (A′(u∗)v)u∗ − F′(u∗)v,v ∈ X1, we assume that the operator A0 ∈ RS(X0) is invertible with ϕRA0

< π−απ2 ,i. e. the operator A0 has the property of maximal Lp-regularity on R+.

Then the equilibrium u∗ is asymptotically stable in Xγ for the quasilinearequation (6) with an algebraic decay rate with exponent between 0 and α − 1/p .

Under the additional assumption (A,F) ∈ C2(V ;B(X1,X0)×X0) the decay ratein Xγ is algebraic with exponent α.

Also in the quasilinear case the main idea for the proof rests on time-weights and the study of a suitably reformulated equation for the weightedsolution. As before this involves a commutator term for which we now needLp-estimates. The main difference to the classical result is the fact that for theoptimal decay rate a higher regularity of the maps A and F is required. Thisadditional regularity allows us to get better estimates for the non-linearityand to guarantee the known decay rate from the linear problem. It is subjectto further research whether this additional regularity assumption is really nec-essary or if it can be weakened. We point out that for the proof of asymptoticstability with a weaker algebraic rate this additional regularity assumption isnot needed.

Overall, this result is the only known generalisation to quasilinear frac-tional evolution equations of the known principle of linearised stability forquasilinear evolution equations. Up to the additional technical regularityassumption for the optimal decay rate it generalises the classical result in theexpected manner.

xi

Introduction

Organization of the Thesis. The thesis is organized as follows.

Chapter 1. The Laplace transform is the main tool for the treatment ofabstract Volterra equations. In the first chapter we introduce the Laplacetransform for vector-valued functions, summarise their most important prop-erties and link to the convolution on R+. In order to examine the asymptoticbehaviour of the Laplace transform of holomorphic functions we prove anAnalytic Representation Theorem and a Tauberian Theorem. These theoremsare fundamental for the following considerations.

Chapter 2. In this chapter basic definitions and properties concerningscalar Volterra equations and their kernels are summarised. We start withthe definition of the scalar resolvent and integral resolvent and give a repre-sentation formula for the solutions of Volterra equations. We introduce theclass of completely positive kernels and the resulting properties for the solu-tions of Volterra equations with these kernels. Next, we restrict our interestto completely monotonic kernels since their Laplace transform is very well-understood, which is a convenient circumstance with regard to our used tools.In addition, for completely monotonic kernels we prove that properties likesectoriality and regularity can be extended to a slightly larger sector then theright complex half-plan. We close this chapter with permanence propertiesfor completely monotonic kernels and give examples of such kernels.

Chapter 3. For Volterra equations the concept of resolvent and inte-gral resolvent families is paramount, the definitions of these objects aregiven in this chapter and we quote generation theorems for these objects.In case of parabolic Volterra equations the existence of resolvent familiesis well-understood. For these resolvent families we prove decay estimatesand examine their stability properties. Furthermore, we give sufficient condi-tions which ensure the existence and integrability of integral resolvents forparabolic Volterra equations. We close this chapter with a time-weightingargument for linear Volterra equations and the calculation of the raisingcommutator term for a special choice of time-weights. These arguments areessential for the proofs of the decay rates for asymptotic stability in case ofsemilinear Volterra equations in Part II and quasilinear fractional evolutionequations in Part III.

Chapter 4. In this chapter we introduce semilinear Volterra equations.On the basis of the resolvent and integral resolvent family we define mildsolutions and prove for locally Lipschitz continuous non-linearities the exis-tence of unique mild solutions. Moreover, we explain how to continue these

xii

solutions and define maximal solutions. The main result of this chapter isthe stability result for semilinear Volterra equations. Assuming the existenceof a bounded resolvent and an integrable integral resolvent for the Volterraequation we can prove the stability of the equilibrium. Moreover, in case ofasymptotic stability we can show that the decay rate is similar to the asymp-totic behaviour of the corresponding scalar resolvent. Using the sufficientconditions for the existence of a bounded resolvent and integrable integralresolvent from Chapter 3 we formulate a further version of the stability re-sult. Finally, we pay special attention to the situation of parabolic Volterraequations with standard kernels.

Chapter 5. The aim of this chapter is to prove an instability result forsemilinear parabolic problems. In order to do this we consider the spectralprojection of an operator associated with a compact spectral set. We showthe compatibility of this projection with the resolvent and integral resolventfamily and their resulting decomposition. Since the part of the operator whichis associated with the compact spectral set is a bounded operator, the corre-sponding resolvent and integral resolvent family can be represented usingDunford’s functional calculus. To understand these objects, it is necessary tostudy the scalar resolvents and integral resolvents, as well as the mappingbehaviour of the Laplace transform of the kernel in detail. We consider appro-priate exponential weights for the scalar resolvents and integral resolventsand prove the convergence and the decay rate of these weighted objects. Animportant issue during these considerations is the uniformity of the estimatesalong compact sets. Finally, we can prove our instability result for semilinearparabolic Volterra equations. This chapter closes with two corollaries whichconsider the situation of parabolic Volterra equations with standard kernels.

Chapter 6. To treat quasilinear fractional evolution equations in theframework of maximal Lp-regularity it is necessary to introduce the mostimportant definitions and underlying objects. Next, we give sufficient condi-tions for maximal Lp-regularity of linear fractional evolution equations andquote the necessary trace space theorem. This chapter is for informationalpurposes only since we use the framework of maximal Lp-regularity merelyas a concept.

Chapter 7. In this chapter we prove the local well-posedness of quasilin-ear fractional evolution equations under the assumption that an appropriatedlinear problem has the property of maximal Lp-regularity on a compact in-terval. Together with the contraction mapping principle we are able to drawconclusions about the solution of the quasilinear problem. This approach

xiii

Introduction

is exactly the same as in the case of classical quasilinear evolution equation.On the basis of the existence of a unique local solution we explain how tocontinue such a local solution and define a maximal solution.

Chapter 8. In the last chapter of this thesis we turn to the stability studyof quasilinear fractional evolution equations. Similar to our approach for thesemilinear equations we use time-weights and study a suitable reformulatedequation for the weighted solution. Here an additional commutator termarises, which has to be studied in detail. For two special choices of time-weighting functions we show Lp-estimates for the commutator term. Underthe same assumptions as in the classical case we can prove the asymptoticstability of an equilibrium of the quasilinear fractional evolution equation.Assuming an additional regularity of the non-linearities the optimal algebraicdecay rate with exponent α can be shown.

We close this chapter with the application of our results to a fractionaldiffusion equation on bounded domains.

Basic Notations. We fix some notations used throughout this thesis,which are fairly standard in the modern mathematical literature. By N, Z, R,C we denote the set of natural numbers, integers, real and complex numbers,respectively, N0 = N∪ 0, R+ = [0,∞) and C+ = z ∈C : Rez > 0. We denoteby Σϕ = z ∈ C \ 0 : |argz| < ϕ the open sector in C symmetric to R+ withopening angle ϕ ∈ (0,π]. The Euclidean norm in R

n is denoted by |·|. For ametric space (M,d) and N ⊂M we designate by N c, N , ∂N the complement,closure and boundary of N , respectively.

X, Y , Z will always be Banach spaces endowed with norms ‖·‖X , ‖·‖Y , ‖·‖Z .B(X,Y ) denotes the space of all bounded linear operators from X to Y , we

write B(X) = B(X,X) for short. BXε (x0) and BXε (x0) are the open and closed

balls, respectively, with centre x0 and radius ε in the Banach space X; thesuperscript is dropped in case no confusion is likely.

IfA is a linear operator inX,D(A), Rg(A), Ker(A) denote the domain, rangeand kernel of the operator A, respectively, while σ (A) and ρ(A) designatethe spectrum and resolvent set of the operator A. For a closed operator Awe denote by XA the domain of A equipped with the graph norm, ‖x‖A =‖x‖X + ‖Ax‖X , x ∈ D(A).

If (Ω,A,µ) is a measure space and X a Banach space, then Lp((Ω,A,µ);X),p ∈ [1,∞), denotes the space of all Bochner-measurable functions f : Ω→ Xsuch that ‖f (·)‖pX is integrable. This space is a Banach space, when it is

endowed with the norm ‖f ‖Lp((Ω,A,µ);X) =(∫

Ω‖f (t)‖pX dµ(t)

)1/p, and functions

equal almost everywhere are identified.

xiv

For Ω ⊂Rn open, A the Lebesgue σ -algebra, and µ the Lebesgue measure,

we abbreviate Lp((Ω,A,µ);X) to Lp(Ω;X), p ∈ [1,∞). In this case Hmp (Ω;X) is

the space of all functions f : Ω→ X having distributional derivatives ∂αf ∈Lp(Ω;X) of order |α| ≤ m. The norm in Hm

p (Ω;X) is given by ‖f ‖Hmp (Ω;X) =(∑

|α|≤m ‖∂αf ‖pLp(Ω;X)

)1/p.

For Ω ⊂ Rn open, Cm(Ω;X) denotes the space of all functions f : Ω→ X

which admit continuous partial derivatives ∂αf in Ω, for each multi-index αwith |α| ≤m.

For f ∈ C(Ω;X) the support of f is defined by suppf = x ∈Ω : f (x) , 0.We call a map F : X → Y locally Lipschitz continuous, if for all x ∈ X

there is some r = r(x) > 0 and L = L(x,r) > 0 such that for all y,z ∈ BXr (x) wehave the estimate

∥∥∥f (x)− f (y)∥∥∥Y≤ L

∥∥∥x − y∥∥∥X

. We define C1−(X;Y ) = f : X→Y is locally Lipschitz continuous, note that C1(X;Y ) ⊂ C1−(X;Y ).

The subscript ’loc’ assigned to any of the above function spaces meansmembership to the corresponding space when restricted to compact subsetsof its domain. In the scalar case X = R or X = C we usually omit the imagespace in the function space notation.

Acknowledgement. First of all I want to thank my supervisor Prof. Dr. RicoZacher for his support and guidance over the past years. Moreover, I wantto thank Prof. Dr. Wolfgang Arendt for accepting to be the second refereeof this thesis. I also want to thank all the former and current members ofthe Institute of Applied Analysis at the University of Ulm for creating such apleasant working atmosphere.

In particular, I want to thank Jochen Glück for many interesting discus-sions and his valuable hints. A special thanks goes to Marcel Kreuter andManfred Sauter for their moral support, particularly in the last phase of thisdoctorate.

Above and all, I want to thank my parents, my brother and Lars for theirenduring support over all the years.

xv

Part I

Preliminaries

1

1The Laplace TransformIn the first part of this chapter we introduce the Laplace transform for vector-valued functions and summarise important general properties of the Laplacetransform. Moreover, we respond to the connection between the Laplacetransform and the convolution of R+, which is introduced as well.

In the further course of this chapter we deal with the Laplace transform ofholomorphic functions and their asymptotic behaviour. We prove importantresults which are fundamental for the following considerations.

Here, (X,‖·‖X) always denotes a complex Banach space.

1.1 Definition and General Properties

We start with the definition of the Laplace integral of locally integrable vector-valued functions.

Definition 1.1.1 (Laplace Transform and Abscissa of Convergence). Letf ∈ L1,loc(R+;X) and λ ∈C. If the improper integral

f (λ) =∫ ∞

0e−λtf (t)dt,

exists in the sense of Bochner, we call it the Laplace integral or the Laplacetransform of the function f at the point λ.

The abscissa of convergence of f is given by abs(f ) = infReλ : f (λ) exists .

Remark 1.1.2. a) We know from Arendt, Batty, Hieber and Neubrander[ABHN01, Proposition 1.4.1] that the Laplace integral f (λ) convergesif Reλ > abs(f ) and diverges if Reλ < abs(f ). Moreover, according toSection 1.4 from Arendt et al. [ABHN01] we have: If f (λ) converges forall λ ∈C, then abs(f ) = −∞, if the domain of convergence is empty, thenabs(f ) =∞. In the case of abs(f ) <∞, the function f is called Laplacetransformable.

b) If the function f ∈ L1,loc(R+;X) is of exponential growth, i. e. there issome ω ∈ R such that

∫∞0 e−ωt ‖f (t)‖X dt <∞, then f (λ) is well-defined

for all Reλ ≥ω. In particular, we have abs(f ) < ω.

c) For more detailed information about the vector-valued Laplace trans-form we refer to Arendt, Batty, Hieber and Neubrander [ABHN01]as well as Hille and Phillips [HP57, Chapter VI]. The standard ref-erences for the classical Laplace transform are Widder [Wid41] andDoetsch [Doe50].

3

1. The Laplace Transform

For a proof of the following theorem about the analytic behaviour of thevector-valued Laplace transform we refer Arendt, Batty, Hieber and Neubran-der, see [ABHN01, Theorem 1.5.1].

Theorem 1.1.3 (Analytic Behaviour of the Laplace Transform). Let f ∈L1,loc(R+;X) with abs(f ) <∞. Then λ 7→ f (λ) is holomorphic for Reλ > abs(f )and, for all n ∈N0 and Reλ > abs(f ) we have

f (n)(λ) =∫ ∞

0e−λt(−t)nf (t)dt,

as an improper Bochner integral.

We have the following uniqueness theorem for the vector-valued Laplacetransform, cf. [ABHN01, Theorem 1.7.3].

Theorem 1.1.4 (Uniqueness of the Laplace Transform). Assume that f ,g ∈L1,loc(R+;X) with abs(f ) <∞ and abs(g) <∞, and let λ0 >maxabs(f ),abs(g).Suppose that f (λ) = g(λ) whenever λ > λ0. Then f (t) = g(t) a. e. on R+.

Now, we introduce the convolution on R+ of a scalar and a locally inte-grable vector-valued function.

Definition 1.1.5 (Convolution). For k ∈ L1,loc(R+) and f ∈ L1,loc(R+;X) wedefine the convolution for t ∈R+ by

(k ∗ f )(t) =∫ t

0k(t − s)f (s)ds,

and understand the integral in the sense of Bochner.

Remark 1.1.6. One can be shown that (k ∗ f )(t) exists for almost all t ∈R+ and(k ∗ f ) ∈ L1,loc(R+). Moreover, we have (k ∗ f )(t) =

∫ t0 k(s)f (t − s)ds, hence one

can write (f ∗ k) instead of (k ∗ f ). For further information, we refer to Arendt,Batty, Hieber and Neubrander, Section 1.3 about the convolution of vector-and operator-valued functions, cf. [ABHN01, Section 1.3].

Similarly the classical case, the Laplace integral of the convolution ofa scalar-valued function k with a vector-valued function f is given by thethe product of the Laplace transform of the single functions, if they exist,see [ABHN01, Proposition 1.6.4].

Proposition 1.1.7. Let k ∈ L1,loc(R+) and f ∈ L1,loc(R+;X) and suppose thatReλ >maxabs(|k|),abs(f ). Then (k ∗ f )(λ) exists and (k ∗ f )(λ) = k(λ)f (λ).

4

1.2. About the Laplace Transform of Holomorphic Functions

1.2 About the Laplace Transform of HolomorphicFunctions

In this section we give an generalisation of the Analytic Representation The-orem from [ABHN01, Theorem 2.6.1]. In particular, the decay behaviour ofthe considered functions is weakened. While the Analytic RepresentationTheorem from Arendt, Batty, Hieber and Neubrander requires a behaviour ofthe Laplace transform as s−1, we allow a weaker decay as s−α, with α ∈ (0,1].

On the basis of this result we are able to prove a Tauberian Theorem interms of the Tauberian Theorem of [ABHN01, Theorem 2.6.4]. All argumentswhich are used in the proofs of our results are quite similar to the argumentsin the corresponding proofs of the theorems in [ABHN01].

1.2.1 An Analytic Representation Theorem

The following theorem extends the Analytic Representation Theorem 2.6.1from [ABHN01], the proof uses similar arguments as the proof of the men-tioned theorem from Arendt, Batty, Hieber and Neubrander.

Theorem 1.2.1 (Analytic Representation Theorem). Let g : (0,∞)→ R+ bea non-negative, non-increasing function with limt→∞ g(t) = 0, which satisfies∫ 1

0 g(1/t)dtt <∞ and supt∈(0,1] tg(t) <∞. Furthermore, let 0 < α ≤ π2 and ω ∈ C.

We suppose that the function q : ω+Σα+π/2→ X is holomorphic and that for allγ ∈ (0,α) there is a constant M =M(γ) > 0 satisfying for all λ ∈ω+Σγ+π/2

‖q(λ)‖X ≤Mg(|λ−ω|).

Then there exists a holomorphic function f : Σα→ X such that for each γ ∈ (0,α)and all z ∈ Σγ there holds

‖e−ωzf (z)‖X ≤C

|z|g(1/ |z|)

with some constant C = C(γ) > 0 and f (λ) = q(λ) for all λ > Reω.

Proof. Let γ ∈ (0,α) and δ > 0 arbitrary, but fixed. There exists M = M((α +γ)/2) > 0 such that ‖q(λ)‖X ≤ Mg(|λ−ω|) for all λ ∈

(ω+Σγ+π/2

)\ ω. We

consider an oriented path Γ = Γ0∪ Γ+∪ Γ− (depending on δ and γ) consisting of

Γ0 =λ ∈C : λ =ω+ δeiθ ,−γ − π

2≤ θ ≤ γ +

π2

,

Γ± =λ ∈C : λ =ω+ re±i(γ+ π

2 ),δ ≤ r.

5


Let ε ∈ (0,γ), as well as z ∈ Σγ−ε and λ ∈ Γ±. Then λz = ωz + rze±i(γ+ π2 ) and

thus

Re(λz) = Re(ωz) + r |z|cos(argz ± (γ +π/2)) ≤ Re(ωz)− r |z|sinε.

For λ ∈ Γ± we have:

∥∥∥eλzq(λ)∥∥∥X

=∥∥∥∥∥e(ω+re±i(γ+ π2 ))zq

(ω+ re±i(γ+ π

2 ))∥∥∥∥∥X≤ eRe(ωz)−|z|r sinεMg(r); (1.1)

and for λ ∈ Γ0 we get the estimate∥∥∥eλzq(λ)∥∥∥X

=∥∥∥∥eωz+zδeiθq

(ω+ δeiθ

)∥∥∥∥X≤ eRe(ωz)+|z|δcos(argz+θ)Mg(δ). (1.2)

We consider for z ∈ Σγ−ε the expression∫Γ

eλzq(λ)dλ =∫Γ0∪Γ+∪Γ−

eλzq(λ)dλ

=∫ γ+ π

2

−γ− π2eωz+zδeiθq

(ω+ δeiθ

)iδeiθ dθ

+∫ ∞δeωz+zre

i(γ+ π2 )q(ω+ rei(γ+ π

2 ))ei(γ+ π

2 )dr

+∫ δ

∞eωz+zre

−i(γ+ π2 )q(ω+ re−i(γ+ π

2 ))e−i(γ+ π

2 )dr.

We have∥∥∥∥∥∥∫Γ±

eλzq(λ)dλ

∥∥∥∥∥∥X

≤∫ ∞δ

∥∥∥∥∥eωz+zre±i(γ+ π2 )

q(ω+ re±i(γ+ π

2 ))e±i(γ+ π

2 )∥∥∥∥∥Xdr,∥∥∥∥∥∥

∫Γ0

eλzq(λ)dλ

∥∥∥∥∥∥X

≤∫ γ+ π

2

−γ− π2

∥∥∥∥eωz+zδeiθq(ω+ δeiθ

)iδeiθ

∥∥∥∥Xdθ.

With the above estimate (1.1) along Γ± and the property that the functions g isnon-increasing on (0,∞) we obtain that∥∥∥∥∥∥

∫Γ±

eλzq(λ)dλ

∥∥∥∥∥∥X

≤∫ ∞δ

eRe(ωz)−|z|r sinεMg(r)dr

≤MeRe(ωz)g(δ)e−|z|δ sinε

|z|sinε. (1.3)

The estimate (1.2) along Γ0 yields

6


∥∥∥∥∥∥∫Γ0

eλzq(λ)dλ

∥∥∥∥∥∥X

≤∫ γ+ π

2

−γ− π2eRe(ωz)+|z|δcos(argz+θ)Mg(δ)δdθ

≤ 2πMeRe(ωz)e|z|δg(δ)δ. (1.4)

All in all we have the estimate∥∥∥∥∥∫Γ

eλzq(λ)dλ∥∥∥∥∥X≤2πMeRe(ωz)e|z|δg(δ)δ+ 2MeRe(ωz)g(δ)

e−|z|δ sinε

|z|sinε.

Since the exponential function and the function q are holomorphic functionson ω + Σα+ π

2these estimates show that f (z) = 1

2πi

∫Γ

eλzq(λ)dλ is absolutelyconvergent, uniformly for z in compact subsets of Σγ , and therefore defines aholomorphic function in Σγ .

We claim that by Cauchy’s theorem the function f (given by the aboveintegral) is independent of δ > 0, and also independent of γ ∈ (0,α) as longas |argz| < γ (use the assumptions on q for λ ∈ω+Σγ+ π

2with large norm). In

fact, let δ1,δ2 > 0 and γ1,γ2 ∈ (0,α). W. l. o. g. we assume that γ1 ≤ γ2. Wedefine for k ∈ 1,2 an oriented path Γ k = Γ k0 ∪ Γ k+ ∪ Γ k− consisting of

Γ k0 =λ ∈C : λ =ω+ δke

iθ , −γk −π2≤ θ ≤ γk +

π2

,

Γ k± =λ ∈C : λ =ω+ re±i(γk+

π2 ), δk ≤ r

,

as well as an oriented path Γ kR = Γ k0 ∪ ΓkR+ ∪ Γ

kR− consisting of Γ k0 as above and

Γ kR± =λ ∈C : λ =ω+ re±i(γk+

π2 ), δk ≤ r ≤ R

.

Moreover, we set

Γ +R =

λ ∈C : λ =ω+Reiθ , γ1 +

π2≤ θ ≤ γ2 +

π2

,

Γ −R =λ ∈C : λ =ω+Re−iθ , γ1 +

π2≤ θ ≤ γ2 +

π2

.

The function eλzq(λ) is holomorphic on ω + Σα+ π2, hence Cauchy’s integral

theorem yields ∫(Γ 1R+Γ −R )−(Γ 2

R+Γ +R )

eλzq(λ)dλ = 0.

Now, let z ∈ Σγ1−ε for some ε ∈ (0,γ1). We have that∫Γ +R

eλzq(λ)dλ =∫ γ2+ π

2

γ1+ π2

eωz+zReiθq(ω+Reiθ

)iReiθ dθ,∫

Γ −R

eλzq(λ)dλ =∫ γ2+ π

2

γ1+ π2

eωz+zRe−iθq(ω+Re−iθ

)(−i)Re−iθ dθ,

7


and hence∥∥∥∥∥∥∫Γ ±R

eλzq(λ)dλ

∥∥∥∥∥∥X

≤∫ γ2+ π

2

γ1+ π2

∥∥∥∥ez(ω+Re±iθ)q(ω+Re±iθ

)(±i)Re±iθ

∥∥∥∥Xdθ

≤Mg(R)Re−|z|Rsinε |eωz| (γ2 −γ1)→ 0,

for R→∞. This implies∫Γ 1

eλzq(λ)dλ = limR→∞

∫Γ 1R

eλzq(λ)dλ

= limR→∞

∫Γ 2R

eλzq(λ)dλ =∫Γ 2

eλzq(λ)dλ,

and hence the function f (z) is independent of the parameters δ and γ of theintegral along the oriented path Γ . Hence 1

2πi

∫Γ

eλzq(λ)dλ defines a holomor-phic function f on Σα.

Let z ∈ Σα be arbitrary, but fixed. We want to estimate f (z). Since thesector Σα is open there is some γ < α and some sufficiently small 0 < ε < γsuch that z ∈ Σγ−ε. Let M = M(γ) > 0 the constant from the requirements.Choosing δ = |z|−1 we obtain with the aid of (1.4) for all λ ∈ Γ0 the estimate∥∥∥∥∥∥

∫Γ0

eλzq(λ)dλ

∥∥∥∥∥∥X

≤ 2πeRe(ωz)+1M

|z|g(1/ |z|),

and for λ ∈ Γ± we have together with estimate (1.3) that∥∥∥∥∥∥∫Γ±

eλzq(λ)dλ

∥∥∥∥∥∥X

≤ eRe(ωz) e−sinε

sinεM

|z|g(1/ |z|).

These estimates establish that for all z ∈ Σγ−ε we have


|z|g(1/ |z|), (1.5)

with C =Mmaxe−sinε/ sinε,2πe.Next, we will show that f (λ) = q(λ) whenever λ > Reω. Using relation

(1.5) we have for all λ > Reω that∫ ∞0

e−λt ‖f (t)‖X dt ≤ C∫ ∞

0e−λteReωt g(1/t)

tdt

≤ C∫ 1

0e−(λ−Reω)t g(1/t)

tdt +C

∫ ∞1

e−(λ−Reω)t g(1/t)t

dt

≤ C∫ 1

0

g(1/t)t

dt +C sups∈(0,1]

sg(s)∫ ∞

1e−(λ−Reω)t dt <∞.

8


This means that f is Laplace transformable for all Reλ > Reω. Given λ > Reω,choose 0 < δ < λ−Reω, and γ ∈ (0,α). Since λ is on the right of the path Γ itfollows that∫

Γ

∫ ∞0

∣∣∣e−(λ−µ)t∣∣∣∥∥∥q(µ)

∥∥∥Xdt

∣∣∣dµ∣∣∣ =∫Γ

∥∥∥q(µ)∥∥∥X

λ−Reµ

∣∣∣dµ∣∣∣≤M

∫Γ

g(∣∣∣µ−ω∣∣∣)λ−Reµ

∣∣∣dµ∣∣∣ <∞,because ∫

Γ±


∣∣∣dµ∣∣∣ =∫ ∞δ

g(r)λ− [Reω+ r cos(γ + π

2 )]dr

≤ 1sin(γ)

∫ ∞δ

g(r)rdr <∞,

since we have for all δ ∈ (0,1)∫ ∞δ

g(r)rdr =

∫ 1

δ

g(r)rdr +

∫ ∞1

g(r)rdr

≤ sups∈[δ,1]

sg(s)∫ 1

δ

ds

s2+∫ 1

0

g(1/t)t

dt <∞,

by assumptions on the map g, as well as∫Γ0


∣∣∣dµ∣∣∣ =∫ γ+ π

2

−γ− π2

g(δ)λ− (Reω+ δcosθ)

δdθ

≤ δg(δ)2π

λ− (Reω+ δ)<∞.

Fubini’s theorem yields∫Γ

∫ ∞0

e−λteµtq(µ)dt dµ =∫ ∞

0

∫Γ

e−λteµtq(µ)dµdt.

Due to the choice of δ ∈ (0,λ−Reω), we know λ is to the right of the path Γ ,and Fubini’s theorem and Cauchy’s residue theorem imply that

f (λ) =∫ ∞

0e−λt

12πi

∫Γ

eµtq(µ)dµdt

=1

2πi

∫Γ

q(µ)λ−µ

dµ

= q(λ) + limR→∞

12πi

∫ΓR

q(µ)λ−µ

dµ,

9


where ΓR =λ ∈C : λ =ω+Reiθ ,−γ − π2 ≤ θ ≤ γ + π

2

. Then

∥∥∥∥∥∥∫ΓR

q(µ)λ−µ

dµ

∥∥∥∥∥∥X

≤∫ γ+ π

2

−γ− π2

∥∥∥∥q (ω+Reiθ)iReiθ

∥∥∥∥X∣∣∣|λ−ω| − ∣∣∣Reiθ

∣∣∣∣∣∣ dθ,

for sufficient large R we have R > 2 |λ−ω| and hence∥∥∥∥∥∥∫ΓR

q(µ)λ−µ

dµ

∥∥∥∥∥∥X

≤∫ γ+ π

2

−γ− π2M

g(R)RR− |λ−ω|

dθ

≤ 2πMg(R)

1− |λ−ω|R

≤ 4πMg(R),

and this expression tends to zero as R→∞ by assumption on the function g.This proves that f (λ) = q(λ) for λ > Reω.

Remark 1.2.2. For the proof of the above theorem it would be enough torequire that

∫∞1 e−εtg(1/t)dt < ∞ for all ε > 0, instead of our requirement

supt∈(0,1] tg(t) < ∞. But for our application in the following the slightlystronger assumption is always satisfied and from our point of view easier tocheck.

The next converse statement is for our following considerations non-essential, but we insert it into this treatise for the sake of completeness.

Theorem 1.2.3. Let g : (0,∞)→R+ be a non-negative, non-increasing functionwith limt→∞ g(t) = 0 and limt→∞ g(1/t)e−ηt = 0 for all η > 0. Furthermore, lettg(t) be non-decreasing and g(t) ≤ −ctg ′(t) for a. e. t > 0 and some c > 0. Inaddition, let α ∈ (0, π2 ), ω ∈C and q : (Reω + |Imω| tanα,∞)→ X. Suppose thereis a holomorphic function f : Σα → X such that for all γ ∈ (0,α) and all z ∈ Σγthere holds


|z|g(1/ |z|)

with some constant C = C(γ) > 0 and f (λ) = q(λ) for all λ > Reω+ |Imω| tanα.Then the function q has a holomorphic extension q : ω+Σα+ π

2→ X such that for

all γ ∈ (0,α) and all λ ∈ω+Σγ+ π2

we have

‖q(λ)‖X ≤Mg(|λ−ω|)

with some constant M =M(γ) > 0.

10


Proof. Let γ ∈ (0,α) be arbitary. Then there exists C = C((α +γ)/2) > 0 suchthat ‖f (z)‖X ≤

C|eωz ||z| g(1/ |z|) for all Σγ \ 0. We define the both paths Γ± by

Γ± = se±iγ : s > 0 and let λ > Reω + |Imω| tanα be arbitrary. By Cauchy’stheorem we have for all R > 0:∫ R

0e−λtf (t)dt = −

∫ γ

0e−λRe±iθf (Reiθ)(±i)Re±iθ dθ −

∫ 0

Re−λse

±iγf (se±iγ )e±iγ ds.

Moreover, we get∥∥∥∥∥∫ γ

0e−λRe±iθf

(Re±iθ

)(±i)Re±iθ dθ

∥∥∥∥∥X

≤∫ γ

0

∣∣∣∣e−λRe±iθeωRe±iθ∣∣∣∣ C∣∣∣Re±iθ

∣∣∣g (1/ ∣∣∣Re±iθ∣∣∣) ∣∣∣Re±iθ

∣∣∣dθ= Cg(1/R)

∫ γ

0e−Re[(λ−ω)Re±iθ]dθ

≤ Cg(1/R)π2

e−[(λ−Reω)−|Imω| tanγ]Rcosγ → 0,

as R→∞. We deduce∫ ∞0

e−λtf (t)dt =∫ ∞

0e−λse

±iγf (se±iγ )e±iγ ds =

∫Γ±

e−λzf (z)dz,

for all λ > Reω+ |Imω| tanα > Reω+ |Imω| tanγ , whenever γ ∈ (0,α).Let ε ∈

(0, π2 −γ

)and λ ∈ C with −π2 +γ + ε < arg(λ−ω) < π

2 −γ − ε. Then

−π2 + ε < arg((λ−ω)e±iγ

)< π

2 − ε, thus

Re(λ−ω)e±iγ = |λ−ω|cos(arg(λ−ω)∓γ)

≥ |λ−ω|cos(π

2− ε

)= |λ−ω|sinε.

For all s > 0 we get the estimate∥∥∥∥eλse±iγf (se±iγ )

∥∥∥∥X≤ e−Re((λ−ω)se±iγ )C

sg(1/s) ≤ e−s|λ−ω|sinεC

sg(1/s),

and hence we have for∫Γ±

e−λzf (z)dz = e±iγ∫ ∞

0e−λse

±iγf (se±iγ )ds,

the estimate

11


∥∥∥∥∥∥∫Γ±

e−λzf (z)dz

∥∥∥∥∥∥X

≤ C∫ ∞

0e−s|λ−ω|sinε g(1/s)

sds

= C∫ 1/ |λ−ω|

0e−s|λ−ω|sinε g(1/s)

sds

+C∫ ∞

1/ |λ−ω|e−s|λ−ω|sinε g(1/s)

sds.

For the first integral we use the estimate g(s) ≤ −csg ′(s) and the resulting in-equality 1

s g(1/s) ≤ c ddsg(1/s). Furthermore, the function sg(s) is non-decreasing,hence 1

s g(1/s) in non-increasing, this property is used for the second integral.We obtain that∥∥∥∥∥∥

∫Γ±

e−λzf (z)dz

∥∥∥∥∥∥X

≤ Cc∫ 1/ |λ−ω|

0e−s|λ−ω|sinε d

dsg(1/s)ds

+Cg (|λ−ω|) |λ−ω|∫ ∞

1/ |λ−ω|e−s|λ−ω|sinε ds.

Since the function g is non-increasing, we have that s 7→ g(1/s) is non-decreasing.With integration by parts it follows that∥∥∥∥∥∥

∫Γ±

e−λzf (z)dz

∥∥∥∥∥∥X

≤ Cce−s|λ−ω|sinεg(1/s)

∣∣∣∣∣1/ |λ−ω|0

−∫ 1/ |λ−ω|

0(−|λ−ω|sinε)e−s|λ−ω|sinεg(1/s)ds

]+Cg (|λ−ω|) |λ−ω|

e−s|λ−ω|sinε

−|λ−ω|sinε

∣∣∣∣∣∞1/ |λ−ω|

≤ Cc[e−sinεg (|λ−ω|)− lim

s→0g(1/s)− g (|λ−ω|)

(e−sinε − 1

)]+Cg (|λ−ω|) |λ−ω|

e−sinε

|λ−ω|= C

(c+ e−sinε

)g (|λ−ω|) =Mg (|λ−ω|) .

Consequently, the integrals∫Γ±

e−λzf (z)dz are absolutely convergent and de-fine a holomorphic function in the region −π2 ∓γ + ε < arg(λ−ω) < π

2 ∓γ − ε,with some constant M > 0 such that∥∥∥∥∥∥

∫Γ±

e−λzf (z)dz

∥∥∥∥∥∥X

≤Mg (|λ−ω|) .

For λ ∈C with −π2 ∓γ + ε < arg(λ−ω) < π2 ∓γ − ε we set

q±(λ) =∫Γ±

e−λzf (z)dz,

12


both q+ and q− are extensions of q, and together they define a holomorphicextension (again denoted by q) to ω+Σ π

2 +γ−ε, satisfying

‖q(λ)‖X ≤Mg(|λ−ω|),

in the sector ω+Σ π2 +γ−ε. Since γ ∈ (0,α) and ε ∈

(0, π2 −γ

)are arbitrary, this

proves the claim.

For ω ∈R we have a quite similar result.

Corollary 1.2.4. Let g : (0,∞)→R+ be a non-negative, non-increasing functionwith limt→∞ g(t) = 0 and limt→∞ g(1/t)e−ηt = 0 for all η > 0. Furthermore, lettg(t) be non-decreasing and g(t) ≤ −ctg ′(t) for a. e. t > 0 with some c > 0. Inaddition, let α ∈ (0, π2 ], ω ∈R and q : (ω,∞)→ X. Suppose there is a holomorphicfunction f : Σα→ X such that for all γ ∈ (0,α) and all z ∈ Σγ we have


|z|g(1/ |z|)

with some constant C = C(γ) > 0 and f (λ) = q(λ) for all λ > ω.Then the function q has a holomorphic extension q : ω+Σα+ π

2→ X such that for

all γ ∈ (0,α) and all λ ∈ω+Σγ+ π2

holds

‖q(λ)‖X ≤Mg(|λ−ω|),

with some constant M =M(γ) > 0.

Proof. The proof is quite similar to the proof of the Theorem 1.2.3. The onlydifference is the following estimate for λ > ω:∣∣∣∣e−(λ−ω)Re−±iθ

∣∣∣∣ = e−Re((λ−ω)Re±iθ) = e−(λ−ω)Rcosθ ,

and hence we have for all θ ∈ [0,γ] the estimate∥∥∥∥∥∫ γ

0e−λRe±iθf (Re±iθ)(±i)Re±iθ dθ

∥∥∥∥∥X≤ Cg(1/R)

π2

e−(λ−ω)Rcosγ → 0,

as R→∞ by assumptions on the function g. Thus, we have that∫ ∞0

e−λtf (t)dt =∫ ∞

0e−λse

±iγf (se±iγ )e±iγ ds =

∫Γ±

e−λzf (z)dz,

for all λ > ω. The subsequent arguments are all the same as in the proof ofTheorem 1.2.3.

Remark 1.2.5. The assumptions on the function g in Theorem 1.2.3 andCorollary 1.2.4 are technical. Nevertheless, these assumptions are satisfied forthe function g(s) = s−α, α ∈ (0,1]. Thus, we can understand these both resultsas an extension of the corresponding part of the Analytic RepresentationTheorem from Arendt et. al., cf. [ABHN01, Theorem 2.6.1].

13


1.2.2 Asymptotic Behaviour of Vector-Valued HolomorphicFunctions

For the proof of our Tauberian Theorem we use the following two auxiliaryresults, taken from [ABHN01, Proposition A. 2 and Theorem A. 5].

Proposition 1.2.6 (Identity Theorem for Holomorphic Functions). Let Ybe a closed subspace of a Banach space X. Let Ω be a connected open set in C

and f : Ω→ X be holomorphic. Assume that there exists a convergent sequence(zn)n∈N ⊂Ω such that limn→∞ zn ∈Ω and f (zn) ∈ Y for all n ∈N.

Then f (z) ∈ Y for all z ∈Ω.

Theorem 1.2.7 (Vitali’s Convergence Theorem). Let Ω ⊂ C be open and con-nected. For every n ∈N let fn : Ω→ X be holomorphic such that

supn∈N,z∈Br (z0)

‖fn(z)‖X <∞

whenever Br(z0) ⊂Ω. Assume that the set

Ω0 = z ∈Ω : limn→∞

fn(z) exists

has a limit point in Ω. Then there exists a holomorphic function f : Ω→ X suchthat

f (k)(z) = limn→∞

f(k)n (z)

uniformly on all compact subsets of Ω for all k ∈N0.

We start with the following adapted version of Proposition 2.6.3 from[ABHN01]. The difference to the known result is, that only the boundednessin the neighbourhood of the origin is required and not on the whole sector.The proof uses the same arguments as in the book from Arendt, Batty, Hieberand Neubrander [ABHN01].

Proposition 1.2.8. Let 0 < α ≤ π and let f : Σα → X be holomorphic such thatfor some r > 0 and all 0 < β < α we have

supz∈Σβ∩Br (0)

‖f (z)‖X <∞.

Let x ∈ X. If limt→0 f (t) = x, then lim|z|→0, z∈Σβ∩Br (0) f (z) = x for all 0 < β < α.

Proof. We set Ω = Σα ∩Br(0) and for n ∈N we define fn(z) = f(zn

), z ∈Ω. We

have for all n ∈N, 0 < β < α and z ∈ Σβ ∩ Br(0) that z/n ∈ Σβ ∩ Br/n(0), thisimplies

14


supn∈N


‖fn(z)‖X = supn∈N


∥∥∥∥∥f ( zn

)∥∥∥∥∥X

= supn∈N

supz∈Σβ∩Br/n(0)

‖f (z)‖X

≤ supn∈N


‖f (z)‖X

= supz∈Σβ∩Br (0)

‖f (z)‖X <∞,

by assumption.

In particular, let Bρ(z0) ⊂ Ω, then there is β′ ∈ (0,α) such that Bρ(z0) ⊂Σβ′ ∩Br(0) and hence

supn∈N,z∈Bρ(z0)

‖fn(z)‖X ≤ supn∈N,z∈Σβ′∩Br (0)

‖fn(z)‖X <∞.

The set Ω0 = z ∈Ω : limn→∞ fn(z) exists ⊃ (0, r) has a limit point in the openand connected set Ω. Now, Vitali’s Theorem 1.2.7 implies that that there is aholomorphic function f : Ω→ X such that f (z) = limn→∞ fn(z) uniformly oncompact subsets of Ω.

LetK ⊂Ω be a connected and open set with [a,b] ⊂ K for some 0 < a < b < r.We set zn = a+ 1/n for n ≥ k0 ∈N with a+ 1/k0 < b. Then zn→ a as n→∞ and

f (zk) = limn→∞

fn(zk) = limn→∞

f (zk/n) = limt→0

f (t) = x,

hence f (zk) = x for all k ≥ k0; and thus we have f (z) = x for all z ∈Ω by theidentity theorem for holomorphic functions Proposition 1.2.6. All in all, weget limn→∞ fn(z) = x uniformly on compact subsets of Ω.

Let 0 < β < α, 0 < η < r and ε > 0 arbitrary, but fixed. There is some k0 =k0(η) ∈N such that ‖fk(z)− x‖X ≤ ε for all z ∈ Σβ with η

2 ≤ |z| ≤ η, k ≥ k0 ≥ 2.Now, let z ∈ Σβ with |z| ≤ η

k0≤ η

2 , choose k ∈N (k ≥ k0) such that ηk+1 < |z| ≤

ηk .

This implies that ‖f (z)− x‖X = ‖fk(k z)− x‖X ≤ ε. Hence, the claim follows.

Finally, we are in the situation to formulate our Tauberian Theorem sim-ilar to the Tauberian Theorem from Arendt, Batty, Hieber and Neubrander,cf. [ABHN01, Theorem 2.6.4]. Note, that the required decay behaviour isweakened in the same sense as for the Analytic Representation Theorem 1.2.1.The idea of the proof is analogue to the book [ABHN01].

15


Theorem 1.2.9 (Tauberian Theorem). Let g : (0,∞) → R+ be non-negative,non-increasing such that sups∈(0,1] sg(s) < ∞ and

∫ 10 g(1/t)dtt < ∞. Suppose

q : Σα+ π2→ X is holomorphic for some α ∈ (0, π2 ] and satisfies for all γ ∈ (0,α) and

all λ ∈ Σγ+ π2

the estimate

‖q(λ)‖X ≤Mg(|λ|),

with some constant M = M(γ) > 0. Let f be such as in Theorem 1.2.1 withf (λ) = q(λ) for all λ > 0.

Then limt→∞ f (t) = x if and only if limλ→0λq(λ) = x.

Proof. Assume that limt→∞ f (t) = x. The claim follows by [ABHN01, Theorem4.1.2], since f ∈ L1,loc(R+;X). Indeed, let T > 0 be arbitrary but fixed, Theorem1.2.1 yields the estimate∫ T

0‖f (t)‖X dt ≤ C

∫ T

0

g(1/t)t

dt

≤ C∫ 1

0

g(1/t)t

dt +C∫ 1

1/T

g(s)sds

≤ C∫ 1

0

g(1/t)t

dt +C sups∈[1/T ,1]

sg(s)∫ 1

1/T

ds

s2<∞,

by the assumptions on the function g.For the converse implication we assume w. l. o. g. that x = 0, otherwise

we replace f (t) by f (t) − x. Assume that limλ→0λq(λ) = x. Let γ ∈ (0,α) bearbitrary, but fixed. By assumption we have for all λ ∈ Σγ+ π

2the estimate

‖λq(λ)‖X ≤ M |λ|g(|λ|), with some constant M = M((α + γ)/2) > 0. Sincesup(0,1] sg(s) <∞ we have

supλ∈Σγ+ π2

∩B1(0)‖λq(λ)‖X ≤M sup

s∈(0,1]sg(s) <∞.

In particular, we have by Proposition 1.2.8 that

lim|λ|→0,λ∈Σγ+ π2

∩B1(0)λq(λ) = x.

Let ε > 0 be arbitrary, but fixed. There exists δ0 ∈ (0,1] with ‖λq(λ)‖X ≤ εwhenever λ ∈ Σγ+ π

2and |λ| ≤ δ0. Let 0 < 1

t ≤ δ0. Now, we choose the contourΓ as in the proof of Theorem 1.2.1 with δ = 1

t and f (t) = 12πi

∫Γ

eλtq(λ)dλ. Wehave

16


∥∥∥∥∥∥ 12πi

∫Γ0

eλtq(λ)dλ

∥∥∥∥∥∥X

=

∥∥∥∥∥∥ 12πi

∫ γ+ π2

−γ− π2eeiθq

(eiθ

t

)ieiθ

tdθ

∥∥∥∥∥∥X

≤ ε2π

∫ γ+ π2

−γ− π2ecosθ dθ

≤ εe,

since ‖λq(λ)‖X ≤ ε for |λ| =∣∣∣∣ eiθt

∣∣∣∣ = 1t ≤ δ0. Furthermore, we have

12πi

∫Γ±

eλtq(λ)dλ =1

2πi

∫ ∞1t

etre±i(γ+ π2 )

q(re±i(γ+ π2 )re±i(γ+ π

2 )drr

=1

2πi

∫ ∞1

ese±i(γ+ π2 )

q(ste±i(γ+ π

2 )) ste±i(γ+ π

2 )dss→ 0,

as t →∞ by the dominated convergence theorem. Indeed, for s ≥ 1 we sett0 = t0(s) = max1/δ0,1/s. For all t > t0 we have s

t e±i(γ+ π

2 ) ∈ Σγ+ π2∩B1(0) and

hence limt→∞

∥∥∥∥q ( st e±i(γ+ π2 ))st e±i(γ+ π

2 )∥∥∥∥X

= 0. This implies

∥∥∥∥∥ese±i(γ+ π2 )

q(ste±i(γ+ π

2 )) ste±i(γ+ π

2 ) 1s

∥∥∥∥∥X≤ e−s sinγ

s

∥∥∥∥∥q (st e±i(γ+ π2 )) ste±i(γ+ π

2 )∥∥∥∥∥X

≤∥∥∥∥∥q (st e±i(γ+ π

2 )) ste±i(γ+ π

2 )∥∥∥∥∥X→ 0,

as t→∞; the integrand converges pointwise to zero.Moreover, the property that the function g(s) is non-increasing implies for

s ≥ 1 that∥∥∥∥∥ese±i(γ+ π2 )

q(ste±i(γ+ π

2 )) ste±i(γ+ π

2 ) 1s

∥∥∥∥∥X≤Me−s sinγ g(s/t)

t≤Me−s sinγ g(1/t)

t,

Using the fact that sups∈(0,1] sg(s) = M0 <∞ and that t > 1 we obtain for alls ≥ 1 the estimate∥∥∥∥∥ese

±i(γ+ π2 )q(ste±i(γ+ π

2 )) ste±i(γ+ π

2 ) 1s

∥∥∥∥∥X≤Me−s sinγM0.

Thus, we have a domination by an integrable function. The dominated con-vergence theorem implies the convergence of the integral to zero. It followsfrom the representation of f given above that limsupt→∞ ‖f (t)‖X ≤ εe. Sinceε > 0 was chosen arbitrary it follows the claim.

17

2Scalar Volterra Equations andKernelsIn this chapter we summarise basic definitions and properties concerningscalar Volterra equations and the associated kernels. At first we introducethe scalar resolvent and integral resolvent for linear scalar Volterra equationsand discuss a representation of solutions. On the basis of these considera-tions we give a motivation why we restrict ourselves in the following to theclass of completely positive kernels. We define this class of kernels, givea characterisation for unbounded, completely positive kernels and collectgeneral properties of these kernels and the corresponding scalar resolventand integral resolvent.

Next, we turn to the more special class of completely monotonic kernels.Kernels of this class are of special interest, since their Laplace transform isvery well understood, and they are characterised by the famous theorem ofBernstein, as Laplace transform of positive measures with support on thenon-negative real axis.

An important part of this section is the paragraph about the properties ofthe Laplace transform of completely monotonic kernels. Beside some knownfacts, we are able to extend the properties of sectoriality and regularity to asector which is larger than the right complex half-plane.

Finally, we summarise some permanence properties of completely mono-tonic kernels. The so-called standard kernel is an important example forsuch kernels. We also give an explicit expression for the corresponding scalarresolvent and integral resolvent in terms of the Mittag-Leffler function.

2.1 Scalar Volterra Equations

We are interested in linear scalar Volterra equations of the type

u(t) +µ(a ∗u)(t) = f (t), t ∈R+, (2.1)

where a : R+→R is a given kernel, µ ∈C is a complex number and f : R+→C

is a given function. A more general setting can be found in Gripenberg,Londen and Staffans [GLS90] as well as Clément and Nohel [CN81]. Weintroduce the scalar resolvent and integral resolvent as solutions of linearVolterra equations with special right-hand side. These functions are essentialfor the treatment of linear Volterra equations.

19

2. Scalar Volterra Equations and Kernels

Definition 2.1.1 (Scalar Resolvent and Integral Resolvent). Let µ ∈ C anda ∈ L1,loc(R+). The solution sµ : R+→C of the scalar Volterra equation

sµ(t) +µ(a ∗ sµ)(t) = 1, t ∈R+,

is called scalar resolvent of the Volterra equation (2.1). The solution rµ : R+→C

of the scalar Volterra equation

rµ(t) +µ(a ∗ rµ)(t) = a(t), t ∈R+,

is called scalar integral resolvent of the Volterra equation (2.1).

We know from Gripenberg, Londen and Staffans [GLS90, Theorem 2.3.1]that for each locally integrable kernel a ∈ L1,loc(R+) there is a unique andlocally integrable scalar integral resolvent rµ ∈ L1,loc(R+). With the aid of theintegral resolvent we can express the solution of our Volterra equation via thevariation of constants formula. We have the following existence and uniquenesstheorem. For the proof we refer to [GLS90, Theorem 2.3.5].

Theorem 2.1.2. Let a ∈ L1,loc(R+) and µ ∈ C. Then, for each f ∈ L1,loc(R+), thereis a unique solution u ∈ L1,loc(R+) of the Volterra equation (2.1). This solution isgiven by

u(t) = f (t)−µ(rµ ∗ f )(t), t ∈R+.

In particular, if f ∈ Lp,loc(R+), with 1 ≤ p ≤∞, or f ∈ C(R+), then u ∈ Lp,loc(R+)or u ∈ C(R+), respectively.

This theorem implies the existence and uniqueness of the scalar resolventsµ ∈ C(R+) whenever a ∈ L1,loc(R+) and the following relation between thescalar resolvent and integral resolvent:

sµ(t) = 1−µ(rµ ∗ 1)(t), t ∈R+. (2.2)

In particular, we know that the scalar resolvent sµ is absolutely continuous onR+ and differentiable almost everywhere on R+ with sµ(t) = −µrµ(t) for almostall t ∈R+.

In the following we consider a special case of Volterra equations

u(t) +µ(a ∗u)(t) = u0 + (a ∗ g)(t), t ∈R+, (2.3)

with some g ∈ L1,loc(R+). The unique solution can be expressed in terms ofthe scalar resolvent and integral resolvent and is given by

u(t) = sµ(t)u0 + (rµ ∗ g)(t), t ∈R+. (2.4)

20

2.2. Completely Positive Kernels

Hence, if one understands the behaviour of the scalar resolvent and integralresolvent very well one understands in particular the solution of the Volterraequation (2.3). We want to motivate in the following, why we restrict ourselvesto Volterra equations with a special class of kernels. J. Levin proved in the1970s the following properties of solutions of the Volterra equation (2.1),cf. [Lev77, Lemma 1.3].

Lemma 2.1.3. Let µ ∈R+, a ∈ L1,loc(R+) be non-negative and non-increasing onR+ and f ∈ L1,loc(R+) be non-negative and non-decreasing. Then, the solution uof the Volterra equation (2.1) is non-negative on R+ and we have 0 ≤ u(t) ≤ f (t)a. e. on R+.

The representation of the solution of (2.3) via the scalar resolvent andintegral resolvent suggests that their properties are primarily responsible forthe properties of the solution. In particular, positivity of the solution dependson the positivity of the scalar resolvent and integral resolvent. For this reason,we restrict our considerations onto a class of kernels which guarantees thepositivity for the scalar resolvent and integral resolvent at least for non-negative parameters µ ∈R+.

2.2 Completely Positive Kernels

We start this paragraph with the definition of completely positive kernels,cf. [CN81, Definition 1.1]

Definition 2.2.1 (Completely Positive Kernel). A kernel a ∈ L1,loc(R+) iscalled completely positive if for all µ ∈ R+ the functions rµ and sµ are non-negative on R+.

It is known from Clemént and Nohel [CN81] that the kernel a ∈ L1,loc(R+)is completely positive if a is non-negative, non-increasing and loga is convexon R+. Furthermore, Clemént and Nohel give in [CN81, Theorem 2.2] thefollowing characterisation of unbounded, completely positive kernels:

Theorem 2.2.2. Let a ∈ L1,loc(R+) \L∞,loc(R+) and a . 0. Then a is a completelypositive kernel if and only if there exists b ∈ L1,loc(R+) non-negative and non-increasing satisfying: (a ∗ b)(t) = 1 for all t ∈ (0,∞).

Remark 2.2.3. N. Sonine was the first in the 1880s who considered so-calledAbel integral equations with kernels a ∈ L1,loc(R+) satisfying the conditionthat there is another kernel b ∈ L1,loc(R+) with (a ∗ b)(t) = 1 for all t ∈ (0,∞),see [SKM93, p. 85]. For this reason we speak in this situation of the Soninecondition and call the kernel b the corresponding Sonine kernel (to the kernela).

21


On the basis of the Sonine condition it is possible to deduce some generalproperties of the kernels, cf. [SC03, Lemma 4.3]. Note that these propertiesare satisfied for any pair of completely positive kernels.

Lemma 2.2.4. Let a, b ∈ L1,loc(R+) non-negative and non-increasing kernels sat-isfying the Sonine condition (a ∗ b)(t) = 1 for all t ∈ (0,∞). Then the followingrelations hold for all t > 0:

(i) ta(t)b(t) ≤ 1,

(ii) a(t)∫ t

0 b(τ)dτ ≤ 1, b(t)∫ t

0 a(τ)dτ ≤ 1,

(iii) a(t)∫ t

0 b(τ)dτ + b(t)∫ t

0 a(τ)dτ ≥ 1,

(iv)∫ t

0 a(τ)dτ∫ t

0 b(τ)dτ ≥ t,

(v) limτ→0+ a(τ) = limτ→0+ b(τ) = +∞,

(vi) limτ→0+ τa(τ) = limτ→0+ τb(τ) = 0.

In the following proposition we summarise some important facts aboutthe scalar resolvent and the scalar integral resolvent of Volterra equationswith completely monotonic kernels. For the proof we refer to Clemént andNohel [CN81, Section 2].1

Proposition 2.2.5. Let a ∈ L1,loc(R+) be a completely positive kernel. Then:

(i) The kernel a is non-negative on R+ and, for all µ ∈R+ the scalar resolventsµ is non-negative and non-increasing on R+ and we have the estimatesµ(t) [1 +µ(1 ∗ a)(t)] ≤ 1 for all t ∈R+.

(ii) For each µ > 0, the scalar integral resolvent rµ is itself completely positive onR+.

(iii) If a < L1(R+), then for all µ > 0 we have limt→∞ sµ(t) = 0 and the equationµ∥∥∥rµ∥∥∥L1(R+)

= 1.

(iv) If a < L∞,loc(R+), then for all µ ∈C we have sµ(t) = (b ∗ rµ)(t), for all t ∈R+,where b ∈ L1,loc(R+) denotes the corresponding Sonine kernel to the kernela ∈ L1,loc(R+).

The following lemma gives a lower pointwise estimate for the scalar resol-vent in the case µ ∈R+.

1We are mainly interested in unbounded kernels which are not integrable on R+ but havean integrable singularity at zero. Therefore we focus on results in this situation. For furtherconclusions we refer to [CN81].

22

2.3. Completely Monotonic Functions and Kernels

Lemma 2.2.6. Let a ∈ L1,loc(R+) be unbounded and completely positive. Foreach µ ∈ R+ we have the estimate [µ+ b(t)]sµ(t) ≥ b(t), for all t ∈ R+, whereb ∈ L1,loc(R+) denotes the corresponding Sonine kernel.

Proof. Let µ ∈ R+ be arbitrary, but fixed. Since the kernel a ∈ L1,loc(R+) iscompletely positive, we know from Proposition 2.2.5 that we have for allt ∈ R+ the relation µsµ(t) = µ(b ∗ rµ)(t). The corresponding Sonine kernelb ∈ L1,loc(R+) is non-increasing, so we can estimate µsµ(t) ≥ µb(t)(1 ∗ rµ)(t).The relation between the scalar resolvent and integral resolvent (2.2) yieldsµsµ(t) ≥ b(t)

[1− sµ(t)

]and the claim follows immediately.

This estimate in combination with Proposition 2.2.5 (i) yields a very precisedescription of the behaviour of the scalar resolvent in case of completelypositive kernels and µ ∈R+:

b(t)µ+ b(t)

≤ sµ(t) ≤ 11 +µ(1 ∗ a)(t)

, t ∈R+. (2.5)

2.3 Completely Monotonic Functions and Kernels

A large class of completely positive kernels is the class of completely mono-tonic kernels. These kernels are of special interest, since their Laplace trans-form is very well understood. We start with the definition of completelymonotonic kernels and subsequently we discuss their properties.

Definition 2.3.1 (Completely Monotonic Function). A function a : (0,∞)→R is completely monotonic if a ∈ C∞((0,∞)) and for all n ∈N0 and all t > 0 wehave (−1)na(n)(t) ≥ 0 .

By [Mil68, Lemma 1] we know that a completely monotonic function eithervanishes or is positive on (0,∞). Furthermore, a non-vanishing completelymonotonic function is log-convex, see [Mil68, Lemma 2].2 In particular, acompletely monotonic kernel a ∈ L1,loc(R+) is completely positive. More-over, if a < L∞,loc(R+), then the corresponding Sonine kernel b ∈ L1,loc(R+) iscompletely monotonic, too, cf. [GLS90, Theorem 5.5.4].

Gripenberg, Staffans and Londen give us the following characterisationfor the scalar integral resolvent of completely monotonic kernels, see [GLS90,Theorem 5.3.1].

Theorem 2.3.2. Let a ∈ L1,loc(R+) and for µ > 0 we consider the correspondingscalar integral resolvent rµ : R+→R, the solution of the scalar Volterra equation

rµ(t) +µ(a ∗ rµ)(t) = a(t), t ∈R+.

2In the following we always exclude the case of vanishing functions whenever we speakabout complete monotonicity.

23


Then the kernel a is completely monotonic if and only if the scalar integral resolventrµ is completely monotonic, rµ ∈ L1(R+), and

∥∥∥rµ∥∥∥L1(R+)≤ 1/µ.

Remark 2.3.3. In particular, the corresponding scalar resolvent sµ, the solu-tion of the scalar Volterra equation

sµ(t) +µ(a ∗ sµ)(t) = 1, t ∈R+,

is completely monotonic, too. Since we have the relation sµ(t) = 1−µ(rµ ∗ 1)(t),t ∈R+, the differentiation of this expression yields the claim. Moreover, therelation sµ(t) = −µrµ(t) is valid for all t > 0.

A characterisation of completely monotonic functions is given by Bern-stein’s Theorem. For a proof of this result we refer to [Wid41, Chapter IV.12].

Theorem 2.3.4 (Bernstein’s Theorem). A function a is completely monotonic ifand only if there is a non-decreasing function β : R+→R+ such that

a(x) =∫ ∞

0e−xt dβ(t), x > 0.

In other words: Completely monotonic functions are Laplace transformsof positive measures supported on R+; the characterisation is unique if onenormalises the function β by the assumptions β(0) = 0 and left-continuity,cf. [Prü12, p. 90].

2.3.1 Properties of the Laplace Transform of CompletelyMonotonic Kernels

Many aspects in the studies of Volterra equations base on the Laplace trans-form of the Volterra equation. The Laplace transform of completely monotonickernels has special properties, which will summarised in this section.

Firstly, we introduce the term of kernels with subexponential growth.

Definition 2.3.5 (Subexponential Growth). A kernel a ∈ L1,loc(R+) is ofsubexponential growth if we have

∫∞0 e−εt |a(t)|dt <∞ for all ε > 0.

This property ensures that the Laplace transform of the kernel a exists theopen right half-plane C+, a important fact since we want to use the Laplacetransform to study our Volterra equations.

Remark 2.3.6. a) For a kernel a ∈ L1,loc(R+) of subexponential growth wehave abs |a| ≤ 0. In particular, the map λ → a(λ) is holomorphic forall λ ∈ C+, as well as for all n ∈ N0 and all λ ∈ C+ we have a(n)(λ) =∫∞

0 e−λt(−t)na(t)dt, see Theorem 1.1.3.

24


Moreover, if the kernel a ∈ L1,loc(R+) is non-negative and of subexpo-nential growth, then its Laplace transform a : (0,∞)→R is completelymonotonic.

b) For a kernel a ∈ L1,loc(R+) of subexponential growth, we have for allλ ∈ C+ that a(λ) = a(λ). Hence, for the examination of the Laplacetransform it is enough to study λ ∈C with Imλ > 0.

We refer to Gripenberg, Londen and Staffans [GLS90, Theorem 5.2.6] forthe following theorem which summarises properties of the Laplace transformof completely monotonic and locally integrable kernels.

Theorem 2.3.7. Let a ∈ L1,loc(R+) be completely monotonic. The Laplace trans-form a has the following properties:

(i) The Laplace transform a has a holomorphic extension to Σπ via

a(λ) =∫ ∞

0

dβ(t)λ+ t

,

where β is the uniquely determined function from Bernstein’s Theorem.

(ii) The Laplace transform a(x) is real and non-negative for x > 0.

(iii) We have limx→∞ a(x) = 0.

(iv) Im a(z) ≤ 0 for all Imz > 0.

Moreover, a ∈ L1(R+) if and only if limsupx→0 |a(x)| <∞.

The representation as Laplace transform of a positive measure shows thata(λ) , 0 for all λ ∈ Σπ, see [Prü12, p. 56].

Now, we study the mapping behaviour of the Laplace transform a of acompletely monotonic kernel a ∈ L1,loc(R+), where we use the representationof a from Theorem 2.3.7 (i). At first we prove that a is injective.

Lemma 2.3.8. Let a ∈ L1,loc(R+) be a completely monotonic kernel.Then a : C+→C is injective.

Proof. Suppose, to the contrary, that there are λ1,λ2 ∈ C+ with λ1 , λ2 suchthat a(λ1) = a(λ2).

In the first instance we know that Imλ1 and Imλ2 have the same sign.3

Indeed, if not, that is sgnImλ1 , sgnImλ2, then sgnIm(λ1+σ ) , sgnIm(λ2+σ )for all σ ∈R+. Thus, we have

sgnIm∫ ∞

0

dβ(σ )λ1 + σ

, sgnIm∫ ∞

0

dβ(σ )λ2 + σ

, (2.6)

3We set for convenience sgn0 = 0.

25


where β is the positive measure supported on R+ from the representation of ain Theorem 2.3.7 (i). But relation (2.6) is a contradiction to the assumptionthat a(λ1) = a(λ2).

In case of Imλ1 = Imλ2 = 0, the assumption a(λ1) = a(λ2) implies directlythat λ1 = λ2, since a is itself completely monotonic on (0,∞), cf. Remark 2.3.6a), and thus injective on (0,∞).

Now, we assume w. l. o. g, that Imλ1 > 0. By the representation of a inTheorem 2.3.7 (i) we obtain

0 = a(λ1)− a(λ2) =∫ ∞

0

λ2 −λ1

(λ1 + σ )(λ2 + σ )dβ(σ ).

Due to the assumption that λ1 , λ2 we deduce that∫ ∞0

dβ(σ )(λ1 + σ )(λ2 + σ )

= 0. (2.7)

Since sgnImλ1 = sgnImλ2 > 0, we have for all σ ∈ R+ that Im(λi + σ ) > 0and λi + σ ∈ C+, i ∈ 1,2. Moreover, we obtain (λ1 + σ )(λ2 + σ ) ∈ Σπ andIm[(λ1 + σ )(λ2 + σ )] > 0 for all σ ∈R+ and hence

Im∫ ∞

0

dβ(σ )(λ1 + σ )(λ2 + σ )

, 0,

in contradiction to relation (2.7). Hence, we have a(λ1) , a(λ2) for all λ1,λ2 ∈C+ with λ1 , λ2.

In a certain manner we can extend the property of injectivity of the Laplacetransform of completely monotonic kernel to some slightly larger sector.

Lemma 2.3.9. Let a ∈ L1,loc(R+) be a completely monotonic kernel and choosesome δ ∈ (0, π2 ].

Then for all ρ ∈ Σ π2 −δ and all λ ∈

Σ π

2 + δ2− ρ

\ 0 we have a(λ+ ρ) , a(ρ).

Proof. We set ε = δ/2 and suppose, to the contrary, that there is some ρ0 ∈ Σ π2 −δ

and some λ0 ∈Σ π

2 +ε − ρ0

\ 0 such that a(λ0 + ρ0) = a(ρ0). Analogously to

the proof of the previous Lemma 2.3.8 it follows that Im(λ0 + ρ0) and Imρ0have the same sign. It is not possible that λ0 + ρ0 and ρ0 are real, since a isitself completely monotonic on (0,∞), cf. Remark 2.3.6 a), and thus injectiveon (0,∞).

W. .l. o. g. we assume that Imρ0 > 0. By the representation of a in Theorem2.3.7 (i) we obtain

0 = a(ρ0)− a(λ0 + ρ0) =∫ ∞

0

λ0

(ρ0 + σ )(λ0 + ρ0 + σ )dβ(σ ).

26


Due to the assumption λ0 , 0 we deduce that∫ ∞0

dβ(σ )(ρ0 + σ )(λ0 + ρ0 + σ )

= 0. (2.8)

We have sgnIm(ρ0) = sgnIm(λ0 + ρ0) > 0, thus we obtain for all σ ∈ R+ thatIm(ρ0 + σ ) > 0 and ρ0 + σ ∈ Σ π

2 −δ, as well as Im(λ0 + ρ0 + σ ) > 0 and thatλ0 + ρ0 + σ ∈ Σ π

2 +ε. Moreover, we deduce that (ρ0 + σ )(λ0 + ρ0 + σ ) ∈ Σπ−ε andIm[(ρ0 + σ )(ρ0 +λ0 + σ )] > 0 for all σ ∈R+. It follows that

Im∫ ∞

0

dβ(σ )(ρ0 + σ )(λ0 + ρ0 + σ )

, 0,

in contradiction to relation (2.8). Hence, we have for all ρ ∈ Σ π2 −δ and all

λ ∈Σ π

2 +ε − ρ\ 0 that a(λ+ ρ) , a(ρ).

To prove the following theorems we need a technical result for estimatesin the complex plane.

Lemma 2.3.10. Let η ∈ (0,π]. Then there exists a constant cη > 0 such that wehave for all λ ∈ Σπ−η the estimate

1 + |λ| ≤ cη |1 +λ| .

Proof. Let λ ∈ Σπ−η . Then there are r ∈ (0,∞) and ϕ ∈ (−π+η,π−η) such thatλ = reiϕ.

|1 +λ|2 = (1 + r cosϕ)2 + r2 sin2ϕ

≥ 1 + r2 − 2r cosη

≥1− cosη

2(1 + r)2 =

1− cosη2

(1 + |λ|)2.

The claim follows with c2η = (1 − cosη)/2. For a illustration of the proof we

refer to [HJ11, p. 4].

An easy consequence is the next corollary.

Corollary 2.3.11. Let θ1,θ2 ∈ [0,π) be such that θ1 + θ2 ∈ [0,π). Then, for allλ ∈ Σθ1

and all µ ∈ Σθ2there is a constant cη > 0 such that∣∣∣λ+ η

∣∣∣ ≥ cη (|λ|+ ∣∣∣µ∣∣∣) ,with η = π − (θ1 +θ2) ∈ (0,π].

The following theorem gives a upper and lower estimate for the Laplacetransform of a completely monotonic function along a sector in the complexplane by the Laplace transform along the positive axis.

27


Theorem 2.3.12. Let a ∈ L1,loc(R+) be completely monotonic. For each ε ∈ (0,π/4]there are constants C1 = C1(ε),C2 = C2(ε) > 0 such that for all λ ∈ Σ3π/4−ε wehave

C1 a(|λ|) ≤ |a(λ)| ≤ C2 a(|λ|).

Proof. The statement of this theorem is true for λ ∈C+\0, cf. [Prü12, Lemma8.1], since completely monotonic kernel are in particular 1-regular. Letε ∈ (0,π/4) be arbitrary, but fixed and let λ ∈ Σ3π/4−ε \C+ be arbitrary, i. e.Reλ < 0. We have

a(λ) =∫ ∞

0

λ+ σ

|λ+ σ |2dβ(σ ),

and hence

Re a(λ) =∫ ∞

0

Reλ+ σ

|λ+ σ |2dβ(σ ), and Im a(λ) = −

∫ ∞0

Imλ

|λ+ σ |2dβ(σ ).

Because of Reλ < 0 we can write Reλ = −|Reλ| and − Imλ = ∓|Imλ|. We have

|a(λ)|2 = (Re a(λ))2 + (|Im a(λ)|)2

≥ 12

(Re a(λ) + |Im a(λ)|)2

=12

(∫ ∞0

−|Reλ|+ σ + |Imλ||λ+ σ |2

dβ(σ ))2

.

Since λ ∈ Σ3π/4−ε\C+ there is some γ ∈ (0,π/4−ε) such that tanγ |Imλ| = |Reλ|.With this relation it follows that

−|Reλ|+ |Imλ|+ σ = (1− tanγ) |Imλ|+ σ ≥ 0

for all σ ≥ 0, since γ ∈ (0,π/4) . Thus we get for all γ ∈ (0,π/4− ε) that

−|Reλ|+ |Imλ|+ σ|λ|+ σ

≥(− tanγ + 1) |Imλ|√

tan2γ + 1 |Imλ|

≥ 1− tan(π/4− ε)√1 + tan2(π/4− ε)

= C(ε) > 0.

With this estimate and the triangle inequality we have for each σ ≥ 0 that

−|Reλ|+ |Imλ|+ σ|λ+ σ |2

≥ C(ε)|λ|+ σ

(|λ|+ σ )2 = C(ε)1

|λ|+ σ.

This implies for the integral with positive integrand that

|a(λ)|2 ≥C(ε)2

2

(∫ ∞0

dβ(σ )|λ|+ σ

)2

=C(ε)2

2|a(|λ|)|2 .

28


For the other estimate we use Lemma 2.3.10. There is a constant C∗ > 0 suchthat for all λ ∈ Σ3π/4 and all σ ∈ R+ we have |λ| + σ ≤ C∗ |λ+ σ |. Thus, weobtain for all λ ∈ Σ3π/4 that

|a(λ)| ≤∫ ∞

0

dβ(σ )|λ+ σ |

≤ C∗∫ ∞

0

dβ(σ )|λ|+ σ

= C∗ a(|λ|).

The claim follows with C1 = minC(ε)/√

2, c−1 and C2 = maxC∗, c, wherec > 0 is the constant from [Prü12, Lemma 8.1].

Further Properties of Kernels. We introduce two special properties of ker-nels and their Laplace transform. These properties are important for oursubsequent work with parabolic Volterra equations, cf. [Prü12, Chapter 3,Definition 3.1 and Definition 3.2].

The first property describes a special mapping property of the Laplacetransform of kernels with subexponential growth.

Definition 2.3.13 (Sectorial Kernel). Let a ∈ L1,loc(R+) be of subexponentialgrowth and suppose a(λ) , 0 for all Reλ > 0. The kernel a is called θ-sectorialif we have for all λ ∈C+ that |arg a(λ)| ≤ θ.

Here arg a(λ) is defined as the imaginary part of a fixed branch of log a(λ),and θ in the definition of sectorial kernels can be greater π. If the kernel ais sectorial, the branch of log a which gives the smallest angle θ is chosen. Ifa(λ) is real for real λ we choose the principal branch.

The second definition we introduce is a notation of regularity of kernels.

Definition 2.3.14 (Regular Kernel). Let a ∈ L1,loc(R+) be of subexponentialgrowth and k ∈N. The kernel a is called k-regular if there is a constant c > 0such that we have for 0 ≤ n ≤ k and all λ ∈C+ that∣∣∣λna(n)(λ)

∣∣∣ ≤ c |a(λ)| .

We know from [Prü12, p. 69] that for a k-regular kernel a, k ∈N arbitrary,its Laplace transform a has no zeros in the open right half-plane. Furthermore,each completely monotonic kernel is k-regular for all k ∈N, cf. [Prü12, p. 72].

In our next step we extend for completely monotonic kernels the propertiesof sectoriality and the 1-regularity onto a larger sector Σ π

2 +ε, with ε > 0.

Extended Sectoriality for Completely Monotonic Kernels. We showthat the property of θ-sectoriality can be continuously extended for com-pletely monotonic kernels.

29


Lemma 2.3.15. Let a ∈ L1,loc(R+) be completely monotonic and θ-sectorial withθ ∈ (0, π2 ]. For each ε ∈ (0, π2 ] there is some δ = δ(ε) ∈ (0, π2 ) such that we have forall λ ∈ Σ π

2 +δ the property a(λ) ∈ Σθ+ε.

Proof. Since the kernel a is θ-sectorial, we have for all λ ∈ Σ π2

that a(λ) ∈ Σθ.We are done if we can prove that a(λ) depends continuously on rotations ofthe argument λ. That is, for all λ ∈ Σ π

2and all ε > 0 there is a δ = δ(ε) > such

that for all |ϕ| ≤ δ we have∣∣∣∣a(λ)− a

(λe±iϕ

)∣∣∣∣ ≤ ε.

By continuity of a on Σπ, the claim is clear for all λ ∈ Σ π2∩B1(0). Indeed,

for each ε > 0 there is some δ0 = δ0(ε) ∈ (0, π2 ) such that for |ϕ| ≤ δ0 and all

λ ∈ Σ π2∩B1(0) we have

∣∣∣∣a(λ)− a(λe±iϕ

)∣∣∣∣ ≤ ε.Now, let λ ∈ Σ π

2with |λ| ≥ 1. Then we have by the representation of a as

Laplace transform of a positive measure that∣∣∣a(λ)− a(λe±iϕ)∣∣∣ ≤ ∣∣∣e±iϕ − 1

∣∣∣∫ ∞0

|λ||λ+ t|

∣∣∣λe±iϕ + t∣∣∣ dβ(t),

For all |ϕ| ≤ δ0 there is a constant Cδ0> 0 such that t + |λ| ≤ Cδ0

|t +λ| for allλ ∈ Σ π

2 +δ0and all t ∈R+, cf. Lemma 2.3.10. This implies the estimate

∣∣∣a(λ)− a(λe±iϕ)∣∣∣ ≤ C2

δ0

∣∣∣e±iϕ − 1∣∣∣∫ ∞

0

|λ|(|λ|+ t)2 dβ(t).

Since |λ| /(|λ|+ t)2 ≤ 1/(|λ|+ 2t) and a is decreasing on (0,∞) we get∣∣∣a(λ)− a(λe±iϕ)∣∣∣ ≤ C2

δ0sin(|ϕ| /2)a(1/2). (2.9)

Obviously, there is some δ ≤ δ0 such that this expression is smaller ε for allλ ∈ Σ π

2with |λ| ≥ 1, if |ϕ| ≤ δ. The claim follows with the continuity of the

complex logarithm.

Extended 1-Regularity for Completely Monotonic Kernels. Finally, weextend the property of 1-regularity for completely monotonic kernels onto alarger sector.

Lemma 2.3.16. Let a ∈ L1,loc(R+) be completely monotonic. For each η ∈ (0,π]there is a constant Cη > 0 such that for all λ ∈ Σπ−η we have the estimate∣∣∣λa′(λ)

∣∣∣ ≤ Cη |a(λ)| .

Proof. For all λ ∈C \R− we have

a(λ) =∫ ∞

0

dβ(σ )λ+ σ

,

30


as well as

a′(λ) = −∫ ∞

0

dβ(σ )(λ+ σ )2 .

For each η ∈ (0,π] there is a constant cη > 0 such that cη |λ+ σ | ≥ |λ|+ σ for allσ ∈R+ and all λ ∈ Σπ−η , cf. Lemma 2.3.10. Hence, we have

∣∣∣λa′(λ)∣∣∣ ≤ ∫ ∞

0

|λ||λ+ σ |2

dβ(σ ) ≤ c2η

∫ ∞0

|λ|(|λ|+ σ )2 dβ(σ )

≤ c2η

∫ ∞0

dβ(σ )|λ|+ σ

= c2η a(|λ|).

Remark 2.3.17. a) As already mentioned, the Laplace transform of a com-pletely monotonic kernel a ∈ L1,loc(R+) has no zeros in Σπ. On the basisof the representation as the Laplace transform of a positive measure itfollows that for every n ∈N and all λ ∈ Σπ we have

a(n)(λ) = (−1)nn!∫ ∞

0

dβ(σ )(λ+ σ )n+1 .

Hence, the n-th derivative a(n) has no zeros in Σπ.

b) We know from Remark 2.3.6 a) that a : (0,∞)→R is completely mono-tonic. Together with the property that a′ has no zero in Σπ it is obviousthat a is strictly decreasing on (0,∞), in particular a is injective on (0,∞).

If additionally a < L1(R+) we have limt→0 a(t) =∞, and limt→∞ a(t) = 0by Theorem 2.3.7 (iii). Thus, the Laplace transform a : (0,∞)→ (0,∞) isbijective.

2.3.2 Permanence Properties of Completely Monotonic Functions

It is a useful result to know that the sum and the product of completelymonotonic functions is completely monotonic, too. For a proof of this resultwe refer to Schilling, Song and Vondraček [SSV12, Corollary 1.6].

Corollary 2.3.18. The set of all completely monotonic functions is a convex conewhich is closed under multiplication and under pointwise convergence.

31


Remark 2.3.19. Examples for completely monotonic functions are

f (t) = tα , α ≤ 0,

g(t) = e−ct , c ∈R+,

h(t) = ln(1 + 1/t),

k(t) = e1/t ,

for t > 0, see [SSV12].

Observe that the composition of two completely monotonic functions isin general not completely monotonic. For this we introduce another class offunctions.

Definition 2.3.20 (Bernstein Function). A function f : (0,∞)→R is a Bern-stein function if f ∈ C∞((0,∞)), f (t) ≥ 0 for all t > 0 and f ′ is completelymonotonic.

The class of Bernstein functions has similar properties like the class ofcompletely monotonic functions, cf. [SSV12, Corollary 3. 8].

Corollary 2.3.21. The set of all Bernstein functions is a convex cone, whichis closed under pointwise limits and composition. Furthermore, for Bernsteinfunctions f , g ∈ C∞((0,∞)) and α, β ∈ (0,1) with α + β ≤ 1 is t 7→ f (tα)g(tβ)again a Bernstein function.

The following structural characterisation for Bernstein functions origi-nates from Bochner [Boc55, Chapter 4.1]. For a proof we refer to [SSV12, The-orem 3.7].

Theorem 2.3.22. Let f ∈ C∞((0,∞)) be a positive function. Then the followingstatements are equivalent:

(i) The function f is a Bernstein function.

(ii) For each completely monotonic function g ∈ C∞((0,∞) the compositiong f ∈ C∞((0,∞)) is completely monotonic.

(iii) For all α > 0 the function e−αf ∈ C∞((0,∞)) is completely monotonic.

In particular, the composition of a completely monotonic function witha Bernstein function always yields a completely monotonic function. Fur-thermore, Corollary 3.8 (iv) from Schilling, Song and Vondraček [SSV12]tells us, that for all Bernstein functions f ∈ C∞((0,∞)) the map t 7→ f (t)/t iscompletely monotonic. Thus, we can construct other completely monotonicfunctions with the aid of Bernstein functions.

32


Remark 2.3.23. Examples for Bernstein functions are

f (t) = tα , α ∈ (0,1),

g(t) =t

1 + t,

h(t) = ln(1 + t),

for t > 0, see [SSV12, Remark 3.13].

2.3.3 Examples for Completely Monotonic Kernels

The possibly most important example for an unbounded, locally integrableand completely monotonic kernel is the so-called standard kernel, which isintroduced as first example in this paragraph. The following examples baseon this kernel.

Standard Kernel. For α > 0 we define gα : (0,∞)→R+ for t > 0 by

gα(t) = tα−1/Γ (α),

and call gα the standard kernel (with exponent α); here Γ denotes the Gammafunction. Then gα is locally integrable on R+ and of sub-exponential growth,and its Laplace transform is given for all λ ∈ C+ by gα(λ) = λ−α. For allα, β > 0 we have the property gα ∗ gβ = gα+β on (0,∞).

Remark 2.3.24. For α ∈ (0,1) the standard kernel a = gα is unbounded, locallyintegrable and completely monotonic, and the corresponding Sonine kernel isgiven by b = g1−α. Their Laplace transforms are given by

a(λ) = λ−α , b(λ) = λα−1, λ ∈C+.

In the situation of the standard kernel we can express the scalar resolventand integral resolvent in terms of the (generalized) Mittag-Leffler function.

Definition 2.3.25. Let α ∈C+ and β ∈C be arbitrary.

(i) The entire function Eα : C→C given by

Eα(z) =∞∑k=0

zk

Γ (αk + 1),

for all z ∈C, is called Mittag-Leffler function.

(ii) The entire function Eα,β : C→C given by

Eα,β(z) =∞∑k=0

zk

Γ (αk + β),

for all z ∈C, is called generalised Mittag-Leffler function.

33


Remark 2.3.26. a) The functions Eα were introduced by G. M. Mittag-Leffler in the 1900s as generalisation of the exponential function. Forexample, we have E1(z) = ez and E2(z2) = cosh(z), cf. [BE55, ChapterXVII], as well as E2(−z2) = cos(z), see [GKMR14, Proposition 3.2]. Spe-cial cases of the generalised Mittag-Leffler function are for instanceE2,1(z) = cosh(

√z) and E2,2(z) = sinh(

√z)/√z, see [GKMR14, Chapter

4.2]. For further information about the order and type of the entirefunctions Eα and Eα,β , we refer to Gorenflo et al. [GKMR14].

b) For α ∈ (0,1] the completely monotonicity of the Mittag-Leffler functionEα(−x) , x > 0, was proved by Pollard [Pol48] without using methods ofprobability theory. A short proof for the completely monotonicity of thegeneralised Mittag-Leffer function Eα,β(−x), x > 0, for all α ∈ (0,1] andβ ≥ α, can be found in the article of Miller and Samko [MS97].

We consider the scalar Volterra equation with the standard kernel gα

u(t) +µ(gα ∗u)(t) = u0 + (gα ∗ f )(t), t ∈R+, (2.10)

where f ∈ L1,loc(R+). The scalar resolvent sµ and integral resolvent rµ aregiven for all t ∈R+ by

sµ(t) = Eα(−µtα),

rµ(t) = tα−1Eα,α(−µtα).

The solution of the scalar Volterra equation (2.10) is given by u(t) = sµ(t)u0 +(rµ ∗ f )(t), t ∈R+. In particular, for α ∈ (0,1] this solution is non-negative forall µ,u0 ∈R+ and non-negative f ∈ L1,loc(R+).

Note that for the standard kernel the scalar resolvent sµ, µ ∈R+, decays tozero as t→∞ like t−α. This rate is optimal, see estimate (2.5).

Standard Kernel with Exponential Weight. The standard kernel with ex-ponential weight is a further example for an unbounded, locally integrableand completely monotonic kernel. Let γ ≥ 0 and α ∈ (0,1), then we have thepair of Sonine kernels

a(t) = gα(t)e−γt +γ(1 ∗ [gαe−γ ·])(t),

b(t) = g1−α(t)e−γt ,

for t > 0. Their Laplace transforms are given by

a(λ) =1 +γ/λ(λ+γ)α

, b(λ) =1

(λ+γ)1−α , λ ∈C+,

cf. [Zac10, p. 3].

34


Sum of Standard Kernels. We can consider arbitrary sums of standardkernels. Let n ∈ N and αi ∈ (0,1) for i ∈ 1, . . . ,n. For t > 0 we define thekernel

b(t) =n∑i=1

g1−αi (t),

which is unbounded, locally integrable and completely monotonic. Hence,there is a completely monotonic corresponding Sonine kernel a ∈ L1,loc(R+).Their Laplace transforms are given by

b(λ) =n∑i=1

λαi−1, a(λ) =

n∑i=1

λαi

−1

, λ ∈C+.

Distributed Order Kernel. It is also possible to consider integrals overstandard kernels. Let 0 < c < d < 1, for t > 0 we define the kernel

b(t) =∫ d

cg1−α(t)dα,

which unbounded, locally integrable and completely monotonic. Thus, thereis completely monotonic corresponding Sonine kernel a ∈ L1,loc(R+). TheirLaplace transforms are given by

b(λ) =

λd−λcλ logλ for λ ∈C+ \ 1,d − c for λ = 1,

a(λ) =

logλλd−λc for λ ∈C+ \ 1,

1d−c for λ = 1.

Ultraslow Diffusion Kernel. If one consider distributed order kernels inthe limiting case of c→ 0 and d→ 1 we get the so-called ultraslow diffusionkernel, which is an unbounded, locally integrable and completely monotonickernel. We have

b(t) =∫ 1

0gα(t)dα, a(t) =

∫ ∞0

e−st

1 + sds, t > 0,

and their Laplace transforms are given by

b(λ) =

λ−1λ logλ for λ ∈C+ \ 1,1 for λ = 1,

a(λ) =

logλλ−1 for λ ∈C+ \ 1,1 for λ = 1,

see [VZ15, Example 6.5].

35

3Linear Volterra EquationsIn this section we summarise the most important definitions and resultsabout the concepts of resolvents and integral resolvents for Banach spacevalued linear Volterra equations of scalar type introduced in the first chapterof [Prü12].

Afterwards we focus on parabolic Volterra equations. For this setting wequote a fundamental theorem about the existence and regularity of resolvents.With regards to Part III of this work we analyse the asymptotic behaviour ofthe resolvent and introduce terms of stability of a resolvent. In particular,we pay attention to the situation of the standard kernel and the resultingestimates.

In the following we give sufficient conditions for the existence and inte-grability of the integral resolvent for parabolic Volterra equations. Here, thestandard kernel situations is discussed, too.

This chapter closes with a time-weighting argument for linear Volterraequations and the calculation of the commutator term for special choice oftime-weights. These arguments are essential components in the proofs ofour stability results for semilinear equations in Part II and for quasilinearequations in Part III.

Throughout this chapter we assume:

Assumptions. Let X be a complex Banach space equipped with the norm‖·‖X , and let A be a closed linear unbounded operator in X with dense domainD(A). We denote by XA = (D(A),‖·‖A) the domain of the operator A equippedwith the operator norm defined for all x ∈ D(A) by ‖x‖A = ‖x‖X + ‖Ax‖X . Sincethe operator A is closed the space XA is a Banach space which is continuouslyand densely embedded into X. Furthermore, let a ∈ L1,loc(R+) be a scalarnon-vanishing kernel, and J = [0,T ] with some T > 0 or J = R+.

3.1 Resolvents and Integral Resolvents

For f ∈ C(J ;X) we consider the linear Volterra equation

u(t) + (a ∗Au)(t) = f (t), t ∈ J. (3.1)

Following Prüss, we introduce the concepts of strong solutions and well-posedness of the linear Volterra equation (3.1), see [Prü12, Definition 1.1 and1.2].

Definition 3.1.1 (Strong Solutions of Volterra Equations). A function u ∈C(J ;X) is called strong solution of (3.1) on J if u ∈ C(J ;XA) and (3.1) holds on J .

37

3. Linear Volterra Equations

Definition 3.1.2 (Well-Posedness of Volterra Equations). The Volterra equa-tion (3.1) is called well-posed if for each x ∈ D(A) there is a unique strongsolution u(t;x) on R+ of

u(t) + (a ∗Au)(t) = x, t ∈R+,

and (xn) ⊂ D(A), xn → 0 imply u(t;xn) → 0 in X, uniformly on compactintervals of R+.

We will only consider well-posed Volterra equations. It can be shownthat the well-posedness of the linear Volterra equation is equivalent to theexistence of a unique resolvent family, defined below, cf. [Prü12, pp. 31 andDefinition 1.3].

Definition 3.1.3 (Resolvent). A family S(t)t∈R+⊂ B(X) of bounded linear

operators in X is called resolvent for the Volterra equation (3.1) if the followingconditions are satisfied:

(S1) S(·) is strongly continuous on R+ and S(0) = Id;

(S2) S(t)D(A) ⊂ D(A) and AS(t)x = S(t)Ax for all x ∈ D(A) and t ∈R+;

(S3) S(t)x+ (a ∗ [ASx])(t) = x, for all x ∈ D(A) and t ∈R+;the resolvent equation is satisfied.

Remark 3.1.4. a) If the Volterra equation (3.1) is well-posed, there is aunique resolvent family S(t)t∈R+

and we have for all x ∈ X and all t ∈R+that (a ∗ [Sx])(t) ∈ D(A) as well as S(t)x +A(a ∗ [Sx])(t) = x, see [Prü12,Proposition 1.1 and Corollary 1.1].

b) If there exists a resolvent S(t)t∈R+for the linear Volterra equation (3.1),

we call u(·;u0) = S(·)u0 ∈ C(R+;X), u0 ∈ X, a mild solution of the linearVolterra equation u(t) + (a ∗Au)(t) = u0, t ∈R+.

We want to discuss the stability of the equilibrium u∗ = 0 for a well-posedlinear Volterra equation

u(t) + (a ∗Au)(t) = u0, t ∈R+, (3.2)

with u0 ∈ X. According to the classical theory of dynamical systems weintroduce the terms of stable and asymptotically stable equilibria.

Definition 3.1.5 (Stability). (i) We call the equilibrium u∗ = 0 stable forthe linear Volterra equation (3.2) if for each ε > 0 there is some δ = δ(ε) >0 such that for all u0 ∈ Bδ(0) the corresponding mild solution u = u(·;u0)exists on R+ and for all t ∈R+ we have ‖u(t)‖X < ε.

38

3.1. Resolvents and Integral Resolvents

(ii) The equilibrium u∗ = 0 is called asymptotically stable for the linearVolterra equation (3.2) if u∗ = 0 is stable and there exists some δ0 > 0such that for all u0 ∈ Bδ0

(0) the corresponding mild solution u = u(·;u0)of the Volterra equation (3.2) satisfies ‖u(t)‖X → 0 as t→∞.

(iii) The equilibrium u∗ = 0 is called unstable for the linear Volterra equation(3.2) if u∗ is not stable.

Analogous to classical linear evolution equations it is possible to charac-terise the stability of the equilibrium u∗ = 0 of the linear equations in terms ofthe resolvent family.

Theorem 3.1.6 (Stability of Linear Volterra Equations). Let S(t)t∈R+be a

resolvent family for the linear Volterra equation (3.1). Then,

(i) u∗ = 0 is stable for the linear Volterra equation (3.2) if and only if we havesupt∈R+

‖S(t)‖B(X) <∞;

(ii) u∗ = 0 is asymptotically stable for the linear Volterra equation (3.2) if andonly if ‖S(t)‖B(X)→ 0 as t→∞.

Proof. The proof is completely analogous to the known result for the classicalcase, cf. [PW10, p. 93]. Since the linear Volterra equation (3.2) admits aresolvent family, for all u0 ∈ X the corresponding mild solution u(·;u0) isgiven by S(·)u0.

(i) We assume that u∗ = 0 is stable for the linear Volterra equation (3.2),i. e. for each ε > 0 there is some δ = δ(ε) > 0 such that for all u0 ∈ Bδ(0)the corresponding solutions u(·;u0) satisfies for all t ∈R+ the estimate‖u(t;u0)‖X < ε. We fix ε = 1 and the corresponding parameter δ =δ(1) > 0. Then, for all u0 ∈ Bδ(0) and all t ∈ R+ we have ‖u(t;u0)‖X < 1,and hence, ‖S(t)u0‖X < 1. Moreover, let v ∈ X \ 0 be arbitrary, we setv0 = δv/ ‖v‖X ∈ Bδ(0). This implies for all t ∈R+ that

‖S(t)v‖X =‖v‖Xδ‖S(t)v0‖X <

‖v‖Xδ

.

In particular, we have ‖S(t)‖B(X) < 1/δ, for all t ∈R+. Thus, the resolventfamily is bounded on R+.

Now, we suppose that ‖S(t)‖B(X) ≤M for all t ∈R+. For each ε > 0 we setδ = ε/M such that for all u0 ∈ Bδ(0) and all t ∈R+ we have

‖u(t,u0)‖X = ‖S(t)u0‖X ≤M ‖u0‖X < ε,

i. e. u∗ = 0 is stable for the Volterra equation (3.2).

39


(ii) Initially, we assume that u∗ = 0 is asymptotically stable for the linearVolterra equation (3.2). That is, u∗ = 0 is stable for (3.2) and there issome δ0 > 0 such that for all u0 ∈ Bδ0

the corresponding solution u(·;u0)exists on R+ and ‖u(t;u0)‖X = ‖S(t)u0‖X → 0 for t→∞. Analogous tothe proof of part (i), we deduce for all v ∈ X that ‖S(t)v‖X → 0. Thisimplies ‖S(t)‖B(X)→ 0 as t→∞.

Conversely, suppose that ‖S(t)‖B(X) → 0. Obviously, we have for allδ0 > 0 and u0 ∈ Bδ0

(0) that ‖u(t;u0)‖X = ‖S(t)u0‖X ≤ ‖S(t)‖B(X) ‖u0‖X → 0as t →∞. Clearly, in this situation the resolvent is bounded. Hence,u∗ = 0 is asymptotically stable for the linear Volterra equation (3.2).

Remark 3.1.7. The above theorem shows that the stability properties of theequilibrium u∗ = 0 for the linear Volterra equation (3.2) only depend on thebehaviour of the resolvent family. For this reason, we speak in the situation ofa bounded resolvent family on X also of a stable resolvent family on X. In thesituation that the B(X)-norm of the resolvent family tends to zero as t→∞,we call the resolvent family asymptotically stable on the Banach space X.

We refer to Section 3.3 where we examine the existence and asymptoticbehaviour of the resolvent family in detail.

In order to treat linear Volterra equations with arbitrary right hand sides,we define the integral resolvent family, cf. [Prü12, Definition 1.6], in a similarway to the resolvent family.

Definition 3.1.8 (Integral Resolvent). A family R(t)t∈R+⊂ B(X) of bounded

linear operators in X is called integral resolvent for the Volterra equation (3.1)if the following conditions are satisfied:

(R1) R(·)x ∈ L1,loc(R+;X) for each x ∈ X, and ‖R(t)‖B(X) ≤ ϕ(t) a. e. on R+, forsome ϕ ∈ L1,loc(R+);

(R2) R(t)D(A) ⊂ D(A) and AR(t)x = R(t)Ax for all x ∈ D(A) and a. a. t ∈R+;

(R3) R(t)x+ (a ∗ [ARx])(t) = a(t)x, for all x ∈ D(A) and a. a. t ∈R+;the integral resolvent equation is satisfied.

Remark 3.1.9. a) If there exists an integral resolvent R(t)t∈R+then we

have for all x ∈ X and almost all t ∈ R+ that (a ∗ [Rx])(t) ∈ D(A) andR(t)x+A(a ∗ [Rx])(t) = a(t)x. For a proof, we refer to Lizama’s work aboutk-regularized resolvents, see [Liz00, Lemma 2.2].

b) For u ∈ C(J ;X) the convolution (R ∗u) is well-defined on J and we havethe estimate ‖(R ∗u)(t)‖X ≤ (ϕ ∗ ‖u‖X)(t), t ∈ J .

40

3.1. Resolvents and Integral Resolvents

c) Assuming the existence of an integral resolvent family R(t)t∈R+, one can

prove the uniqueness of this object. In case of f = g ∗ a with g ∈ C(J ;XA)a representation of the strong solution of the linear Volterra equation(3.1) is given by u(t) = f (t)−A(R ∗ f )(t), t ∈ J , cf. [Prü12, p. 46].

d) If both resolvent and integral resolvent exist for the Volterra equation(3.1), they are related by the following identities:

R(t)Ax = −∂t S(t)x, for all x ∈ D(A), and a. a. t ∈R+,

R(t)x = ∂t (a ∗ S)(t)x, for all x ∈ X, and a. a. t ∈R+.

We will mostly confine ourselves to a special case of the linear Volterraequation (3.1)

u(t) + (a ∗Au)(t) = u0 + (a ∗ f )(t), t ∈ J, (3.3)

with u0 ∈ X and f ∈ C(J ;X).

Definition 3.1.10 (Mild Solution). Let u0 ∈ X and f ∈ C(J ;X). Assume thatthe equation Volterra equation (3.3) admits both a resolvent S(t)t∈R+

and anintegral resolvent R(t)t∈R+

. Then we call

u(t) = S(t)u0 + (R ∗ f )(t), t ∈ J, (3.4)

the (unique) mild solution of the linear Volterra equation (3.3).

Remark 3.1.11. a) Using the properties of the resolvent and integral resol-vent one assures oneself that the mild solution is even a strong solutionif u0 ∈ D(A) and f ∈ C(J ;XA).

b) In the case a ≡ 1 both resolvent and integral resolvent coincides with theC0-semigroup e−At. In this situation the formula (3.4) is nothing elsethan the well-known variation of parameters formula.

Analogous to the classical case there are generation theorems which char-acterise the existence of a resolvent and an integral resolvent. For the proofwe refer to Prüss [Prü12, Theorem 1.3 and 1.4].

41


Theorem 3.1.12 (Generation Theorem for Resolvents and Integral Resol-vents). Let A be a closed linear unbounded operator in X with dense domain D(A)and let a ∈ L1,loc(R+) satisfy

∫∞0 e−ωt |a(t)|dt <∞ for some ω ∈R.

(i) Then the Volterra equation (3.1) admits a resolvent family S(t)t∈R+such

that there is a constant M ≥ 1 with ‖S(t)‖B(X) ≤Meωt , for all t ∈R+ if andonly if the following conditions hold.

(H1) a(λ) , 0 and −1/a(λ) ∈ ρ(A) for all λ > ω;

(H2) H(λ) = (Id+a(λ)A)−1/λ satisfies the estimate∥∥∥H (n)(λ)∥∥∥B(X)

≤Mn!(λ−ω)−(n+1),

for all λ > ω and all n ∈N0, with some constant M ≥ 1.

(ii) The Volterra equation (3.1) admits an integral resolvent family R(t)t∈R+

such that ‖R(t)‖B(X) ≤ eωtϕ(t), for a. a. t ∈R+, holds for some ϕ ∈ L1(R+) ifand only if the following conditions are satisfied.

(K1) a(λ) , 0 and −1/a(λ) ∈ ρ(A) for all λ > ω;

(K2) K(λ) = a(λ)(Id+a(λ)A)−1 satisfies the estimate∞∑n=0

(λ−ω)n∥∥∥K (n)(λ)

∥∥∥B(X)≤M,

for all λ > ω, with some constant M ≥ 1.

In general the conditions in the generation theorem are hard to verify.Thus, we restrict our considerations to the import class of parabolic Volterraequation.

3.2 Parabolic Equations

In this section we summarise the most important definitions and results aboutparabolic Volterra equations taken from chapter 3 in [Prü12].

We assume additionally to the section above:

Assumption. The kernel a ∈ L1,loc(R+) is of subexponential growth, cf. Defi-nition 2.3.5. If this is the case it is obvious that the Laplace transform a(λ) ofthe kernel exists for all λ ∈C+.

Again, we consider the linear Volterra equation

u(t) + (a ∗Au)(t) = f (t), t ∈R+. (3.5)

Following Prüss, we introduce the notion of parabolicity, cf. [Prü12, Definition3.1].

42

3.2. Parabolic Equations

Definition 3.2.1 (Parabolic Volterra Equation). The Volterra equation (3.5)is called parabolic, if the following conditions hold.

(P1) a(λ) , 0, and −1/a(λ) ∈ ρ(A) for all λ ∈C+.

(P2) There is a constant M > 0 such that∥∥∥(Id+a(λ)A)−1

∥∥∥B(X)≤ M for all

λ ∈C+.

The next proposition provides sufficient conditions for the parabolicity ofthe Volterra equation (3.5), see [Prü12, Proposition 3.1].

Proposition 3.2.2. Let a ∈ L1,loc(R+) be θ-sectorial for some θ < π, suppose A isa closed linear densely defined operator, such that ρ(−A) ⊃ Σθ, and for all µ ∈ Σθwe have

∥∥∥(µ+A)−1∥∥∥B(X)

≤M/ |µ|. Then (3.5) is a parabolic Volterra equation.

Additionally, we introduce the notation of sectorial operators, see [DHP03,Definition 1.1].

Definition 3.2.3 (Sectorial Operator). Let X be a complex Banach space, andA be a closed linear operator in X. The operator A is called sectorial if thefollowing conditions are satisfied.

(i) D(A) = X, Rg(A) = X, (−∞,0) ⊂ ρ(A).

(ii)∥∥∥t(t +A)−1

∥∥∥B(X)≤M for all t > 0 and some M <∞.

Remark 3.2.4. It is not difficult to see that for a sectorial operator A there issome θ > 0 such that ρ(−A) ⊃ Σθ and sup‖λ(λ+A)−1‖B(X) : |argλ| < θ <∞.We define the spectral angle ϕA of the sectorial operator A by

ϕA = inf

ϕ : ρ(−A) ⊃ Σπ−ϕ , supλ∈Σπ−ϕ

∥∥∥λ(λ+A)−1∥∥∥B(X)

<∞

,cf. [DHP03, p. 9].

Remark 3.2.5. In particular, if A is a sectorial operator with spectral angleϕA, and the kernel a is θa-sectorial, then (3.5) is a parabolic Volterra equationprovided ϕA +θa < π.

The subsequent technical corollary shows that the additional assumptionof invertibility of the operator A yields an uniform bound for the resolventoperator (µ+A)−1 on a closed sector with smaller opening angle.

Corollary 3.2.6. We assume additionally to the requirements of Proposition 3.2.2that 0 ∈ ρ(A). Then for each ψ < θ there is some constant M0 =M0(ψ) > 0 suchthat for all µ ∈ Σψ we have ∥∥∥(µ+A)−1

∥∥∥B(X)≤M0.

43


Proof. Let ψ < θ be arbitrary, but fixed. For all µ ∈ Σψ with |µ| > 1 the estimatefor the resolvent (µ + A)−1 implies ‖(µ + A)−1‖B(X) ≤ M/ |µ| ≤ M. Moreover,the set Σψ ∩ B1(0) is compact and µ 7→ ‖(µ +A)−1‖B(X) is continuous, hence‖(µ+A)−1‖B(X) ≤ supλ∈Σψ∩B1(0)‖(λ+A)−1‖B(X) ≤M1. The claim follows withM0 = maxM,M1.

3.3 Existence and Estimates for Resolvents andIntegral Resolvents of Parabolic VolterraEquations

3.3.1 Existence and Boundedness of Resolvents

We have the following theorem about the existence and boundedness ofresolvents for parabolic Volterra equations, see [Prü12, Theorem 3.1]

Theorem 3.3.1 (Resolvents for Parabolic Equations). LetX be a Banach space,A a closed linear operator inX with dense domainD(A) and a ∈ L1,loc(R+). Assumethe Volterra equation (3.5) is parabolic, and the kernel a is 1-regular.

Then there is a uniformly bounded resolvent S ∈ C(R+;B(X)) for (3.5).

Remark 3.3.2. A similar existence result for the integral resolvent is notknown. From Comment 2.7.d) in [Prü12] we know that the existence of anintegral resolvent is not clear even in the case of analytic resolvents1. Althoughthe resolvent is differentiable on (0,∞) the local integrability on R+ can onlybe ensured under additional assumptions onto the kernel a ∈ L1,loc(R+).

For Part II of this work it is enough to have knowledge about the existenceof a bounded resolvent. With regard to Part III it is necessary to know theasymptotic behaviour of the resolvent family. The result reads as follows:

Theorem 3.3.3 (Asymptotic Behaviour of Resolvents). Let a ∈ L1,loc(R+) bean unbounded, completely monotonic kernel which is θa-sectorial with θa ∈(0, π2 ]. We assume that the corresponding completely monotonic Sonine kernel

b ∈ L1,loc(R+) satisfies the estimate∫ 1

0 b(1/t)dtt <∞. Furthermore, let A be a closed,linear and densely defined operator on the Banach space X with ρ(−A) ⊃ Σϕ ∪ 0and

∥∥∥(µ+A)−1∥∥∥B(X)

≤M/ |µ| for all µ ∈ Σϕ, with some ϕ > θa.Then, there is a constantC > 0 such that the resolvent S(t)t∈R+

for the Volterraequation (3.5) satisfies for each t > 0 the estimate

‖S(t)‖B(X) ≤ Cb(1/t)/t.1Analogous to the concept of analytic C0-semigroups, it is possible to extend the concept

of resolvents to some sector Σθ where we have an analytic extension of the resolvent S(t)t∈R+ .For further information see chapter 2 in [Prü12].

44

3.3. Existence and Estimates for Resolvents and Integral Resolvents

Remark 3.3.4. The proof of Theorem 3.3.3 as well as the proof of the subse-quent Theorem 3.3.8 heavily relies on the Analytic Representation Theorem1.2.1. For this reason, we explain why the function a and b, respectively,satisfies the assumptions for this theorem.

The kernel a ∈ L1,loc(R+) is unbounded and completely monotonic. Hence,a is a completely monotonic function, see Remark 2.3.6; in particular a isnon-negative and non-increasing on (0,∞). By the properties of the Laplacetransform of completely monotonic functions it follows that limt→∞ a(t) = 0,see Theorem 2.3.7. The same properties hold for the corresponding completelymonotonic Sonine kernel b ∈ L1,loc(R+).

The Laplace transform of the Sonine condition a(λ)b(λ) = 1/λ, for allλ ∈ C+, implies that supt∈(0,1] ta(t) = supt∈(0,1] 1/b(t). Note that limt→0 ta(t)

either vanishes in the case of b < L1(R+) or is finite with limt→0 ta(t) = 1/b(0)in the case of b ∈ L1(R+). Since b is non-increasing, it follows that 1/b isincreasing. In particular, we have supt∈(0,1] ta(t) ≤ 1/b(1) < ∞. The sameargument works if one changes the role of the kernels a and b.

Proof of Theorem 3.3.3. The assumptions on the kernel a imply that the kernela is ϕ-sectorial. Together with the assumptions on the operator A we knowthat the Volterra equation (3.5) is a parabolic, see Proposition 3.2.2.

The Laplace transform a of the kernel a can be extended to the sectorΣπ, cf. Theorem 2.3.7. Furthermore, the mapping property of the Laplacetransform a onto the sector Σθa is continuously extendable, see Lemma 2.3.15.In particular, we may choose ε0 < minϕ − θa, π2 , and fix δ0 ∈ (0,π/4) suchthat a(λ) ∈ Σθa+ε0

for all λ ∈ Σ π2 +δ0

.Moreover, the Laplace transform of the corresponding completely mono-

tonic kernel b(λ) has a holomorphic extension to Σπ and we have for allλ ∈ Σ π

2 +δ0the lower and upper estimate of |b(λ)| by b(|λ|), see Theorem 2.3.12.

Due to the choice of ε0 it is obvious that for all λ ∈ Σ π2 +δ0

we have 1/a(λ) ∈Σθa+ε0

⊂ Σϕ. We set

H(λ) = (1/a(λ) +A)−1/(λa(λ)) = b(λ)(1/a(λ) +A)−1,


. Obviously, H is holomorphic on Σ π2 +δ0

with values in B(X).Since θa + ε0 < ϕ and because of the invertibility of the operator A it followsthat the expression ‖(1/a(λ) +A)−1‖B(X) is uniformly bounded on Σθa+ε0

, cf.Corollary 3.2.6. Together with the upper estimate of |b(λ)| by b(|λ|), seeTheorem 2.3.12, we deduce for all λ ∈ Σ π

2 +δ0the estimate

‖H(λ)‖B(X) ≤Mb(|λ|),

with some constant M > 0. The map b : (0,∞)→R+ satisfies the condition inTheorem 1.2.1. It follows that there is a holomorphic function h : Σδ0

→B(X)

45


satisfying the estimate ‖h(t)‖B(X) ≤ Cb(1/t)/t on (0,∞) and h(λ) =H(λ) for allλ > 0. The uniqueness of the Laplace transform, Theorem 1.1.4, together withthe uniqueness of the resolvent, implies the desired estimate for the resolventfamily. For each t > 0 we have ‖S(t)‖B(X) ≤ Cb(1/t)/t.

In the following we give sufficient conditions for the stability and asymp-totic stability of the resolvent family S(t)t∈R+

, cf. Remark 3.1.7 and Theorem3.1.6.

Lemma 3.3.5. Let a ∈ L1,loc(R+) be an unbounded, completely monotonic kernelwhich is θa-sectorial with θa ∈ (0, π2 ]. Furthermore, let A be a closed, linearand densely defined operator on the Banach space X with ρ(−A) ⊃ Σϕ ∪ 0 and∥∥∥(µ+A)−1

∥∥∥B(X)≤M/ |µ| for all µ ∈ Σϕ, with some ϕ > θa.

Then, the resolvent family S(t)t∈R+is stable on X.

Moreover, if a < L1(R+) and the corresponding completely monotonic Soninekernel satisfies the estimate

∫ 10 b(1/t)dtt <∞, then the resolvent family S(t)t∈R+

isasymptotically stable on the Banach space X.

Proof. The conclusion about the stability follows immediately from the exis-tence of a bounded resolvent, cf. Theorem 3.3.1.

Using the estimate from Theorem 3.3.3, we have in case of a < L1(R+) thatlimt→∞ b(1/t)/t = limt→∞1/a(1/t) = 1/a(0) = 0. The claim follows.

We conclude with an estimate of the resolvent family S(t) on XA =(D(A),‖·‖A), which is densely embedded inX, and on various real interpolationspaces betweenXA andX, the spaces (X,XA)γ,q for γ ∈ (0,1) and q ∈ [1,∞]. Forconvenience we set (X,XA)0,q = X and (X,XA)1,q = XA, for all values of q. Werefer to Bergh and Löfström [BL76], Lunardi [Lun09] as well as Triebel [Tri78]and [Tri83] for further information about real interpolation spaces and generalinterpolation theory.

Corollary 3.3.6. Let a ∈ L1,loc(R+) be an unbounded, completely monotonic kernelwhich is θa-sectorial with θa ∈ (0, π2 ]. We assume that the corresponding completely

monotonic Sonine kernel b ∈ L1,loc(R+) satisfies the estimate∫ 1

0 b(1/t)dtt <∞.Furthermore, let A be a closed, linear and densely defined operator on the

Banach space X with ρ(−A) ⊃ Σϕ∪0 and∥∥∥(µ+A)−1

∥∥∥B(X)≤M/ |µ| for all µ ∈ Σϕ,

with some ϕ > θa.Then, for each γ ∈ [0,1] and all q ∈ [1,∞] there is some constantM =M(γ,q) >

0 such that for all t ∈R+ we have

‖S(t)‖B((X,XA)γ,q

) ≤Mmin1, b(1/t)/t.

In particular, if a < L1(R+), then the resolvent family is asymptotically stable on(X,XA)γ,q, for all γ ∈ [0,1] and all q ∈ [1,∞].

46


Proof. Since equation (3.5) is parabolic, the claim for the Banach space Xfollows by Theorem 3.3.1 and Theorem 3.3.3. For the Banach space XA wehave for x ∈ D(A) and all t ∈R+ that ‖S(t)x‖A = ‖S(t)x‖X + ‖AS(t)x‖X . On D(A)the resolvent commutes with the operator A, thus ‖S(t)x‖A ≤ ‖S(t)‖B(X) ‖x‖X +‖S(t)‖B(X) ‖Ax‖X . This implies ‖S(t)‖B(XA) ≤ ‖S(t)‖B(X) and thus the desiredpointwise estimate.

The claim for the real interpolation spaces (X,XA)γ,q between X and XAfollows by the fact that we have an exact interpolation pair of exponent γ .

Finally, we consider the case of the standard kernel.

Corollary 3.3.7. Let X be a Banach space, p ∈ (1,∞) and α ∈ (0,1). Assume, thatthe operator A is invertible and sectorial with spectral angle ϕA < π −απ2 .

We denote by S(t)t∈R+the resolvent family of the linear Volterra equation

with standard kernel

u(t) + (gα ∗Au)(t) = f (t), t ∈ [0,T ],

where f ∈ C([0,T ];X). The following statements hold true.

(i) For each γ ∈ [0,1] and all q ∈ [1,∞] the resolvent family S(t)t∈R+is asymp-

totically stable on (X,XA)γ,q.

(ii) For all α ∈ (1/p,1) we have S ∈ Lp(R+;B

((X,XA)γ,q

)).

(iii) There is some constant M > 0 such that for all t > 0 we have the estimate‖S(t)‖B(X;XA) ≤Mt−α.

Proof. (i) We know from Remark 3.2.5 and Proposition 3.2.2 that the oper-ator A together with the standard kernels a = gα and b = g1−α satisfy theconditions of Corollary 3.3.6. Since gα < L1(R+) we obtain the asymp-totic stability of the resolvent family on (X,XA)γ,q for all γ ∈ [0,1] andall q ∈ [1,∞].

(ii) In view of b(1/t)/t = t−α we know that t 7→min1, b(1/t)/t ∈ Lp(R+) forall α ∈ (1/p,1).

(iii) For the estimate of the resolvent family as bounded operator from X toXA we refer to Corollary 3 in [Prü96a].

3.3.2 Existence and Integrability of Integral Resolvents

In this section we give sufficient conditions for the existence of an integralresolvent and their integrability. Following Prüss [Prü12, Definition 10.2], wecall an integral resolvent R(t)t∈R+

integrable if there is a ϕ ∈ L1(R+) such that‖R(t)‖B(X) ≤ ϕ(t) for a. a. t ∈R+.

47


Theorem 3.3.8 (Existence and Integrability of Integral Resolvents). Let a ∈L1,loc(R+) be an unbounded, completely monotonic kernel which is θa-sectorial

with θa ∈ (0, π2 ] and satisfies the estimate∫ 1

0 a(1/t)dtt <∞. Furthermore, let A be aclosed, linear and densely defined operator on the Banach space X with ρ(−A) ⊃ Σϕ

and∥∥∥(µ+A)−1

∥∥∥B(X)≤M/ |µ| for all µ ∈ Σϕ, with some ϕ > θa.

(i) Then there exists an integral resolvent R(t)t∈R+for the Volterra equation

(3.5), and we have for all t > 0 the estimate

‖R(t)‖B(X) ≤ Caa(1/t)/t,

with some constant Ca > 0.

Assume additionally that the corresponding Sonine kernel b ∈ L1,loc(R+) satisfies

the estimate∫ 1

0 b(1/t)dtt <∞ and that 0 ∈ ρ(A).

(ii) Then the integral resolvent for the Volterra equation (3.5) satisfies for allt > 0 the estimate

‖R(t)‖B(X) ≤ Cbb(1/t)/t2,

with some constant Cb > 0.

(iii) If the map t 7→mina(1/t)/t, b(1/t)/t2 is integrable on R+, then the integralresolvent is an integrable integral resolvent.

In particular, if∫∞

1 b(1/t)dtt2 <∞, then the integral resolvent is integrable.

Proof of Theorem 3.3.8. Analogous to the proof of the above Theorem 3.3.3we know that the assumptions on the kernel a imply that the kernel a isϕ-sectorial. Together with the assumptions on the operator A we know thatthe Volterra equation (3.5) is a parabolic, see Proposition 3.2.2.

The Laplace transform a of the kernel a can be extended to the sectorΣπ, cf. Theorem 2.3.7. Furthermore, the mapping property of the Laplacetransform a onto the sector Σθa is continuously extendable, see Lemma 2.3.15.In particular, we may choose ε0 < minϕ − θa, π2 , and fix δ0 ∈ (0,π/4) suchthat a(λ) ∈ Σθa+ε0


. Moreover, we have for all λ ∈ Σ π2 +δ0

thelower and upper estimate of |a(λ)| by a(|λ|), see Theorem 2.3.12, as well as theextended 1-regularity on Σ π

2 +δ0, cf. Lemma 2.3.16.

(i) Due to the choice of ε0 it is obvious, that for all λ ∈ Σ π2 +δ0

we have1/a(λ) ∈ Σθa+ε0

⊂ Σϕ. We set K(λ) = (1/a(λ) +A)−1 for λ ∈ Σ π2 +δ0

. Obvi-ously, K is holomorphic on Σ π

2 +δ0with values in B(X). The assumed

estimate for the resolvent operator (µ+A)−1 implies for all λ ∈ Σ π2 +δ0

that ‖K(λ)‖B(X) = ‖(1/a(λ) +A)−1‖B(X) ≤M |a(λ)|. Since |a(λ)| behaves as

48


a(|λ|) on Σ π2 +δ0

, we have for all λ ∈ Σ π2 +δ0

that there is some constantMa > 0 such that

‖K(λ)‖B(X) ≤Maa(|λ|).

The map a : (0,∞) → R+ satisfies the conditions for Theorem 1.2.1.Therefore, there is a holomorphic function f : Σδ0

→ B(X) which sat-isfies the estimate ‖f (t)‖B(X) ≤ Caa(1/t)/t on (0,∞) and f (λ) = K(λ) forall λ > 0. In particular, f (·)x ∈ L1,loc(R+;X) and f ∈ L1,loc(R+;B(X)),hence f satisfies condition (R1) from Definition 3.1.8. The commutationproperty (R2) is obviously satisfied, since the operator A commutes withits resolvent operator on D(A). Due to the uniqueness of the Laplacetransform, Theorem 1.1.4, the property f (λ) = K(λ) for λ > 0 impliesthat f satisfies the integral resolvent equation (R3). Since there is onlyone integral resolvent for the Volterra equation (3.5) we know that on(0,∞) the map f is the integral resolvent R(t)t∈R+

we searched for andwe have for all t > 0 the estimate

‖R(t)‖B(X) ≤ Caa(1/t)/t;

obviously, the map t 7→ a(1/t)/t is locally integrable on R+.

(ii) Assuming that the operatorA is invertible, there is some constantM0 > 0such that for all λ ∈ Σ π

2 +δ0we have the uniform estimate ‖K(λ)‖B(X) ≤

M0, cf. Corollary 3.2.6.

The map K is holomorphic on Σ π2 +δ0

, so we can consider the holo-morphic function K ′ on Σ π

2 +δ0with values in B(X). We have K ′(λ) =

K2(λ)a′(λ)/a2(λ), and with the corresponding Sonine kernel b ∈ L1,loc(R+)we can rewrite this expression as K ′(λ) = K2(λ)λa′(λ)b(λ)/a(λ) on Σ π

2 +δ0.

Using the 1-regularity of the kernel a as well as the uniform bound-edness of ‖K(λ)‖B(X) on the sector Σ π

2 +δ0we obtain for all λ ∈ Σ π

2 +δ0

the estimate ‖K ′(λ)‖B(X) ≤M20C|b(λ)|. The corresponding Sonine kernel

b ∈ L1,loc(R+) is itself completely monotonic. We can estimate |b(λ)| onΣ π

2 +δ0by b(|λ|), see Theorem 2.3.12. So, we see that for all λ ∈ Σ π

2 +δ0that

there is some constant Mb > 0 such that∥∥∥K ′(λ)∥∥∥B(X)

≤Mbb(|λ|).

The map b : (0,∞)→R+ satisfies the properties of Theorem 1.2.1. Hence,there is a holomorphic function g : Σδ0

→ X satisfying the estimate‖g(t)‖B(X) ≤ Cbb(1/t)/t for all t > 0, with g(λ) = K ′(λ) for all λ > 0. Again,via the uniqueness of the Laplace transform, Theorem 1.1.4, this impliesg(t) = −tR(t) for all t > 0. In particular, we have for all t > 0 the estimate

‖R(t)‖B(X) ≤ Cbb(1/t)/t2.

49


(iii) Combining the estimates from part (i) and (ii) implies for each t > 0 that

‖R(t)‖B(X) ≤ Cmina(1/t)/t, b(1/t)/t2,

with some constant C > 0. The map t 7→mina(1/t)/t, b(1/t)/t2 is inte-grable on R+, by assumption. This implies directly the integrability ofthe integral resolvent on R+.

Moreover, if∫∞

1 b(1/t)dtt2 <∞, then we have∫∞

0 mina(1/t)/t, b(1/t)/t2dt ≤∫ 10 a(1/t)

dtt +

∫∞1 b(1/t)dtt2 <∞, and the integrability of the integral resol-

vent on R+ follows.

In the situation of Volterra equations with standard kernel and a compati-ble sectorial operator there always exists an integrable integral resolvent.

Corollary 3.3.9. Let α ∈ (0,1) and A be a sectorial operator on the Banach spaceX with spectral angle ϕA < π −απ2 .

Then the Volterra equation with standard kernel

u(t) + (gα ∗Au)(t) = f (t), t ∈ [0,T ], (3.6)

where f ∈ C([0,T ];X), admits an integral resolvent R(t)t∈R+. Moreover, if the

operator A is invertible, then the integral resolvent is integrable on R+

Proof. We consider the situation of the standard kernel with a = gα andb = g1−α, α ∈ (0,1). Their Laplace transforms are given by a(λ) = λ−α andb(λ) = λ−(1−α), for λ ∈C+. In particular, we have

∫ 10 a(1/t)dtt =

∫ 10 t

α dtt = 1

α <∞,

as well as∫ 1

0 b(1/t)dtt =∫ 1

0 t1−α dt

t = 11−α <∞ and

∫∞1 b(1/t)dtt2 =

∫∞1 t1−α dtt2 = 1

α <∞. Together with the assumptions on the operator A it follows by Theorem3.3.8 that there is an integral resolvent R(t)t∈R+

for the Volterra equation(3.6). Assuming the invertibility of the operator A, the integral resolvent isintegrable on R+.

The following corollary gives sufficient conditions for the integrabilityrequirements in the above Theorem 3.3.8. These conditions are integrabilityconditions for the given kernel and not for its Laplace transform.

Corollary 3.3.10. Let c ∈ L1,loc(R+) be a non-negative kernel of subexponentialgrowth.

(i) If −∫ 1

0 c(t) ln t dt <∞, then∫ 1

0 c(1/t)dtt <∞.

(ii) The condition∫∞

1 c(t)dtt <∞ implies∫∞

1 c(1/t)dtt2 <∞.

50


Proof. (i) By substitution we have∫ 1

0 c(t)dtt =

∫∞1 c(r)drr . The definition of

the Laplace transform yields∫ 1

0c(t)

dtt

=∫ ∞

1

∫ ∞0

e−rtc(t)dtdrr.

Note that the integrand is non-negative for all (r, t) ∈ [1,∞)× (0,∞). Wechange the order of integration and show that the integral exists. Thus,∫ ∞

0c(t)

∫ ∞1

e−rtdrrdt ≤

∫ ∞0c(t)e−t ln

(1 +

1t

)dt,

here we applied the estimate for the exponential integral∫∞

1 e−rt drr <

e−t ln(1 + 1

t

)for t > 0, see [AS64, p. 229, 5.1.10]. On the interval [0,1]

we have the estimate ln(1 + 1

t

)= ln(t + 1)− ln t ≤ ln2− ln t, and on the

interval [1,∞) that ln(1 + 1

t

)≤ ln2. This yields∫ ∞

0c(t)

∫ ∞1

e−rtdrrdt ≤ ln2

∫ ∞0c(t)e−t dt −

∫ 1

0c(t) ln t dt <∞,

by the assumption that −∫ 1

0 c(t) ln t dt < ∞. By Tonelli’s Theorem we

know that both repeated integrals coincide and thus∫∞

1 c(t)dtt <∞.

(ii) An easy substitution yields that∫∞

1 c(1/t)dtt2 =∫ 1

0 c(r)dr. With the defini-tion of the Laplace transform it follows that∫ ∞

1c(1/t)

dt

t2=

∫ 1

0

∫ ∞0

e−rtc(t)dt dr.

Observe that the integrand is non-negative for all (r, t) ∈ [0,1]× (0,∞).Once more we change the order of integration and prove that the integralexists, thus ∫ ∞

0c(t)

∫ 1

0e−rt dr dt =

∫ ∞0c(t)

[1− e−t

] dtt.

We can estimate the term[1− e−t

]/t on the interval [0,1] uniformly by

one. Together with the triangle inequality we deduce∫ ∞0c(t)

∫ 1

0e−rt dr dt ≤

∫ 1

0c(t)dt +

∫ ∞1c(t)

dtt

+∫ ∞

1e−tc(t)dt <∞,

since c ∈ L1,loc(R+) is of subexponential growth and∫∞

1 c(t)dtt <∞, byassumption.

51


Remark 3.3.11. Observe that the idea for the condition in Corollary 3.3.10 (i)originates from Comment 2.7 c) in [Prü12]. Here, Prüss gives similar sufficientconditions for the integrability of the integral resolvent without any reasoning.Our sufficient conditions on the kernel a for the existence of an integrableintegral resolvent correspond to the conditions in the monograph of Prüss.

3.4 Time-Weights for Linear Volterra Equations

An essential step in the proof of our stability result in Part II is the time-weighting of the Volterra equation and the rewriting of the correspondingmild solution. For this reason we introduce the used time-weighting argumentfor linear Volterra equations. At first, we have the following auxiliary result.

Lemma 3.4.1. Let b ∈ L1,loc(R+)∩C1((0,∞)) such that b ≥ 0 with limt→0 tb(t) = 0and −b ≥ 0 is non-increasing on (0,∞). For some T > 0 we consider ϕ ∈ C([0,T ])∩C1((0,T )) with ϕ > 0 on [0,T ], and u ∈ C([0,T ];X).

Then the map t 7→ ϕ(t)(b ∗ u)(t) − (b ∗ [ϕu])(t) is continuous on [0,T ] anddifferentiable on (0,T ) with

∂t [ϕ(b ∗u)− (b ∗ [ϕu])] (t) = ϕ(t)(b ∗u)(t) +∫ t

0b(t − τ)[ϕ(t)−ϕ(τ)]u(τ)dτ,

for all t ∈ (0,T ).

Remark 3.4.2. a) The assumptions of Lemma 3.4.1 imply that [t 7→ tb(t)] ∈L1,loc(R+). Indeed, for arbitrary T > 0 we have by integration by partsthat ∣∣∣∣∣∣

∫ T

0tb(t)dt

∣∣∣∣∣∣ ≤ T b(T ) + limt→0

tb(t) +∫ T

0b(t)dt <∞,

since limt→0 tb(t) = 0 and b ∈ L1,loc(R+) is non-negative.

b) Note, that for all completely monotonic kernels b ∈ L1,loc(R+), whichadmit a corresponding Sonine kernel, the assumptions of Lemma 3.4.1are satisfied.

Proof. The continuity of the map t 7→ ϕ(t)(b∗u)(t)−(b∗[ϕu])(t) is obvious, sincethe convolution of a locally integrable kernel b with the continuous function uand ϕu, respectively, is continuous on [0,T ]. We prove the differentiability byshowing the limit of the differential quotient exists. Let t ∈ (0,T ) be arbitrary,but fixed, and |h| ∈ (0,mint/4,T − t) be sufficiently small. We have

52

3.4. Time-Weights for Linear Volterra Equations

ϕ(t + h)(b ∗u)(t + h)− (b ∗ [ϕu])(t + h)− [ϕ(t)(b ∗u)(t)− (b ∗ [ϕu])(t)]h

=[ϕ(t + h)−ϕ(t)] (b ∗u)(t + h)

h

+ϕ(t)(b ∗u)(t + h)− (b ∗ [ϕu])(t + h)− [ϕ(t)(b ∗u)(t)− (b ∗ [ϕu])(t)]

h. (3.7)

The function ϕ is differentiable on (0,T ) and the convolution (b ∗u) is contin-uous on [0,T ]. This implies

[ϕ(t + h)−ϕ(t)] (b ∗u)(t + h)h

→ ϕ(t)(b ∗u)(t),

as |h| → 0.

Right-Hand Differential Quotient. We consider the second summandin (3.7) with h ∈ (0,mint/4,T − t) and have

1hϕ(t)(b ∗u)(t + h)− (b ∗ [ϕu])(t + h)− [ϕ(t)(b ∗u)(t)− (b ∗ [ϕu])(t)]

=∫ t

0

b(t + h− τ)− b(t − τ)h

[ϕ(t)−ϕ(τ)]u(τ)dτ

+∫ t+h

tb(t + h− τ)

ϕ(t)−ϕ(τ)h

u(τ)dτ.

Since ϕ is continuously differentiable on (0,T ) we have for all τ ∈ (t, t + h)that there is some ξ ∈ (t,τ) ⊂ (t, t + h) such that |ϕ(t)−ϕ(τ)| ≤ |ϕ(ξ)| |t − τ | ≤‖ϕ‖C([t,min5t/4,T ])h. This yields∫ t+h

t|b(t + h− τ)|

|ϕ(t)−ϕ(τ)|h

‖u(τ)‖X dτ

≤∫ t+h

t|b(t + h− τ)|dτ ‖ϕ‖C([t,min5t/4,T ]) ‖u‖C([0,T ];X) ;

since b ∈ L1((0,T )), this expression tends to zero as h→ 0.Now, we want to apply the vector-valued version of the dominated con-

vergence theorem from Arendt et al. [ABHN01, Theorem 1.1.8] to deducethat∫ t

0

b(t + h− τ)− b(t − τ)h

[ϕ(t)−ϕ(τ)]u(τ)dτ→∫ t

0b(t − τ) [ϕ(t)−ϕ(τ)]u(τ)dτ,

as h→ 0. We define fh : [0, t]→ X by

fh(τ) =

b(t−τ+h)−b(t−τ)h [ϕ(t)−ϕ(τ)]u(τ), τ ∈ [0, t),

0, τ = t.

53


Obviously, for all τ ∈ [0, t) we have fh(τ)→ b(t − τ)[ϕ(t)−ϕ(τ)]u(τ) as h→ 0.We will show that the map g : [0, t]→R+, given by

g(τ) =

|b(t − τ)|2‖ϕ‖C([0,T ]) ‖u‖C([0,T ];X) , τ ∈ [0, t/2],

|b(t − τ)(t − τ)| ‖ϕ‖C([t/2,t]) ‖u‖C([0,T ];X) , τ ∈ (t/2, t),

0, τ = t,

dominates fh(τ) for almost all τ ∈ [0, t] and all h > 0. The assumptions on b andb ensure the integrability of s 7→ sb(s) on (0, t). Hence, we have g ∈ L1((0, t)).

Using the continuous differentiability of the kernel b we deduce that foreach τ ∈ [0, t) there is some ξ ∈ (0,h) with |b(t − τ + h)− b(t − τ)| ≤ |b(t−τ +ξ)|h.Since −b = |b| is non-increasing, we have |b(t − τ + h)− b(t − τ)| ≤ h|b(t − τ)|.

For τ ∈ [0, t/2] we have the uniform estimate

‖fh(τ)‖X ≤ |b(t − τ)|2‖ϕ‖C([0,T ]) ‖u‖C([0,T ];X) .

For τ ∈ (t/2, t) we use the continuous differentiability of the map ϕ. Thereis some ξ ∈ (τ, t) ⊂ (t/2, t) such that |ϕ(t)−ϕ(τ)| ≤ |ϕ(ξ)| |t − τ |. This yields forτ ∈ [t/2, t) that

‖fh(τ)‖X ≤ |b(t − τ)(t − τ)| ‖ϕ‖C([t/2,t]) ‖u‖C([0,T ];X) .

Finally, the vector-valued dominated convergence theorem implies that∫ t

0fh(τ)dτ→

∫ t


as h→ 0. Thus, ϕ(t)(b ∗u)(t)− (b ∗ [ϕu])(t) is differentiable from the right.

Left-Hand Differential Quotient. Now, we consider the second sum-mand in (3.7) with h ∈ (0,mint/4,T − t) and have

1hϕ(t)(b ∗u)(t)− (b ∗ [ϕu])(t)− [ϕ(t)(b ∗u)(t − h)− (b ∗ [ϕu])(t − h)]

=∫ t−h

0

b(t − τ)− b(t − h− τ)h


+∫ t

t−hb(t − τ)

ϕ(t)−ϕ(τ)h

u(τ)dτ.

Analogous to the situation above, we use the continuous differentiability of ϕon (0,T ). For all τ ∈ [t − h, t] there is some ξ ∈ (τ, t) ⊂ (t − h, t) ⊂ (3t/4, t) suchthat |ϕ(t)−ϕ(τ)| ≤ | ˙ϕ(ξ)| |t − τ | ≤ ‖ϕ‖C([3t/4,t])h. This implies∫ t

t−h|b(t − τ)|dτ ‖ϕ‖C([3t/4,t]) ‖u‖C([0,T ];X) ;

54


since b ∈ L1((0,T )) this expression tends to zero as h→ 0.Once more we want to apply the vector-valued version of the dominated

convergence theorem from Arendt et al. [ABHN01, Theorem 1.1.8] to provethat ∫ t−h

0

b(t − τ)− b(t − h− τ)h


=∫ t

h

b(t − τ + h)− b(t − τ)h

[ϕ(t)−ϕ(τ − h)]u(τ − h)dτ

→∫ t

0b(t − τ) [ϕ(t)−ϕ(τ)]u(τ)dτ,

as h→ 0. We define kh : [0, t]→ X by

kh(τ) =

b(t−τ+h)−b(t−τ)h [ϕ(t)−ϕ(τ − h)]u(τ − h), τ ∈ [h, t)

0, τ ∈ [0,h)∪ t.

Obviously, for all τ ∈ [0, t) we have kh(τ)→ b(t − τ)[ϕ(t)−ϕ(τ)]u(τ) as h→ 0.We will show that the map l : [0, t]→R+, given by

l(τ) =

|b(t − τ)|2‖ϕ‖C([0,T ]) ‖u‖C([0,T ];X) , τ ∈ [0, t/2],[2 |b(t − s)|+ |b(t − τ)(t − τ)|

]‖ϕ‖C([t/2,t]) ‖u‖C([0,T ];X) , τ ∈ (t/2, t),

0, τ = t,

dominates kh(τ) for almost all τ ∈ [0, t] and all h > 0. By the assumptions on band b we obtain l ∈ L1((0, t)).

The continuous differentiability of the kernel b implies that for eachτ ∈ [0, t) there is some ξ ∈ (0,h) such that |b(t − τ + h)− b(t − τ)| ≤ |b(t − τ + ξ)|h.Since −b = |b| is non-increasing, we have |b(t − τ + h)− b(t − τ)| ≤ h|b(t − τ)|.

For τ ∈ [0, t/2] it follows the uniform estimate

‖kh(τ)‖X ≤ |b(t − τ)|2‖ϕ‖C([0,T ]) ‖u‖C([0,T ];X) .

For τ ∈ (t/2, t) we employ the continuous differentiability of the map ϕ.There is some ξ ∈ (t − h, t) ⊂ (3t/4, t) such that |ϕ(t)−ϕ(t − h)| ≤ |ϕ(ξ)|h. Thesame statement remains true for all τ ∈ (t/2, t), there is some η ∈ (τ −h, t −h) ⊂(t/4, t) such that |ϕ(t − h)−ϕ(τ − h)| ≤ |ϕ(η)| |t − τ |. Combining these estimateswith the known estimates for b from above, we have for τ ∈ (t/2, t) that

‖kh(τ)‖X ≤ |b(t + h− τ)− b(t − τ)||ϕ(t)−ϕ(t − h)|

h‖u‖C([0,T ];X)

+|b(t + h− τ)− b(t − τ)|

h|ϕ(t − h)−ϕ(τ − h)| ‖u‖C([0,T ];X)

≤[2 |b(t − τ)| ‖ϕ‖C([3t/4,t]) + |b(t − τ)(t − τ)| ‖ϕ‖C([t/4,t])

]‖u‖C([0,T ];X) .

55


Finally, the vector-valued dominated convergence theorem yields that∫ t

0kh(τ)dτ→

∫ t


as h→ 0. Hence, ϕ(t)(b ∗u)(t)− (b ∗ [ϕu])(t) is differentiable from the left.

Since the right-hand limit and the left-hand limit of the differential quo-tient coincide, the map t 7→ ϕ(t)(b ∗ u)(t) − (b ∗ [ϕu])(t) is differentiable att ∈ (0,T ). In particular,

∫ t0 b(t − τ)[ϕ(t)−ϕ(τ)]u(τ)dτ exists for all t ∈ (0,T ) as

a Bochner integral.

Corollary 3.4.3. Let the assumptions of Lemma 3.4.1 be satisfied.If for some t ∈ (0,T ) it is known that (b ∗u)(t) or (b ∗ [ϕu])(t) is differentiable

at the point t, then we have

ϕ(t)∂t(b ∗u)(t)−∂t(b ∗ [ϕu])(t) =∫ t

0b(t − τ)[ϕ(t)−ϕ(τ)]u(τ)dτ. (3.8)

Remark 3.4.4. In the situation of Corollary 3.4.3 we call the expression∫ t

0

[−b(t − τ)

][ϕ(t)−ϕ(τ)]u(τ)dτ

commutator term.

Proof of Corollary 3.4.3. Firstly, we consider the case that (b ∗u)(t) is differen-tiable at the point t ∈ (0,T ). The differentiability of ϕ(t)(b ∗u)(t)− (b ∗ [ϕu])(t)and (b ∗ u)(t) at the point t ∈ (0,T ) yields the differentiability of (b ∗ [ϕu])(t)at t. Now, we consider the case that (b ∗ [ϕu])(t) is differentiable at the pointt ∈ (0,T ). The differentiability of ϕ(t)(b ∗ u)(t)− (b ∗ [ϕu])(t) and (b ∗ [ϕu])(t)implies the differentiability of ϕ(t)(b ∗ u)(t) at t. Since ϕ(t) > 0 and ϕ(t) isdifferentiable it follows that (b ∗u)(t) is differentiable at t.

The relation (3.8) follows by Lemma 3.4.1 and the product rule for differ-entiable functions.

Suppose that u ∈ C([0,T ];XA) is a strong solution of the linear Volterraequation

u(t) + (a ∗Au)(t) = u0 + (a ∗ f )(t), t ∈ [0,T ], (3.9)

where A is a closed, linear and densely defined operator on the Banach spaceX, u0 ∈ D(A) and f ∈ C([0,T ];XA). Applying the above Corollary 3.4.3 wededuce a Volterra equation for the product ϕu, with ϕ ∈ C([0,T ])∩C1((0,T )).

56


Lemma 3.4.5. Let a ∈ L1,loc(R+) be a unbounded, completely monotonic kerneland b ∈ L1,loc(R+) denote the corresponding completely monotonic Sonine kernel.Furthermore, let A be a closed, linear and densely defined operator in the Banachspace X. We assume that the Volterra equation (3.9) admits a resolvent and anintegral resolvent. Moreover, let u ∈ C([0,T ];XA) be a strong solution of the linearVolterra equation (3.9) for some T > 0, f ∈ C([0,T ];XA) and u0 ∈ D(A), as well asϕ ∈ C([0,T ])∩C1((0,T )) with ϕ > 0 on [0,T ].

Then we have for the product ϕu the following Volterra equation

[ϕu](t) + (a ∗A[ϕu])(t) =u0 + (a ∗ [b[ϕ − 1]u0]) (t) + (a ∗ [ϕf ])(t) (3.10)

+(a ∗

∫ ·0

[−b(· − τ)

][ϕ(·)−ϕ(τ)]u(τ)dτ

)(t), t ∈ [0,T ].

Remark 3.4.6. The strong solution u ∈ C([0,T ];XA) is given by the mild formu-lation u(t) = S(t)u0 + (R ∗ f )(t), t ∈ [0,T ]. According to this representation wehave [ϕu](t) = ϕ(t)S(t)u0 +ϕ(t)(R ∗ f )(t). The equation (3.10) from the aboveLemma 3.4.5 yields for the product ϕu another representation

[ϕu](t) =S(t)u0 + (R ∗ [b[ϕ − 1]u0]) (t) + (R ∗ [ϕf ])(t)

+(R ∗

∫ ·0

[−b(· − τ)

][ϕ(·)−ϕ(τ)]u(τ)dτ

)(t),

for all t ∈ [0,T ].

Proof of Lemma 3.4.5. Let u ∈ C([0,T ];XA) be a strong solution of the linearVolterra equations (3.9). If we convolve the equation (3.9) with the corre-sponding Sonine kernel b ∈ L1,loc(R+) and use the Sonine condition a ∗ b ≡ 1on (0,∞), we obtain the equation

(b ∗u)(t) + (1 ∗Au)(t) = (b ∗ 1)(t)u0 + (1 ∗ f )(t), t ∈ [0,T ].

Since (1∗Au), (1∗f ) ∈ C1([0,T ];X) and (1∗b) ∈ C1((0,T ];X), we know that everyterm of the equation is differentiable on (0,T ]. In particular, b∗u differentiableon (0,T ] with ∂t(b ∗u)(t) = b(t)u0 + f (t)−Au(t). Obviously ∂t(b ∗u)(t)− b(t)u0is continuous on [0,T ]. Hence, we get for t ∈ (0,T ] the equation

∂t(b ∗u)(t) +Au(t) = b(t)u0 + f (t).

Now, we multiply this equation by the scalar function ϕ and apply Corollary3.4.3 to replace the expression ϕ(t)∂t(b ∗u)(t). This yields for all t ∈ (0,T ] that

∂t(b ∗ [ϕu])(t) +A[ϕu](t) =(ϕb)(t)u0 + (ϕf )(t)

+∫ t

0(−b(t − τ))[ϕ(t)−ϕ(τ)]u(τ)dτ.

57


We convolve both sides of this equation with the kernel a ∈ L1,loc(R+) andobtain for all t ∈ [0,T ] the equation

(a ∗∂t(b ∗ [ϕu]))(t) + (a ∗A[ϕu])(t) =(a ∗ [ϕb])(t)u0 + (a ∗ [ϕf ])(t)

+(a ∗

∫ ·0

[−b(· − τ)

][ϕ(·)−ϕ(τ)]u(τ)dτ

)(t).

Finally, we can replace the term a ∗∂t(b ∗ [ϕu]) by ϕu on [0,T ], since we have(b ∗ [ϕu])(0) = 0 and a ∗ b ≡ 1 on (0,∞). Thus,

[ϕu](t) + (a ∗A[ϕu])(t) =(a ∗ [bϕ])(t)u0 + (a ∗ [ϕf ])(t)

+(a ∗

∫ ·0

[−b(· − τ)

][ϕ(·)−ϕ(τ)]u(τ)dτ

)(t), t ∈ [0,T ].

Obviously, we can add a zero u0 − (a ∗ b)(·)u0 on the right hand side of theequation. The linearity of the integral yields the claim.

Lemma 3.4.7. Let a ∈ L1,loc(R+) be an unbounded, completely monotonic kerneland b ∈ L1,loc(R+) denote the corresponding Sonine kernel. Furthermore, let A be aclosed, linear and densely defined operator in the Banach space X. We assume thatthe Volterra equation (3.9) admits a resolvent and an integral resolvent. Moreover,let u0 ∈ X and f ∈ C([0,T ];X) for some T > 0, as well as ϕ ∈ C([0,T ])∩C1((0,T ))with ϕ > 0 on [0,T ]. We denote by u ∈ C([0,T ];X) the mild solution of the Volterraequation (3.9). Then,

[ϕu](t) =S(t)u0 + (R ∗ [b[ϕ − 1]u0]) (t) + (R ∗ [ϕf ])(t)

+(R ∗

∫ ·0

[−b(· − τ)

][ϕ(·)−ϕ(τ)]u(τ)dτ

)(t),

for all t ∈ [0,T ].

Proof. Lemma 3.4.5 and Remark 3.4.6 imply this representation for the prod-

uct only for u0 ∈ D(A) and f ∈ C([0,T ];XA). Let(u

(n)0

)n∈N⊂ D(A) be such that

u(n)0 → u0 in X and

(f (n)

)n∈N⊂ C([0,T ];XA) be such that

∥∥∥f (n) − f∥∥∥C([0,T ];X)

→0 as n → ∞. We denote by u(n) ∈ C([0,T ];XA) the strong solution of theVolterra equation

u(n)(t) +(a ∗Au(n)

)(t) = u(n)

0 +(a ∗ f (n)

)(t), t ∈ [0,T ].

It is given by the mild formulation u(n)(t) = S(t)u(n)0 +

(R ∗ f (n)

)(t), t ∈ [0,T ].

58

3.5. Commutator Term

On one hand, we have∥∥∥u −u(n)∥∥∥C([0,T ];X)

≤ supt∈[0,T ]

‖S(t)‖B(X)

∥∥∥∥u0 −u(n)0

∥∥∥∥X

+ (ψ ∗ 1)(T )∥∥∥f − f (n)

∥∥∥C([0,T ];X)

→ 0,

as n→∞. Here ψ ∈ L1,loc(R+) denotes the locally integrable function from theDefinition 3.1.8 of the integral resolvent. Note that the resolvent is uniformlybounded on compact intervals. On the other hand, we have by Lemma 3.4.5that

[ϕu](n)(t) =S(t)u(n)0 +

(R ∗

[b[ϕ − 1]u(n)

0

])(t) +

(R ∗

[ϕf (n)

])(t)

+(R ∗

∫ ·0

[−b(· − τ)

][ϕ(·)−ϕ(τ)]u(n)(τ)dτ

)(t).

In particular, we obtain for each t ∈ [0,T ] that∥∥∥∥∥S(t)[u0 −u

(n)0

]+(R ∗

[b[ϕ − 1]

(u0 −u

(n)0

)])(t) +

(R ∗

[ϕ

(f − f (n)

)])(t)

+(R ∗

∫ ·0

[−b(· − τ)

][ϕ(·)−ϕ(τ)]

(u(τ)−u(n)(τ)

)dτ

)(t)

∥∥∥∥∥∥X

≤ supt∈[0,T ]

‖S(t)‖B(X)

∥∥∥∥u0 −u(n)0

∥∥∥∥X

+ (ψ ∗ |b[ϕ − 1]|) (T )∥∥∥∥u0 −u

(n)0

∥∥∥∥X

+ (ψ ∗ϕ)(T )∥∥∥f − f (n)

∥∥∥C([0,T ];X)

+(ψ ∗

∫ ·0|b(· − τ)| |ϕ(·)−ϕ(τ)|dτ

)(T )

∥∥∥u −u(n)∥∥∥C([0,T ];X)

→ 0,

as n→∞. The claim follows.

3.5 Commutator Term

For a special choice of the maps ϕ and u we are able to calculate the commu-tator term.

Lemma 3.5.1. Let ε > 0 and a ∈ L1,loc(R+) be an unbounded, completely mono-tonic kernel and b ∈ L1,loc(R+) be the corresponding completely monotonic So-nine kernel. Let sε denote the scalar resolvent, i. e. the solution of the equationsε + ε(sε ∗ a) = 1 on R+. We set ϕε = 1/sε on R+.

Then we have for all t > 0 the equation∫ t

0

[−b(t − τ)

][ϕε(t)−ϕε(τ)]

1ϕε(τ)

dτ = ε+ b(t) [1−ϕε(t)] ≥ 0. (3.11)

59


Remark 3.5.2. By the properties of the scalar resolvent sε, we know for themap ϕε that ϕε(0) = 1, ϕε ≥ 1 is increasing and absolutely continuous onR+, and ϕε ∈ L1,loc(R+), with ϕε(t) = εrε(t)/s2ε (t) for all t ∈ (0,∞). The relation(3.11) implies on (0,∞) the estimate 0 ≤ b(t) [ϕε(t)− 1] ≤ ε. Note that thisestimate is also known from Lemma 2.2.6.

Proof of Lemma 3.5.1. We bring the expression in the bracket down to a com-mon denominator and obtain∫ t

0

[−b(t − τ)

][ 1sε(t)

− 1sε(τ)

]sε(τ)dτ =

1sε(t)

∫ t

0

[−b(t − τ)

][sε(τ)− sε(t)]dτ.

By the fundamental theorem of calculus it follows that

1sε(t)

∫ t

0

[−b(t − τ)

][sε(τ)− sε(t)]dτ =

1sε(t)

∫ t

0

[−b(t − τ)

]∫ t

τ[−sε(x)]dxdτ.

Changing the order of integration yields

1sε(t)

∫ t

0

∫ t

τ

[−b(t − τ)

][−sε(x)]dxdτ =

1sε(t)

∫ t

0[−sε(x)]

∫ x

0

[−b(t − τ)

]dτ dx.

Applying again the fundamental theorem of calculus yields

1sε(t)

∫ t

0[−sε(x)]

∫ x

0

[−b(t − τ)

]dτ dx =

1sε(t)

∫ t

0[−sε(x)] [b(t − x)− b(t)]dx.

Now, we use the relation −sε = εrε on (0,∞), where rε denotes the scalarintegral resolvent, cf. Remark 2.3.3, and get

1sε(t)

∫ t

0[−sε(x)] [b(t − x)− b(t)]dx

=εsε(t)

∫ t

0rε(x)b(t − x)dx − b(t)

sε(t)

∫ t

0[−sε(x)]dx.

We recall the relations rε ∗ b = sε on (0,∞) and sε(0) = 1, see Proposition 2.2.5,and obtain the relation

εsε(t)

∫ t

0rε(x)b(t − x)dx − b(t)

sε(t)

∫ t

0[−sε(x)]dx =

εsε(t)

sε(t)−b(t)sε(t)

[1− sε(t)] .

The preceding calculations imply∫ t

0

[−b(t − τ)

][ 1sε(t)

− 1sε(τ)

]sε(τ)dτ = ε+ b(t)− b(t)

sε(t).

The non-negativity of the integral relies on the non-negativity of −b and thefact that sε is non-increasing on R+. Therefore the integrand is non-negative,too. It is also possible to see the non-negativity of the expression by using theestimate from Lemma 2.2.6.

60

Part II

Semilinear Parabolic VolterraEquations

61

4Stability for Semilinear ParabolicVolterra EquationsIn this chapter we consider semilinear Volterra equations. We define mildsolutions for these equations, prove the local existence and uniqueness of mildsolutions for semilinear problems and show how to continue local solutions.We close this chapter with the principle of linearised stability for semilinearparabolic Volterra equations, Theorem 4.2.2. The simple subsequent corol-laries pay attention to the special case of Volterra equations with standardkernels.

Throughout this chapter we assume the following:

Assumptions. Let X be a complex Banach space equipped with the norm‖·‖X , and let A be a closed, unbounded linear operator in X with dense domainD(A) and XA = (D(A),‖·‖A). Furthermore, let a ∈ L1,loc(R+) be an unbounded,non-vanishing scalar kernel and J = [0,T ] with some T > 0, as well as u0 ∈ X.For T > 0 and some r > 0 the map F : [0,T ] × Br(u0)→ X is continuous andsatisfies a Lipschitz condition in the second component, i. e. there is someL > 0 such that for all u,v ∈ Br(u0) and all t ∈ [0,T ] we have the estimate‖F(t,u)−F(t,v)‖X ≤ L‖u − v‖X .

4.1 Local Existence and Uniqueness of Mild Solutionsand Maximal Solutions

We study the following semilinear Volterra equation

u(t) + (a ∗Au)(t) = u0 + (a ∗F(·,u))(t), t ∈ J, (4.1)

under the additional assumption, that this equation admits both a resolventS(t)t∈R+

and an integral resolvent R(t)t∈R+in the sense of Section 3.1.

Definition 4.1.1 (Mild Solution). We call a function u ∈ C(J ;X) mild solutionof the Volterra equation (4.1) if u solves the equation

u(t) = S(t)u0 + (R ∗F(·,u))(t), t ∈ J.

We start with a result on existence and uniqueness of mild solutions of thesemilinear equation (4.1) under the above assumption on the map F.

63

4. Stability for Semilinear Parabolic Volterra Equations

4.1.1 Local Existence and Uniqueness

We prove that the semilinear Volterra equation admits a unique local mildsolution.

Theorem 4.1.2 (Local Existence and Uniqueness). Let u0 ∈ X and F : [0,T ]×Br(u0)→ X be continuous and Lipschitz continuous in the second component. Weassume that there is a resolvent S(t)t∈R+

and an integral resolvent R(t)t∈R+for

the Volterra equation (4.1).Then there is some δ = δ(u0) > 0 such that there is a unique local mild solution

u ∈ C([0,δ];X) for (4.1).

Proof. For a given u0 ∈ X and some sufficiently small δ > 0, which will bechosen later, we define the map K : C([0,δ];X)→ C([0,δ];X) via

(Ku)(t) = S(t)u0 + (R ∗F(·,u))(t),

for all t ∈ [0,δ] ⊂ [0,T ]. A fixed point of K yields the desired mild solution.We define the set

B(u0,δ, r) = u ∈ C([0,δ];X) : ‖u −u0‖C([0,δ],X) ≤ r,

and we will prove that K maps the set B(u0,δ, r) into itself and K is a contrac-tion.

Self-mapping. Let u ∈ B(u0,δ, r) be arbitrary. We show that for suffi-ciently small δ > 0 the function Ku belongs to B(u0,δ, r), too. Obviously, Kuis continuous on [0,δ] and we have for all t ∈ [0,δ] that

‖(R ∗F(·,u))(t)‖X ≤∫ t

0‖R(t − τ)F(τ,u(τ))‖X dτ

≤ ‖F(·,u)‖C([0,δ];X) ‖ϕ‖L1([0,δ]) .

Via the triangle inequality and the continuity of the map F it is obvious that

‖F(·,u)‖C([0,δ];X) ≤ L‖u −u0‖C([0,δ];X) +M,

with M = ‖F(·,u0)‖C([0,T ];X). Thus, we have

‖Ku −u0‖C([0,δ],X) ≤ ‖S(t)u0 −u0‖C([0,δ];X) + ‖R ∗F(·,u)‖C([0,δ];X)

≤ ‖S(t)u0 −u0‖C([0,δ];X) + (Lr +M)‖ϕ‖L1([0,δ]) .

We choose δ1 > 0 such that for all δ ≤ δ1 we have (Lr +M)‖ϕ‖L1([0,δ]) ≤ r/2.Then, we choose δ2 > 0 such small that we have for all δ ≤ δ2 the estimate‖S(t)u0 −u0‖C([0,δ];X) ≤ r/2. Hence, for all δ ≤ minδ1,δ2 we obtain the self-mapping property of the map K onto the set B(u0,δ, r).

64

4.1. Local Existence and Uniqueness of Mild Solutions and Maximal Solutions

Contraction. Let u, u ∈ B(u0,δ, r) be arbitrary. We show that for suffi-ciently small δ > 0 the map K is a contraction. We have

‖Ku −Ku‖C([0,δ];X) ≤ supt∈[0,δ]

∫ t

0‖R(t − τ) [F(τ,u(τ))−F(τ, u(τ))]‖X dτ,

≤ L‖u − u‖C([0,δ];X) ‖ϕ‖L1([0,δ]) .

We choose δ3 > 0 such small that for all δ ≤ δ3 we have L‖ϕ‖L1([0,δ]) ≤ 1/2 andthe contraction property of the map K is obtained.

Altogether, we have for δ ≤ minδ1,δ2,δ3 that K is a contractive self-mapping of the set B(u0,δ, r). This set is not empty and a complete metricspace when endowed with the sup-norm. Hence, the contraction mappingprinciple yields a unique fixed point of K in the set B(u0,δ, r). It follows theclaim.

4.1.2 Maximal Solutions

In this section we show that a local unique mild solution u ∈ C([0, τ];X) of (4.1)can be continued to a mild solution u ∈ C([0, τ + δ];X) for some sufficientlysmall δ > 0, where u restricted to the interval [0, τ] is equal to u. On the basisof this continuation we are able to define a maximal (mild) solution for thesemilinear problem (4.1).

Definition 4.1.3. We call a function F : R+×X→ X locally Lipschitz continuousin the second component, uniformly on bounded intervals of R+, if for allT > 0 and all u0 ∈ X there is some r = r(u0) > 0 and a constant L = L(u0, r,T ) > 0such that for all t ∈ [0,T ] and all x,y ∈ Br(u0) we have the Lipschitz estimate∥∥∥F(t,x)−F(t,y)

∥∥∥X≤ L

∥∥∥x − y∥∥∥X.

Lemma 4.1.4. Let F : R+×X→ X be continuous and locally Lipschitz continuousin the second component, uniformly on bounded intervals of R+. We assume thatthere is a resolvent S(t)t∈R+

and an integral resolvent R(t)t∈R+for the Volterra

equation (4.1). Let u ∈ C([0, τ];X) be the local mild solution of (4.1) with u0 ∈ X.Then there is some δ > 0 such that there is a mild solution u ∈ C([0, τ + δ];X)

of (4.1) and for all t ∈ [0, τ] we have u(t) = u(t).

Proof. The local mild solution u of (4.1) with u0 ∈ X is continuous on [0, τ],hence we set uτ = u(τ). Due to the local Lipschitz continuity in the secondcomponent of F there is some r = r(uτ ) > 0 and some L = L(uτ , r,τ) > 0 suchthat for all x,y ∈ Br(uτ ) and all t ∈ [τ + 1] we have the Lipschitz estimate∥∥∥F(t,x)−F(t,y)

∥∥∥X≤ L

∥∥∥x − y∥∥∥X

.

65


Once more, using a fixed point argument we will prove that the there is acontinuation on some larger interval. For some sufficiently small δ ∈ (0,1], wedefine the map K : C([0, τ + δ];X)→ C([0, τ + δ];X) via

(Kv)(t) = S(t)u0 + (R ∗F(·,v))(t),

for all t ∈ [0, τ + δ]. A fixed point of K yields the wanted continuation of ourmild solution. Therefore, we define the set

B(τ,δ, r) = v ∈ C([0, τ + δ];X) : v = u on [0, τ] and ‖uτ − v‖C([0,τ+δ];X) ≤ r/2,

and we will show that K maps B(τ,δ, r) into itself and it is a contraction,provided that δ ∈ (0,1] is small enough. We set u(t) = (Kv)(t) for t ∈ [0, τ + δ].

Self-mapping. Let v ∈ B(δ,τ, r) be arbitrary. We will prove that forsufficiently small δ > 0 the function Kv belongs to B(τ,δ, r), too. It is obviousthat the map u is continuous on [0, τ+δ]. We consider u−uτ on [0, τ+δ]. Sinceuτ = u(τ) = S(τ)u0 + (R ∗F(·,u))(τ) we have

u(t)−uτ = S(t)u0 − S(τ)u0 + (R ∗F(·,v)χ[τ,τ+δ])(t).

For all t ∈ [τ,τ + δ] we obtain

‖u(t)−uτ‖X ≤ ‖S(t)u0 − S(τ)u0‖X +∫ t

τ‖R(t − s)F(s,v(s))‖X ds.

We know that S(·)u0 is continuous on R+, in particular S(·)u0 uniformlycontinuous on [0, τ + 1], thus we may choose δ1 such small that for all δ ≤ δ1and all |t − τ | ≤ δ we have ‖S(t)u0 − S(τ)u0‖X ≤ r/4. Moreover, we have for allt ∈ [τ,τ + δ] that v(t) ∈ Br(uτ ) and hence the estimate

‖F(t,v(t))‖X ≤ L‖v(t)−uτ‖X + ‖F(t,uτ )‖X ,

Since v ∈ B(τ,δ, r) we deduce ‖F(·,v)‖C([τ,τ+δ];X) ≤ Lr/2 +N (τ), where N (τ) =‖F(·,uτ )‖C([τ,τ+δ];X). Thus,

‖u(t)−uτ‖X ≤ ‖S(t)u0 − S(τ)u0‖X + ‖F(·,v)‖C([τ,τ+δ];X) ‖ϕ‖L1([0,t−τ])

≤ r/4 + (Lr/2 +N (τ))‖ϕ‖L1([0,δ]) .

We choose δ2 ∈ (0,δ1] such that for all δ ≤ δ2 we have the estimate (Lr/2 +N (τ))‖ϕ‖L1([0,δ]) ≤ r/4, and hence we conclude the self-mapping property ofthe map K onto the set B(τ,δ, r).

66

4.1. Local Existence and Uniqueness of Mild Solutions and Maximal Solutions

Contraction. Let v, v ∈ B(τ,δ, r) be arbitrary. We show that for suffi-ciently small δ ∈ (0,1] the map K is a contraction. Since v(s), v(s) ∈ Br(uτ ) forall s ∈ [τ,τ + δ] we obtain that

‖Kv −Kv‖C([0,τ+δ];X) ≤ supt∈[0,τ+δ]

∫ t

τ‖R(t − s) [F(s,v(s))−F(s, v(s))]‖X ds,

≤ L‖v − v‖C([0,τ+δ];X) ‖ϕ‖L1([0,δ]) .

The choice of δ2 implies in particular that for all δ ≤ δ2 that L‖ϕ‖L1([0,δ]) ≤ 1/2.Hence, we obtain for all δ ≤ δ2 the contraction property of the map K .

Altogether, we have for δ ≤minδ2,1 that K is a contractive self-mappingof the set B(τ,δ, r); of course δ might depend on τ . This set is not empty and itbecomes a complete metric space when endowed with the sup-norm. Hence,the contraction mapping principle yields a unique fixed point of K in the setB(τ,δ, r). This fixed point u ∈ C([0, τ + δ];X) has the property that u = u on[0, τ], and it is a mild solution of our semilinear Volterra equation (4.1) on[0, τ + δ].

Additionally, we introduce the term of Lipschitz continuity on boundedsets.

Definition 4.1.5. A map F : R+ × X → X is called Lipschitz continuous onbounded sets in the second component, uniformly on bounded intervals ofR+, if for each T > 0 and C > 0 there is some L = L(C,T ) such that for allx,y ∈ BC(0) and all t ∈ [0,T ] we have the Lipschitz estimate∥∥∥F(t,x)−F(t,y)

∥∥∥X≤ L

∥∥∥x − y∥∥∥X.

Finally, we can define a maximal mild solution for the semilinear Volterraequation (4.1).

Theorem 4.1.6. Let F : R+ ×X→ X be continuous and locally Lipschitz continu-ous in the second component, uniformly on bounded intervals of R+. Suppose thatthere is a resolvent S(t)t∈R+

and an integral resolvent R(t)t∈R+for (4.1).

(i) Then for each u0 ∈ X there is a τ(u0) ≤∞, such that the Volterra equation(4.1) has a unique mild solution on [0, τ(u0)) which cannot be extendedfurther. Moreover, if τ(u0) <∞ then limt→τ(u0)−u(t) does not exists in X.

Assume additionally that F is Lipschitz continuous on bounded sets in the secondcomponent, uniformly on bounded intervals of R+.

(ii) If τ(u0) <∞ then limt→τ(u0)− ‖u(t)‖X =∞.

67


Remark 4.1.7. We call the interval [0, τ(u0)) maximal interval of existence, andthe corresponding mild solution u on [0, τ(u0)) the maximal mild solution of(4.1) with u0 ∈ X. Moreover, we have

τ(u0) = supT > 0: u ∈ C([0,T ];X) is solution of (4.1).

Proof. (i) Let u0 ∈ X be arbitrary, but fixed. The existence of a maximalmild solution on the maximal interval of existence [0, τ(u0)) follows im-mediately from Lemma 4.1.4 about the continuation of local solutions.

In case of τ(u0) <∞ suppose, to the contrary, that limt→τ(u0)−u(t) = u1exists in X. Thus, u can continuously extended to u ∈ C([0, τ(u0)];X)with u = u on [0, τ(u0)) and u(τ(u0)) = u1. Using Lemma 4.1.4 there issome sufficiently small δ > 0 such that we can extend u to the interval[0, τ(u0) + δ]. Then u ∈ C([0, τ(u0) + δ];X) is an extension of u. Thiscontradicts the definition of τ(u0). Hence, the limit limt→τ(u0)−u(t)cannot exists in X.

(ii) This proof is completely analogous to the classical case ofC0-semigroupspresented by Pazy, see [Paz92, Theorem 6.1.4].

Let [0, τ(u0)) be the maximal interval of the mild solution of (4.1). Ifτ(u0) < ∞ then limt→τ(u0) ‖u(t)‖X = ∞. Since otherwise there is a se-quence (tn)n∈N with tn→ τ(u0) such that ‖u(tn)‖X ≤ r/2, for some r > 0and for all n ∈N.

Let n ∈N and consider the proof of Lemma 4.1.4 where we replace uτby u(tn). Since S(·)u0 is uniformly continuous on [0, τ(u0) + 1] we canchose δ1 independent of tn. The Lipschitz continuity on the boundedset Br(0) with L = L(r,τ(u0)) > 0 implies for all t ∈ [0, τ(u0) + 1] that

‖F(t,u(tn))‖X ≤ ‖F(t,u(tn))−F(t,0)‖X + ‖F(t,0)‖X≤ L‖u(tn)‖X + ‖F(t,0)‖X ,

Thus, we have with the uniform constant N0 = ‖F(·,0)‖C([0,τ(u0)+1];X) that

‖F(·,u(tn)‖C([0,τ(u0)+1];X) ≤N,

whereN = Lr/2+N0 is independent of tn. Thus, analogous to the proof ofLemma 4.1.4, we can choose δ2 ∈ (0,δ1] such that (Lr/2 +N )‖ϕ‖L1([0,δ]) ≤r/4 independent of tn. Hence, we can extend the solution u to [tn, tn + δ],with δ > 0 independent of tn. (By uniqueness, on [tn, τ(u0)) this is ofcourse the solution we already have.)

This would imply by the previous considerations that for each tn, closeenough to τ(u0), u defined on [0, tn] can be extended to [0, tn + δ] whereδ > 0 is independent of tn and hence u can be extended beyond τ(u0)contradicting the definition of τ(u0).

68

4.2. Stability Theorem

4.2 Stability Theorem

In this section we study semilinear Volterra equations

u(t) + (a ∗Au)(t) = u0 + (a ∗ f (u))(t), t ∈ [0,T ], (4.2)

where u0 ∈ X and f ∈ C1(X;X) with f (0) = 0 and f ′(0) = 0.1 At first, weintroduce the notion of stability of the equilibrium u∗ = 0. This concept issimilar to the well-known definitions for classical evolution equations.

Definition 4.2.1 (Stability). (i) We call the equilibrium u∗ = 0 stable forthe Volterra equation (4.2) if for each ε > 0 there is some δ = δ(ε) > 0such that for all u0 ∈ Bδ(0) a mild solution u = u(·;u0) of the Volterraequation (4.2) exists on R+ and ‖u‖C(R+;X) < ε.

(ii) We call the equilibrium u∗ = 0 asymptotically stable for the Volterraequation (4.2) if u∗ = 0 is stable and there is some δ0 > 0 such that for allu0 ∈ Bδ0

(0) the mild solution u = u(·;u0) of the Volterra equation (4.2)satisfies ‖u(t)‖X → 0 as t→∞.

(iii) We call the equilibrium u∗ = 0 asymptotically stable with rate ω if u∗ isasymptotically stable and there is a constant C > 0 such that for eachu0 ∈ Bδ0

(0) we have for the mild solution u(·;u0) the estimate

‖u(t;u0)‖X ≤ C ‖u0‖Xω(t), t ∈ (0,∞).

where ω : (0,∞)→R+ is non-negative with ω(t)→ 0 as t→∞.

(iv) We call the equilibrium u∗ = 0 unstable for the Volterra equation (4.2) ifu∗ is not stable.

Theorem 4.2.2 (Stability Theorem). Let X be a complex Banach space andA be a linear, closed and densely defined operator in X. Furthermore, let a ∈L1,loc(R+) be an unbounded, completely monotonic kernel and b ∈ L1,loc(R+) bethe corresponding Sonine kernel. We assume, that the linear Volterra equation

u(t) + (a ∗Au)(t) = g(t), t ∈ [0,T ],

with g ∈ C([0,T ];X), admits a bounded resolvent S(t)t∈R+and an integrable

integral resolvent R(t)t∈R+. Suppose that f ∈ C1(X;X) with f (0) = 0 , f ′(0) = 0.

Then the equilibrium u∗ = 0 is stable for the semilinear Volterra equation

u(t) + (a ∗Au)(t) = u0 + (a ∗ f (u))(t), t ∈ [0,T ],

with u0 ∈ X. Moreover, if a < L1(R+) the equilibrium u∗ = 0 is asymptoticallystable with rate sε(t) for some sufficiently small ε > 0.

1Note that unlike the previous section we drop the explicit time-dependence of the functionf , the results about local solutions etc. remain true.

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Here, sε denotes as usual the scalar resolvent, i. e. the solution of the scalarVolterra equation

sε(t) + ε(sε ∗ a)(t) = 1, t ∈R+.

Proof. Since the resolvent family is bounded, there is a constant M0 ≥ 1 suchthat ‖S(t)‖B(X) ≤ M0. The assumption of an integrable integral resolventyields a ψ ∈ L1(R+) such that ‖R(t)‖B(X) ≤ ψ(t) for almost all t ∈ R+ and‖ψ‖L1(R+) =N0 <∞.

Since f is continuously differentiable on X we know that f is locallyLipschitz continuous. Thus, there is some Br(0) such that f |Br (0) is Lipschitzcontinuous with constant L > 0. Moreover, the continuous differentiabilityof the map f , together with the assumption f (0) = 0 and f ′(0) = 0, impliesthat for each ρ > 0 there is some η = η(ρ) ∈ (0, r/2] such that for all x ∈ X with‖x‖X ≤ η it follows that ‖f (x)‖X ≤ ρ ‖x‖X . Fix ρ ∈ (0,1/(4N0)] and η = η(ρ) ∈(0, r/2] according to the previously described property. We know by the LocalExistence and Uniqueness Theorem 4.1.2 that for all u0 ∈ Br(0) there is someT = T (u0) > 0 such there is a unique mild solution u = u(·;u0) ∈ C([0,T ];X).This solution with initial value u0 ∈ Br(0) can be extended to a maximalinterval of existence [0, t∗(u0)). If t∗(u0) is finite, then the limit limt→t∗(u0)−u(t)does not exist in X, see Theorem 4.1.6.

Now, let u0 ∈ Bη(0) be arbitrary, but fixed, and let u = u(·;u0) denote thecorresponding mild solution with maximal interval of existence [0, t∗). Wedefine the exit time for the ball Bη(0), that is

t0 = supt ∈ (0, t∗) : ‖u(τ)‖X ≤ η for all τ ∈ [0, t],

and suppose t0 < t∗. We know that u ∈ C([0, t0];X) is the unique local mild solu-tion of (4.2), hence u(t) = S(t)u0+(R∗f (u))(t) for t ∈ [0, t0], and ‖u‖C([0,t0];X) ≤ ηby definition of the exit time.

On [0, t0] we consider for ε ∈ (0,1/(4N0)] the function v = ϕεu, withϕε = 1/sε, where sε denotes the scalar resolvent. Our aim is to prove that v isuniformly bounded.

Lemma 3.4.7 gives us the representation for v(t) = ϕε(t)u(t) on [0, t0] by

v(t) =S(t)u0 + (R ∗ [b[ϕε − 1]]) (t)u0 + (R ∗ [ϕεf (v/ϕε)]) (t)

+(R ∗

∫ ·0

[−b(· − τ)

][ϕε(·)−ϕε(τ)])

v(τ)ϕε(τ)

dτ

)(t).

In addition, we know that v ∈ C([0, t0];X) and ‖v‖C([0,t0];X) ≤ ϕε(t0)η, since ϕεis continuous and increasing on R+.

Now, let t ∈ [0, t0] be arbitrary. Recall that ϕε ≥ 1, thus we have theestimate

‖v(t)‖X ≤‖S(t)‖B(X) ‖u0‖X + (ψ ∗ b[ϕε − 1]) (t)‖u0‖X

70


+(ψ ∗ϕε ‖f (v/ϕε)‖X

)(t)

+(ψ ∗

∫ ·0

[−b(· − τ)

][ϕε(·)−ϕε(τ)]

‖v(τ)‖Xϕε(τ)

dτ

)(t).

The growth property of the map f yields on [0, t0] the uniform estimate

ϕε(t)‖f (v(t)/ϕε(t))‖X ≤ ϕε(t)ρ ‖v(t)/ϕε(t)‖X = ρ ‖v(t)‖X ,

since ‖v(t)/ϕε(t)‖X = ‖u(t)‖X ≤ η for all t ∈ [0, t0]. By the boundedness of theresolvent and the integrability of the integral resolvent it follows that

‖v(t)‖X ≤M0 ‖u0‖X + ε(ψ ∗ 1)(t)‖u0‖X + ρ(ψ ∗ ‖v‖X)(t)

+(ψ ∗

∫ ·0

[−b(· − τ)

][ϕε(·)−ϕε(τ)]

‖v(τ)‖Xϕε(τ)

dτ

)(t),

note that b(t)[ϕε(t)− 1] ∈ [0, ε] for all t ∈ (0,∞) by Remark 3.5.2. Clearly, wecan estimate ‖v(t)‖X uniformly by ‖v‖C([0,t0];X). Hence, we have

‖v(t)‖X ≤M0 ‖u0‖X + ε ‖u0‖X ‖ψ‖L1([0,t0]) + ρ ‖v‖C([0,t0];X) ‖ψ‖L1([0,t0])

+ ‖v‖C([0,t0];X)

(ψ ∗

∫ ·0

[−b(· − τ)

][ϕε(·)−ϕε(τ)]

dτϕε(τ)

)(t).

We are able to calculate the commutator on (0,∞), see Lemma 3.5.1. We have∫ t

0

[−b(t − τ)

][ϕε(t)−ϕε(τ)]

dτϕε(τ)

= ε+ b(t)[1−ϕε(t)] ∈ [0, ε],

and hence we have for all t ∈ [0, t0] that(ψ ∗

∫ ·0

(−b(· − τ))[ϕε(·)−ϕε(τ)]dτϕε(τ)

)(t) ≤ ε‖ψ‖L1([0,t0]).

By the integrability of ψ on R+ we obtain the uniform estimate

‖v(t)‖X ≤ (M0 + εN0)‖u0‖X + (ρ+ ε)N0 ‖v‖C([0,t0];X) ,

and thus

‖v‖C([0,t0];X) ≤ (M0 + εN0)‖u0‖X + (ρ+ ε)N0 ‖v‖C([0,t0];X) .

With the restriction ρ ∈ (0,1/(4N0)] and ε ∈ (0,1/(4N0)] it follows that

‖v‖C([0,t0];X) ≤ C0 ‖u0‖X ,

with C0 = 2M0 + 1/2. We have for all initial values u0 ∈ Bδ(0) with δ ≤ δ1 =η/(2C0) that ‖v‖C([0,t0];X) ≤ η/2. In particular, we have ‖v(t0)‖X ≤ η/2. This

71


contradicts the definition of the exit time t0 for the ball Bη(0). Thus, weconclude t0 = t∗. The above arguments yield for each δ ≤ δ1 and all u0 ∈ Bδ(0)the uniform estimate ‖v‖C([0,T ];X) ≤ η/2 for all T < t∗. This implies t∗ =∞, herewe use that ‖u(t)‖X ≤ ‖v(t)‖X < r for all t ∈ [0, t∗) and that we have a uniformLipschitz estimate for the map f on Br(0). Thus, we can extend the mildsolution to R+, see the proof of Theorem 4.1.6 (ii), which in turn, togetherwith the previous estimates for v gives ‖v‖C(R+;X) ≤ C0 ‖u0‖X . Rephrasing thisinequality in term of u, we obtain

‖u(t)‖X ≤ C0sε(t)‖u0‖X ≤ C0 ‖u0‖X ,

for all t ∈R+. In particular, the equilibrium u∗ = 0 is stable. Indeed, let ε0 > 0be arbitrary, but fixed. With δ <minη/(2C0), ε0/C0, we have ‖u‖C(R+;X) < ε0for all u0 ∈ Bδ(0). Moreover, for the completely monotonic kernel a ∈ L1,loc(R+)we have the estimate 0 ≤ sε(t) ≤ (1+ε(1∗a)(t))−1, see Lemma 2.2.5. Hence, sε(t)tends to zero as t→∞, if a < L1(R+). This implies the asymptotic stability ofthe equilibrium u∗ = 0 with the rate sε(t) in case of a < L1(R+).

Remark 4.2.3. One can weaken the assumption on the map f , it suffices toknow that f is continuously differentiable in a small neighbourhood of zerowith f (0) = 0 and f ′(0) = 0.

Using the sufficient conditions for the existence of a bounded resolventand an integrable integral resolvent, see Section 3.3, the stability result readsas follows.

Theorem 4.2.4. Let a ∈ L1,loc(R+) be an unbounded, completely monotonic kernelwhich is θa-sectorial with some θa ∈ (0, π2 ]. We denote by b ∈ L1,loc(R+) the

corresponding Sonine kernel. Assume that∫ 1

0 a(1/t)dtt <∞,

∫ 10 b(1/t)dtt <∞ and[

t 7→mina(1/t)/t, b(1/t)/t2]∈ L1(R+).

Moreover, let A be an invertible and sectorial operator with spectral angleϕA < π −θa and let f ∈ C1(X;X) such that f (0) = 0 and f ′(0) = 0.

Then the equilibrium u∗ = 0 is stable for the Volterra equation (4.2). If inaddition a < L1(R+) then u∗ = 0 is asymptotically stable with rate sε(t) for somesufficiently small ε > 0.

Proof. The assumptions on the operator A together with the θa-sectoriality ofthe kernel imply that the Volterra equation (4.2) is parabolic. In particular,completely monotonic kernels are 1-regular, hence, we know by Theorem3.3.1 that there is a bounded resolvent S(t)t∈R+

for the Volterra equation(4.2). The integrability requirements on the kernels a and b as well as theassumptions on the operator A ensure the existence of an integrable integralresolvent, see Theorem 3.3.8. Hence, all requirements for Theorem 4.2.2 aresatisfied. It follows the claim.

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It is easy to see that for a Volterra equation with standard kernel and aproper sectorial operator A the equilibrium u∗ = 0 is asymptotically stable.

Corollary 4.2.5. Let α ∈ (0,1) and suppose that the invertible operator A issectorial on the Banach space X with spectral angle ϕA < π −απ2 . Moreover, letf ∈ C1(X;X) with f (0) = 0, f ′(0) = 0.

Then, the equilibrium u∗ = 0 is asymptotically stable for the Volterra equation

u(t) + (gα ∗Au)(t) = u0 + (gα ∗ f (u))(t), t ∈ [0,T ],

where u0 ∈ X; the decay rate is given by t−α.

Proof. We know that the operator A together with the standard kernel gα <L1(R+) satisfy all assumptions from Theorem 4.2.4. In case of the standardkernel we know that sε decays to zero with optimal rate t−α. The claimfollows.

The following corollary is a further easy consequence of Theorem 4.2.2for the finite dimensional case with the standard kernel.

Corollary 4.2.6. Let α ∈ (0,1) and assume that the matrix A ∈Rn×n is invertibleand that every eigenvalue λ ∈ σ (A) satisfies |argλ| < π − απ2 . Moreover, letf ∈ C1(Rn;Rn) with f (0) = 0, f ′(0) = 0.

Then, the equilibrium u∗ = 0 is asymptotically stable for the Volterra equation


where u0 ∈Rn, and the decay rate is given by t−α.

Finally, we use Theorem 4.2.2 to deduce the following further result aboutthe decay behaviour of the resolvent family.

Corollary 4.2.7. Assume the situation of Theorem 4.2.2 with f ≡ 0.Then there is a constant C > 0 such that we have for the resolvent family

S(t)t∈R+the estimate

‖S(t)‖B(X) ≤ Csε(t), t ∈R+,

where sε denotes the scalar resolvent for some sufficiently small ε > 0.Consequently, the decay behaviour of the resolvent family S(t)t∈R+

is the sameas the decay behaviour of the scalar resolvent sε.

Remark 4.2.8. We consider the situation of exponentially weighted standardkernels with

a(λ) =1 +γ/λ(λ+γ)α

, b(λ) =1

(λ+γ)1−α ,

73


where γ ≥ 0 and α ∈ (0,1). Easy calculations show that these completely

monotonic kernels satisfy the assumptions∫ 1

0 a(1/t)dt <∞,∫ 1

0 b(1/t) dt2 <∞ and∫∞1 b(1/t)dt <∞. Thus, we know that the Volterra equation with exponentially

weighted standard kernel and an invertible sectorial operator A with ϕA <π2 admits a bounded resolvent family S(t)t∈R+

and an integrable integralresolvent R(t)t∈R+

. Corollary 4.2.7 implies that we have for the resolventfamily S(t)t∈R+

the estimate

‖S(t)‖B(X) ≤ Csε(t), t ∈R+,

where C > 0 is a constant and sε denotes the scalar resolvent for a sufficientlysmall ε > 0. Vergara and Zacher proved in [VZ17] that in the case of exponen-tially weighted standard kernels the scalar resolvent decays to zero as t→∞with an appropriate exponential rate. Hence, ‖S(·)‖B(X) decays to zero withthe same exponential rate as the scalar resolvent.

74

5Instability for Semilinear ParabolicVolterra EquationsThe aim of this chapter is to prove the instability result, Theorem 5.3.1, forsemilinear parabolic Volterra equations. In order to do this, firstly we intro-duce the spectral projection of an operator associated with a compact spectralset and show its compatibility with the resolvent and integral resolvent, seeSection 5.1. Afterwards, we consider exponential weights for the scalar re-solvents and integral resolvents and determine their long-time behaviourincluding estimates for the convergence rate, cf. Section 5.2. In doing so oneof the major challenges is the proof of the uniformity of the estimates alongcompact sets. These results are interesting in its own and crucial for the proofof our instability result Theorem 5.3.1. Finally, we close this chapter with aneasy consequence for Volterra equations with standard kernel.

5.1 Spectral Projections associated with Spectral Setsand Volterra Equations

5.1.1 Spectral Sets and Projections

In this section we introduce the spectral projection of a linear operator Aassociated with a compact spectral set. The subsequent result can be found inArendt et al. [ABHN01, Proposition B.9].

Proposition 5.1.1. Let A be closed, linear operator on the Banach space X withρ(A) , ∅. We suppose that σ (A) = σ+∪σ− with σ+∩σ− = ∅, where σ+ is a compactsubset and σ− a closed subset of C.

Then there is a bounded projection P+ ∈ B(X) on X such that (λ −A)−1P+ =P+(λ −A)−1, for all λ ∈ ρ(A), RgP+ ⊂ D(A), σ (A+) = σ+ and σ (A−) = σ−, whereA+ denotes the part of A in RgP+ and A− the part of A in Rg(Id−P+). Moreover,the projection P+ is unique and A|RgP+

∈ B(RgP+).

Remark 5.1.2. a) The projection P+ is called the spectral projection of theoperator A associated with the compact spectral set σ+.

b) We set P− = Id−P+. Obviously, P− ∈ B(X) is also a projection and com-mutes with the resolvent operator (λ−A)−1, λ ∈ ρ(A), too.

Moreover, A commutes with P± on D(A), cf. [ABHN01, Proposition B.7] .

c) Introducing X+ = RgP+ and X− = Rg(Id−P+), we know from Kato [Kat66,p. 155] that X± are closed linear manifolds of X and X = X+ ⊕X−.

75

5. Instability for Semilinear Parabolic Volterra Equations

We have A+ : X+ → X+ and A+x = Ax for all x ∈ X+ = D(A+) andA− : D(A−)→ X− and A−x = Ax for all x ∈ D(A−) = X− ∩D(A).

The operators A± are closed operators on X±, cf. [Kat66, p. 172].

For each λ ∈ ρ(A) we have the decomposition of the resolvent operator(λ − A)−1 = (λ − A+)−1 + (λ − A−)−1, where (λ − A+)−1 = P+(λ − A)−1 =(λ −A)−1P+, the part of (λ −A)1− in X+, and (λ −A−)−1 = P−(λ −A)−1 =(λ−A)−1P−, the part of (λ−A)−1 in X−, see [Kat66, p. 179].

d) According to Lunardi [Lun95, Appendix A.1] we know that there existsa bounded open set Ω ⊂ C containing σ+ and such that its closure isdisjoint from σ− (since the distance between σ− and σ+ is positive) andwe may assume that the boundary γ of Ω consists of a finite number ofrectifiable closed Jordan curves, oriented counterclockwise. Then thespectral projection P+ ∈ B(X) of the operator A associated to the compactspectral set σ+ is given by

P+ =1

2πi

∫γ

(ξ −A)−1dξ. (5.1)

5.1.2 Projections of Resolvents and Integral Resolvents

In this section we use the above introduced projections to decompose the re-solvent and integral resolvent of a linear Volterra equation into the respectiveobjects on the subspaces X±. Throughout this section we assume:

Assumptions. The operator A is a linear, closed and densely defined oper-ator on the Banach space X with domain D(A). Let the resolvent set of theoperator A be non-empty and assume that σ (A) = σ+∪σ− with σ+∩σ− = ∅ andσ+ is a compact and σ− is a closed subset of C. The spectral projection of Aassociated with σ+ is denoted by P+ and P− = Id−P+. We have X± = RgP± andA± denotes the part of A in X±. Furthermore, let a ∈ L1,loc(R+) be a locallyintegrable kernel. For T > 0 we set J = [0,T ].

We consider the linear Volterra equation

u(t) + (a ∗Au)(t) = f (t), t ∈ J, (5.2)

where f ∈ C(J ;X).

Lemma 5.1.3. Assume that the Volterra equation (5.2) admits a resolvent S(t)t∈R+.

Then the resolvent operator (λ−A)−1 ∈ B(X), λ ∈ ρ(A), as well as the spectralprojections P± commute with the resolvent S(t)t∈R+

.If the Volterra equation (5.2) admits an integral resolvent R(t)t∈R+

then thesame statement holds true for the integral resolvent R(t)t∈R+

.

76

5.1. Spectral Projections associated with Spectral Sets and Volterra Equations

Proof. Due to condition (S2)/(R3) we know that the resolvent S(t)t∈R+⊂ B(X)

and the integral resolvent R(t)t∈R+⊂ B(X) commute with the operator A on

D(A). Using [ABHN01, Proposition B.7] it follows directly that the resolventand integral resolvent family, respectively, commute with the resolvent opera-tor (λ−A)−1, λ ∈ ρ(A). Let t ∈R+ and x ∈ X be arbitrary. We set T = S(t) andT = R(t), respectively. We have with equation (5.1) that

P+T x =1

2πi

∫γ

(λ−A)−1T xdλ

=1

2πi

∫γT (λ−A)−1xdλ,

here we used the fact that both the resolvent S(t)t∈R+and the integral resol-

vent R(t)t∈R+commute with the resolvent operator. Since λ 7→ (λ−A)−1x is

Bochner integrable along γ , we have by [ABHN01, Proposition 1.1.6] that

P+T x =1

2πiT

∫γ

(λ−A)−1xdλ = T P+x.

In addition, we get P−x = T x − P+T x = T x − T P+x = T P−x.

Lemma 5.1.4. Assume that the Volterra equation (5.2) admit a resolvent S(t)t∈R+

and an integral resolvent R(t)t∈R+.

Then S±(t)t∈R+= P±S(t)t∈R+

is a resolvent and R±(t)t∈R+= P±R(t)t∈R+

is an integral resolvent for the Volterra equation

u(t) + (a ∗A±u)(t) = f±(t), t ∈ J, (5.3)

with f± ∈ C(J ;X±).Furthermore, if both Volterra equations (5.3) admits a resolvent S±(t)t∈R+

and an integral resolvent R±(t)t∈R+, then for all t ∈R+ we have S±(t) = S±(t) and

R±(t) = R±(t) on X±. Moreover, S(t)t∈R+= S+(t) + S−(t)t∈R+

is a resolvent andR(t)t∈R+

= R+(t) +R−(t)t∈R+is an integral resolvent for the Volterra equation

(5.2).

Proof. Let t ∈R+ be arbitrary, but fixed. Lemma 5.1.3 implies that S(t)X± ⊂ X±and R(t)X± ⊂ X±. Furthermore, we have ‖P±S(t)‖B(X±) ≤ ‖P±‖B(X) ‖S(t)‖B(X) <∞, as well as ‖P±R(t)‖B(X±) ≤ ‖P±‖B(X) ‖R(t)‖B(X) < ∞. Hence, S±(t)t∈R+

⊂B(X±) and R±(t)t∈R+

⊂ B(X±) are families of bounded operators on X±.In our next steps, we prove that S±(t)t∈R+

satisfies the properties (S1)−(S3) from Definition 3.1.3 of the resolvent and that R±(t)t∈R+

, satisfies theproperties (R1)− (R3) from Definition 3.1.8 of the integral resolvent.

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Condition (S1). We show that P±S(t) is strongly continuous on R+ andP±S(0) = Id on X±.

Let t ∈ R+ and x± ⊂ X± ⊂ X be arbitrary, but fixed. For each ε > 0 existsδ = δ(ε, t) > 0 such that for all t, s ∈R+ with |t − s| ≤ δ we have

‖P±S(s)x± − P±S(t)x±‖X ≤ ‖P±‖B(X) ‖S(s)x± − S(t)x±‖X ≤ ε,

because S(·)x is strongly continuous on R+ for all x ∈ X. Moreover, we haveP±S(0)x± = S(0)P±x± = S(0)x± = x±, since S(0) = Id on X. This implies P±S(0) =Id on X±. Thus, S±(t)t∈R+

satisfies (S1).

Condition (R1). We prove that P±R(·)x± ∈ L1,loc(R+;X±) for each x± ∈ X±and ‖P±R(t)‖B(X) ≤ ϕ±(t) a. e. t ∈R+ for some ϕ± ∈ L1,loc(R+).

Let x± ∈ X± ⊂ X be arbitrary, but fixed. The integral resolvent commuteswith P±, see Lemma 5.1.3, we have P±R(·)x± = R(·)x± : R+ → X± and thusP±R(·)x± ∈ L1,loc(R+;X±), since R(·)x is locally Bochner integrable for all x ∈ Xand X± are closed subspaces of X. Moreover, for a. e. t ∈R+ we have

‖P±R(t)‖B(X±) ≤ ‖P±‖B(X) ‖R(t)‖B(X) ≤ ϕ±(t),

withϕ± = ‖P±‖B(X)ϕ ∈ L1,loc(R+), sinceϕ ∈ L1,loc(R+) from the property (R1) ofthe definition of the integral resolvent R(t)t∈R+

. Hence, R±(t)t∈R+satisfies

condition (R1).

Condition (S2)/(R2). We show that for each t ∈ R+ commutes P±S(t)and P±R(t) with A± on D(A±).

Let t ∈ R+ and x± ∈ D(A±) be arbitrary, but fixed. We set T = S(t) orT = R(t), respectively. Since D(A±) ⊂ D(A), we know that T commutes withA on D(A±). The closed subspaces X± ⊂ X are invariant for T , i. e. T x± ⊂ X±.Moreover, the domain D(A) is invariant for T , i. e, T x± ∈ D(A). In particular,we have T x± ∈ D(A) ∩X± = D(A±). We know that the operators A and A±coincide on D(A±), by definition. Hence, we have P±TA±x± = P±TAx± =P±AT x±. The operators P± commute with the operator A on D(A), so we getP±AT x± = AP±T x±. Since we have T x± ∈ D(A±) and we know that A andA± coincide on D(A±), we obtain P±TA±x± = A±P±T x±. Thus, S±(t)t∈R+

andR±(t)t∈R+

, respectively, satisfy condition (S2) and (R2), respectively.

Condition (S3). We prove that for each t ∈ R+ and all x± ∈ D(A±) theresolvent equation

S±(t)x± + (a ∗ [A±S±x±])(t) = x±,

is satisfied.

78


Let t ∈ R+ and x± ∈ D(A±) ⊂ D(A) be arbitrary, but fixed. We know thatS(t) satisfies the resolvent equation, so S(t)x± + (a ∗ [ASx±])(t) = x±. On onehand, we have P±S(τ)x± = S(τ)x± ∈ D(A±), τ ∈R+, see proof of condition (S2).On the other hand, we know that A coincide with A± on D(A±). Combiningthese facts, we get for all τ ∈ R+ that AS(τ)x± = A±P±S(τ)x±. In particular,the resolvent equation is equivalent to P±S(t)x± + (a ∗ [A±P±Sx±])(t) = x±, forx± ∈ D(A±). This implies, that S±(t)t∈R+

satisfies condition (S3).

Condition (R3). We show that for almost all t ∈ R+ and all x± ∈ D(A±)the integral resolvent equation

R±(t)x± + (a ∗ [A±R±x±])(t) = a(t)x±,

is satisfied.Let x± ∈ D(A±) be arbitrary, but fixed. Let t ∈R+ be such that R(t) satisfies

the integral resolvent equation, i. e. R(t)x± + (a ∗ [ARx±])(t) = a(t)x±. Anal-ogous to the argumentation for condition (S2), we have for all τ ∈ R+ thatP±R(τ)x± = R(τ)x± ∈ D(A±) as well as AR(τ)x± = A±P±R(τ)x±. Thus, the inte-gral resolvent equation is equivalent to P±R(t)x± + (a ∗ [A±P±Rx±])(t) = a(t)x±,for x± ∈ D(A±). This proves that R±(t)t∈R+

satisfies condition (R3).

We can conclude, that S±(t)t∈R+and R±(t)t∈R+

are resolvent and integralresolvent families for the Volterra equations (5.3). The resolvent family aswell as the integral resolvent family are unique provided they exist. Thus, forall t ∈R+ we know that S±(t) and S±(t), as well as R±(t) and R±(t) coincide onD(A±) and by the density they coincide on X±.

Using the decomposition of the space X and of the operator A, which isintroduced by the spectral projection P+, together with the uniqueness of theresolvent and integral resolvent it follows directly that S(t) = S+(t) + S−(t) andR(t) = R+(t) +R−(t) are the desired objects.

As a direct consequence of Lemma 5.1.4 we obtain the following corollary.

Corollary 5.1.5. Assume that the Volterra equation (5.2) admits a resolventS(t)t∈R+

and an integral resolvent R(t)t∈R+. Suppose further that both Volterra

equations (5.3) admits a resolvent S±(t)t∈R+and an integral resolvent R±(t)t∈R+

.Then, the mild solution u(t) = S(t)u0 + (R ∗ f )(t), t ∈ J , of the Volterra equation

u(t) + (a ∗Au)(t) = u0 + (a ∗ f )(t), t ∈ J,

where u0 ∈ X and f ∈ C(J ;X), decomposes into its components on X± given by themild solutions of the Volterra equations

u±(t) + (a ∗A±u±)(t) = P±u0 + (a ∗ [P±f ])(t), t ∈ J,

79


that is

u±(t) = S±(t)u0 + (R± ∗ f )(t), t ∈ J,

and u(t) = P+u(t) + P−u(t) = u+(t) +u−(t), t ∈ J .

The next result shows the holomorphic dependence of the scalar resolventsµ and integral resolvent rµ on the parameter µ ∈C.

Lemma 5.1.6. Let a ∈ L1,loc(R+) be scalar kernel. Assume that Ω ⊂ C is a non-empty, open, connected and bounded set in C.

Then, for each T > 0 the maps r : Ω → L1([0,T ];C), µ 7→ rµ, as well ass : Ω→ L1([0,T ];C), µ 7→ sµ, are holomorphic on Ω.

Proof. Let T > 0 and µ ∈Ω be arbitrary, but fixed. We consider on [0,T ] thescalar integral resolvent equation

rµ(t) +µ(a ∗ rµ)(t) = a(t), t ∈ [0,T ].

We introduce for σ ∈ C the notation gσ (t) = g(t)e−σt, the multiplicationwith an exponential weight. We choose σ > 0 sufficiently large, such thatsupµ∈Ω‖µaσ ‖L1([0,T ];C) < 1. We multiply the Volterra equation for rµ with e−σt,this yields the equation

rσµ (t) + (rσµ ∗ [µaσ ])(t) = aσ (t), t ∈ [0,T ]. (5.4)

We define for every n ∈N and all t ∈ [0,T ] the expression

rσ,nµ (t) = aσ (t) +n∑k=1

(−1)k[(µaσ )∗k ∗ aσ

](t).

Here, for every k ∈ N we denote by (µaσ )∗k the (k − 1)-fold convolution ofµaσ by itself and (µaσ )∗1 = µaσ . Obviously, the expressions rσ,nµ ∈ L1([0,T ];C)are holomorphic in µ ∈Ω, for all n ∈N. Furthermore, for each fixed µ ∈Ωthe sequence rσ,nµ n∈N is a Cauchy sequence in L1([0,T ];C), since ‖(µaσ )∗k ∗aσ ‖L1([0,T ];C) ≤ ‖µaσ ‖kL1([0,T ];C)‖a

σ ‖L1([0,T ];C), k ∈ N, and ‖µaσ ‖L1([0,T ];C) < 1 bythe choice of σ > 0. The space L1([0,T ];C) is complete, so we know thatthere is a rσµ ∈ L1([0,T ];C) such that rσ,nµ → rσµ in L1([0,T ];C) as n→∞, andrσµ satisfies our equation (5.4). These arguments are completely analogousto the arguments which Gripenberg, Londen and Staffans give in the proofof [GLS90, Theorem 2.3.1].

Now, we want to apply Vitali’s Theorem 1.2.7 to show that rσµ is holo-morphic on Ω. Obviously, we have that Ω = µ ∈Ω : limn→∞ r

σ,nµ exists , ∅.

Furthermore, we have for each ball Br(z0) ⊂Ω that

80


supn∈N,µ∈Br (z0)

∥∥∥rσ,nµ ∥∥∥L1([0,T ];C)

≤ supn∈N,µ∈Br (z0)

‖aσ ‖L1([0,T ];C)

n∑j=0

∥∥∥µaσ∥∥∥jL1([0,T ];C)

≤

‖aσ ‖L1([0,T ];C)

1−∥∥∥µaσ∥∥∥

L1([0,T ];C)

<∞.

Via Vitali’s Theorem 1.2.7 we know that there is a holomorphic functionrσ· : Ω→ L1([0,T ];C) such that rσµ = limn→∞ r

σ,nµ in L1([0,T ];C) uniformly on

all compact subsets of Ω.

Hence, r : Ω → L1([0,T ];C), µ 7→ rµ, is holomorphic on Ω. Since theresolvent sµ is given by sµ(t) = 1 − µ(rµ ∗ 1)(t), t ∈ R+, see (2.2), it is obviousthat s : Ω→ L1([0,T ];C), µ 7→ sµ, is holomorphic on Ω, too.

Corollary 5.1.7. We consider on X+ the Volterra equation

u(t) + (a ∗A+u)(t) = f (t), t ∈ J, (5.5)

where A+ ∈ B(X+) with σ (A+) = σ+.

Then, the resolvent S+(t)t∈R+and the integral resolvent R+(t)t∈R+

of theVolterra equation (5.5) are given by Dunford’s integral representation

S+(t) =1

2πi

∫γsλ(t)(λ−A+)−1dλ, t ∈R+,

R+(t) =1

2πi

∫γrλ(t)(λ−A+)−1dλ, t ∈R+,

where γ is an appropriately chosen path in the complex plane C around σ+, cf.Remark 5.1.2 d).

Proof. Since σ+ is compact in C, there is some non-empty, open, connectedand bounded set Ω ⊂C which contains σ+ ⊂Ω and γ denotes the boundaryof Ω, cf. Remark 5.1.2 d). Note that in this situation it is possible to choose γsufficiently smooth. Moreover, the operator A+ ∈ B(X+) is a bounded operatoron X+, so Dunford’s integral representation, cf. [Yos95, Section VIII.7], yieldsthe claim.

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5.2 Exponentially Weighted Scalar Resolvents andIntegral Resolvents


Assumptions. Let a ∈ L1,loc(R+) \ L1(R+) be an unbounded, completelymonotonic kernel and by b ∈ L1,loc(R+) we denote the corresponding com-pletely monotonic kernel with a ∗ b ≡ 1 on (0,∞). Furthermore, let the kernela be θa-sectorial with θa ∈ (0,π/2].

First, we recall that for arbitrary µ ∈C the scalar resolvent sµ ∈ L1,loc(R+;C)and integral resolvent rµ ∈ L1,loc(R+;C) are the unique solutions of the follow-ing linear Volterra equations on R+

sµ(t) +µ(sµ ∗ a)(t) = 1,

rµ(t) +µ(rµ ∗ a)(t) = a(t).

In the following we use for ω ∈ C the notation kω(t) = k(t)e−ωt, the multi-plication with an exponential weight.

From Lemma 2.3.8 we know that the Laplace transform a of the completelymonotonic kernel a ∈ L1,loc(R+) is injective. Hence, the Laplace transform amaps the right half-plane C+ bijectively to its image set a(C+) ⊂ Σθa .

1 Thisimplies that for all −1/µ ∈ a(C+) ⊂ Σθa there is a unique ρ ∈ C+ such thata(ρ) = −1/µ, i. e. ρ = a−1(−1/µ). Note that the condition −1/µ ∈ a(C+) isequivalent to the condition µ ∈ −1/a(C+), since a has no zeros in Σπ.

For each −1/µ ∈ a(C+) ⊂ Σθa we multiply the Volterra equation for sµ andrµ with the exponential weight e−ρt, where ρ is chosen such that a(ρ) = −1/µ.We deduce for all t ∈R+ that

sρµ(t) +µ(s

ρµ ∗ aρ)(t) = 1ρ(t),

rρµ (t) +µ(r

ρµ ∗ aρ)(t) = aρ(t).

Since µ = −1/a(ρ), we may simplify the notation by writing sρ ≡ sρµ and rρ ≡ rρµ .Thus, we have on R+ the equations

sρ(t)− (sρ ∗ aρ)(t)a(ρ)

= 1ρ(t),

1We denote by a(C+) the image set of the mapping a of the complex right half-plane C+ , i. e.a(C+) = λ ∈C : there is some z ∈C+ such that λ = a(z). Since a is a non-trivial holomorphicfunction a(C+) is an open, non-empty, connected set in C. The assumption, that the kernel a isθa-sectorial, implies that a(C) ⊂ Σθa . Since a has no zeros in Σπ, cf. Remark 2.3.17 a), the set−1/a(C+) = λ ∈C : there is some z ∈C+ such that µ = −1/a(z) ⊂ −Σθa is also well-defined.

82

5.2. Exponentially Weighted Scalar Resolvents and Integral Resolvents

rρ(t)− (rρ ∗ aρ)(t)a(ρ)

= aρ(t).

Evidently, we have aρ,1ρ ∈ L1(R+;C), therefore it is a direct consequence of thePaley-Wiener Theorem on the Half-Line [GLS90, Theorem 4.4.1], that sρ, rρ ∈L1,loc(R+;C) are of subexponential growth. Thus, the Laplace transform of sρ

and rρ exists for all λ ∈C+ given by

sρ(λ) =1

λ+ ρ· 1

1− aρ(λ)a(ρ)

=1

λ+ ρ· 1

1− a(λ+ρ)a(ρ)

,

rρ(λ) = aρ(λ) · 1

1− aρ(λ)a(ρ)

= a(λ+ ρ) · 1

1− a(λ+ρ)a(ρ)

.

For each ρ ∈ C+ there is some δ0 = δ0(ρ) ∈ (0, π2 ) such that ρ ∈ Σ π2 −δ0

. Using

Lemma 2.3.9 we know that for all λ ∈Σ π

2 + δ02− ρ

\ 0 we have a(λ+ρ) , a(ρ).

Moreover, the Laplace transform a possesses a holomorphic extension to Σπ,since a is completely monotonic, cf. Theorem 2.3.7. Due to the structure ofsρ(λ) and rρ(λ) we can conclude that both expressions have a holomorphicextension at least to the set Σ π

2 + δ02− ρ \ 0.

5.2.1 Long-Time Behaviour

We are interested in the long-time behaviour of sρ and rρ. We want to examinetheir behaviour via the Tauberian Theorem 1.2.9. By L’Hospital’s rule we have

limλ→0+

λ

1− a(λ+ρ)a(ρ)

= limλ→0+

1

− a′(λ+ρ)a(ρ)

= −a(ρ)a′(ρ)

,

and hence

limλ→0+

λsρ(λ) = −1ρ·a(ρ)a′(ρ)

=: s?(ρ),

limλ→0+

λrρ(λ) = −a2(ρ)a′(ρ)

=: r?(ρ).

We point out that the kernel a ∈ L1,loc(R+) is completely monotonic, so neithera nor a′ has zeros in C+, see Remark 2.3.17. Thus, we have for all ρ ∈C+ that|s?(ρ)|, |r?(ρ)| <∞, as well as s?(ρ) , 0 and r?(ρ) , 0. In other words, λ = 0 is apole of first order of sρ and rρ. This implies, together with the holomorphicextension properties of a on Σπ, that for fixed ρ ∈ Σ π

2 −δ0the both expressions

sρ(λ)− s?(ρ)/λ and rρ(λ)− r?(ρ)/λ have a holomorphic extension at least to theshifted sector Σ π

2 + δ02− ρ.

To prove the convergence sρ(t)→ s?(ρ) and rρ(t)→ r?(ρ) as t→∞, we usethe following auxiliary result.

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Proposition 5.2.1. For δ0 ∈ (0, π2 ) we consider some ρ ∈ Σπ/2−δ0. We set α0 =

δ0/2. Then we have for all λ ∈ Σπ/2+α0the following estimates.

(i) There are constants Cα0, cα0

> 0 such that cα0a(|λ|) ≤ |a(λ)| ≤ Cα0

a(|λ|).

(ii) There is some constant C = C(δ0) > 0 such that |λ+ ρ| ≥ C |λ|.

(iii) We have the estimate |a(λ+ ρ)| ≤ Cα0a(C |λ|), where C > 0 is the constant

from part (ii).

(iv) There is some ξ0 = ξ0(δ0, |ρ|) > 0 such that for all |λ| ≥ ξ0 we have theestimate ∣∣∣∣∣1− a(λ+ ρ)

a(ρ)

∣∣∣∣∣−1

≤ 2.

(v) For each C > 0 the function t 7→ ta(Ct) is non-decreasing on (0,∞).

Proof. (i) The angle α0 is chosen smaller that π/4, hence the claim followsby Theorem 2.3.12.

(ii) The choice of the angle α0 ensures that we have π/2 +α0 + (π/2− δ0) =π − δ0/2 < π. Corollary 2.3.11 implies that there is some constant C =C(δ0) > 0 such that for ρ ∈ Σπ/2−δ0

and all λ ∈ Σπ/2+α0we have |λ+ ρ| ≥

C (|λ|+ |ρ|). The claim follows immediately.

(iii) We know that a is decreasing on (0,∞), see Remark 2.3.6. Together withthe estimate in part (ii) we obtain for all λ ∈ Σπ/2+ε0

that |a(λ + ρ)| ≤Cα0

a(|λ+ ρ|) ≤ Cα0a(C |λ|), where C > 0 is the constant from part (ii).

(iv) On the whole we have the estimate∣∣∣a(λ+ ρ)∣∣∣∣∣∣a(ρ)

∣∣∣ ≤Cα0

a (|λ+ ρ|)cα0a(|ρ|)

≤Cα0

a(C |λ|)cα0a(|ρ|)

.

Due to the assumption a ∈ L1,loc(R+) \L1(R+) we deduce for the continu-ous and decreasing function a : (0,∞)→R+ that a is bijective on (0,∞),cf. Remark 2.3.17 b).

Now, we choose ξ0 = ξ0(δ0, |ρ|) > 0 such that

ξ0 =1Ca−1

(12cα0a(|ρ|)Cα0

).

Hence, we have for all λ ∈ Σπ/2+α0with |λ| ≥ ξ0 the estimate∣∣∣∣∣ a(λ+ ρ)

a(ρ)

∣∣∣∣∣ ≤ 12,

84


and consequently ∣∣∣∣∣1− a(λ+ ρ)a(ρ)

∣∣∣∣∣−1

≤ 2.

(v) The Sonine condition implies for all t ∈ (0,∞) that ta(t) = 1/b(t). Sincethe corresponding Sonine kernel b ∈ L1,loc(R+) is completely monotonic,its Laplace transform b is decreasing on (0,∞), cf. Remark 2.3.6. Thus, itis obvious that t 7→ ta(Ct) is non-decreasing on (0,∞), for each constantC > 0.

Now, we can prove the following convergence theorem.

Lemma 5.2.2. Assume additionally that∫ 1

0 a(1/t)dtt <∞.

Then we have for each ρ ∈C+ that

(i) sρ(t)→ s?(ρ) = −1ρ ·

a(ρ)a′(ρ) , as t→∞,

(ii) rρ(t)→ r?(ρ) = − a2(ρ)a′(ρ) , as t→∞.

Proof. Let ρ ∈ C+ be arbitrary, but fixed. Then there is some δ0 = δ0(ρ) ∈(0,π/2) such that ρ ∈ Σπ/2−δ0

. We set α0 = δ0/2 < π/4. Obviously, we haveΣπ/2+α0

⊂ Σπ/2+α0− ρ \ 0. Thus, sρ and rρ are holomorphic on the sector

Σπ/2+α0. We want to apply the Tauberian Theorem 1.2.9. In particular, we

show that for all λ ∈ Σπ/2+α0there are constants Ms, Mr ,C > 0 with∣∣∣sρ(λ)∣∣∣ ≤Ms/ |λ| ,∣∣∣rρ(λ)∣∣∣ ≤Mr [1/ |λ|+ a(C |λ|)] .

(i) Firstly, we consider for λ ∈ Σπ/2+α0the expression

∣∣∣sρ(λ)∣∣∣ =

∣∣∣∣∣∣∣∣ 1λ+ ρ

· 1

1− a(λ+ρ)a(ρ)

∣∣∣∣∣∣∣∣ .By Proposition 5.2.1 (ii) there is some constant C = C(δ0) > 0 such thatfor all λ ∈ Σπ/2+α0

we have∣∣∣λ/(λ+ ρ)

∣∣∣ ≤ 1/C(δ0). Together with theestimate from Proposition 5.2.1 (iv), we obtain for all λ ∈ Σπ/2+α0

with|λ| ≥ ξ0 that

∣∣∣λsρ(λ)∣∣∣ =

∣∣∣∣∣∣∣∣ λλ+ ρ

· 1

1− a(λ+ρ)a(ρ)

∣∣∣∣∣∣∣∣ ≤ 2C(δ0)

.

85


Due to the previous considerations we know λsρ(λ)→ s?(ρ) for λ→ 0and sρ(λ) is holomorphic on Σπ/2+α0

−ρ \ 0. Thus, we conclude λsρ(λ)is holomorphic on Σπ/2+α0

− ρ. The set Σπ/2+α0∩Bξ0

(0) is a compact setin Σπ/2+α0

− ρ. Hence

supλ∈Σπ/2+α0∩Bξ0 (0)

∣∣∣λsρ(λ)∣∣∣ ≤M1,

with some constant M1 =M1(α0,ρ,ξ0) =M1(δ0, |ρ|). Thus, we obtain forall λ ∈ Σπ/2+α0

that∣∣∣λsρ(λ)∣∣∣ ≤maxM1,2/C(δ0) =Ms(δ0, |ρ|) <∞.

It is easy to see that g(s) = 1/s, s ∈ (0,∞) satisfies the condition of theTauberian Theorem 1.2.9. The claim follows.

(ii) Now, we estimate for λ ∈ Σπ/2+α0the expression:

∣∣∣rρ(λ)∣∣∣ =

∣∣∣∣∣∣∣∣a(λ+ ρ) · 1

1− a(λ+ρ)a(ρ)

∣∣∣∣∣∣∣∣Just like above, we have with Proposition 5.2.1 for all λ ∈ Σπ/2+α0

with|λ| ≥ ξ0 that ∣∣∣rρ(λ)

∣∣∣ ≤ 2Cα0a(C |λ|),

where C = C(δ0) > 0 is the constant from Proposition 5.2.1 (ii). We knowfrom the previous considerations that limλ→0λrρ(λ) = r?(ρ) and rρ(λ)is holomorphic on Σπ/2+α0

− ρ \ 0. Thus, we conclude that λrρ(λ) isholomorphic on Σπ/2+α0

−ρ. The set Σπ/2+α0∩Bξ0

(0) is a compact subsetof Σπ/2+α0

− ρ. Hence,

supλ∈Σα0∩Bξ0 (0)

∣∣∣λrρ(λ)∣∣∣ ≤M2,

with some constant M2 = M2(α0,ρ,ξ0) = M2(δ0, |ρ|). All together, wehave for each λ ∈ Σπ/2+α0

that∣∣∣rρ(λ)∣∣∣ ≤ M2

|λ|+ 2Cα0

a(C |λ|) ≤maxM3,2Cα0(

1|λ|

+ a(C |λ|)).

This means that we have |rρ(λ)| ≤ Mrg(|λ|) with g(s) = 1/s + a(Cs), fors ∈ (0,∞). The function g is a positive, non-increasing function on(0,∞) and sg(s) = 1 + sa(Cs) is non-decreasing on (0,∞), cf. Proposition5.2.1 (iv). Thus, we have supt∈(0,1] tg(t) ≤ 1 + a(C), and

∫ 10 g(1/t)dtt ≤

1 +∫ 1

0 a(C/t)dtt <∞. The Tauberian Theorem 1.2.9 yields the claim.

86


5.2.2 Convergence Rate

To prove the convergence rate of sρ and rρ to s?(ρ) and r?(ρ), respectively, weneed the following auxiliary estimates which are uniform in ρ on compactsets of C+.

Proposition 5.2.3. Let K ⊂ C+ be a compact set such that for suitable constantsδ0 ∈ (0,π/2) and ρmax ≥ ρmin > 0, depending on K , we have K ⊂ Σπ/2−δ0

∩Bρmax

(0) \Bρmin(0)

. Furthermore, we set α0 = δ0/2.

Then we have for each ρ ∈ K and all λ ∈ Σπ/2+α0− ρ the following statements.

(i) We have a(ρmax) ≤ a(|ρ|) ≤ a(ρmin).

(ii) There are constants Cα0, cα0

> 0 such thatcα0a(|λ+ ρ|) ≤ |a(λ+ ρ)| ≤ Cα0

a(|λ+ ρ|).

(iii) For |λ+ ρ| ≥ 2ρmax we have |λ| ≥ ρmax and |λ+ ρ|/ |λ| ≤ 2.

(iv) For |λ+ ρ| ≤ ρmin/2 we have |λ| ≥ ρmin/2 and|λ+ ρ|/ |λ| ≤ 1 + 2ρmax/ρmin.

(v) There is some ξ1 = ξ1(K) > 0 such that for all |λ+ ρ| ≥ ξ1 we have∣∣∣∣∣1− a(λ+ ρ)a(ρ)

∣∣∣∣∣−1

≤ 2.

(vi) There is some ξ2 = ξ2(K) > 0 such that for all |λ+ ρ| ≥ ξ2 we have∣∣∣λa(λ+ ρ)∣∣∣ ≥ 1

2.

(vii) There is some ξ? = ξ?(K) > 0 such that for all |λ+ ρ| ≥ ξ? we have∣∣∣∣∣ 1a(λ+ ρ)

− 1a(ρ)

∣∣∣∣∣−1≤ 2cα0

a(ρmax).

(viii) There is some ν1 = ν1(K) > 0 such that for all |λ+ ρ| ≤ ν1 we have∣∣∣∣∣1− a(λ+ ρ)a(ρ)

∣∣∣∣∣−1

≤ 12.

(ix) There is some ν2 = ν2(K) > 0 such that for all |λ+ ρ| ≤ ν2 we have∣∣∣λa(λ+ ρ)∣∣∣ ≥ 1

2.

87


(x) There is some ν? = ν?(K) > 0 such that for all |λ+ ρ| ≤ ν? we have∣∣∣∣∣ 1a(λ+ ρ)

− 1a(ρ)

∣∣∣∣∣−1≤ 2Cα0

a(ρmin).

Proof. (i) The Laplace transform a is decreasing on (0,∞), see Remark 2.3.6,the claim follows directly.

(ii) Since λ + ρ ∈ Σπ/2+α0and α0 < π/4 the assertion follows by Theorem

2.3.12.

(iii) We have |λ| ≥ |λ+ ρ| − |ρ| ≥ 2ρmax − ρmax = ρmax, as well as |λ+ ρ|/ |λ| ≤1 + |ρ|/ |λ| ≤ 1 + ρmax/ρmax = 2.

(iv) We have |λ| ≥ |ρ| − |λ+ ρ| ≥ ρmin −ρmin/2 = ρmin/2, as well as |λ+ρ|/ |λ| ≤1 + |ρ|/ |λ| ≤ 1 + 2ρmax/ρmin.

(v) From part (i) and (ii) it follows the estimate∣∣∣a(λ+ ρ)∣∣∣∣∣∣a(ρ)

∣∣∣ ≤Cα0

a(|λ+ ρ|)cα0a(|ρ|)

≤Cα0

a(|λ+ ρ|)cα0a(ρmin)

.

Due to the bijectivity of a on (0,∞), we choose ξ1 = ξ1(α0,ρmin) = ξ1(K) >0 such that

ξ1 = a−1(

12cα0a(ρmin)Cα0

).

Thus, for each ρ ∈ K and all λ ∈ Σπ/2+α0− ρ with |λ+ ρ| ≥ ξ1 we obtain

that ∣∣∣∣∣ a(λ+ ρ)a(ρ)

∣∣∣∣∣ ≤ 12,


∣∣∣∣∣−1

≤ 2.

(vi) The corresponding Sonine kernel b ∈ L1,loc(R+) is completely monotonic,hence b : (0,∞)→ R+ is strictly decreasing. In case of b ∈ L1(R+) andlimt→0 b(t) = β <∞ we assume w. l. o. g. that β > cα0

(if this is not true,we replace in part (ii) cα0

by β/2), then b : (0,∞)→ (0,β) is bijective andb−1(cα0

) > 0 is well-defined. We set ξ2 = maxb−1(cα0),2ρmax = ξ2(K) > 0.

For each ρ ∈ K and λ ∈ Σπ/2+α0− ρ with |λ+ ρ| ≥ ξ2 we obtain, together

with the estimate from part (iii), that

88


∣∣∣λa(λ+ ρ)∣∣∣ ≥ |λ||λ+ ρ|

cα0|λ+ ρ|a(|λ+ ρ|)

≥cα0

2b(|λ+ ρ|)

≥cα0

2b(ξ2)≥ 1/2.

(vii) Now, we choose ξ? = ξ?(α0,ρmax) = ξ?(K) > 0 such that

ξ? = a−1(

23cα0a(ρmax)Cα0

).

For each ρ ∈ K and all λ ∈ Σπ/2+α0− ρ with |λ+ ρ| ≥ ξ? we have on one

hand |a(ρ)| ≥ cα0a(|ρ|) ≥ cα0

a(ρmax) and on the other hand that |a(λ+ρ)| ≤Cα0

a(|λ+ ρ|) ≤ Cα0a(ξ?) = 2cα0

a(ρmax)/3. Thus it follows that∣∣∣∣∣ 1a(λ+ ρ)

− 1a(ρ)

∣∣∣∣∣ ≥ |a(λ+ ρ)|−1 − |a(ρ)|−1

≥ [Cα0a(ξ?)]−1 − [cα0

a(ρmax)]−1

= [2cα0a(ρmax)]−1.

(viii) For each ρ ∈ K and all λ ∈ Σπ/2+α0−ρwe obtain by the previous estimates

from part (i) and (ii) that∣∣∣a(λ+ ρ)∣∣∣∣∣∣a(ρ)

∣∣∣ ≥cα0a(|λ+ ρ|)Cα0

a(|ρ|)≥cα0a(|λ+ ρ|)

Cα0a(ρmin)

.

Using the bijectivity of a on (0,∞), we choose ν1 = ν1(α0,ρmax) = ν1(K) >0 such that

ν1 = a−1(3Cα0

a(ρmax)cα0

).

For each ρ ∈ K and all λ ∈ Σπ/2+α0− ρ with |λ + ρ| ≤ ν1 we obtain the

estimate ∣∣∣∣∣ a(λ+ ρ)a(ρ)

∣∣∣∣∣ ≥ 3,


∣∣∣∣∣−1

≤ 12.

89


(ix) We set ν2 = mina−1(1/(ρmincα0)),ρmin/2 = ν2(K) > 0. For each ρ ∈ K

and all λ ∈ Σπ/2+α0− ρ with |λ + ρ| ≤ ν2 we deduce, together with the

previous estimates from part (iv), that∣∣∣λa(λ+ ρ)∣∣∣ ≥ ρmincα0

a(|λ+ ρ|)/2≥ ρmincα0

a(ν2)/2 ≥ 1/2.

(x) Using the bijectivity of a on (0,∞), we choose ν? = ν?(α0,ρmin) = ν?(K) >0 such that

ν? = a−1(2Cα0

a(ρmin)cα0

).

For ρ ∈ K and all λ ∈ Σπ/2+α0− ρ with |λ+ ρ| ≤ ν? we have the estimates

|a(ρ)| ≤ Cα0a(|ρ|) ≤ Cα0

a(ρmin) and |a(λ + ρ)| ≥ cα0a(|λ + ρ|) ≥ cα0

a(ν?) =2Cα0

a(ρmin). Thus it follows that∣∣∣∣∣ 1a(λ+ ρ)

− 1a(ρ)

∣∣∣∣∣ ≥ |a(ρ)|−1 − |a(λ+ ρ)|−1

≥ [Cα0a(ρmin)]−1 − [cα0

a(ν?)]−1

= [2Cα0a(ρmin)]−1.

After the previous preparations we are able to determine the convergencerate of the exponentially weighted scalar resolvent sρ(t) and integral resolventrρ(t) to s?(ρ) and r?(ρ), respectively, as t→∞.

Lemma 5.2.4. We assume additionally that∫ 1

0 a(1/t)dtt ,

∫ 10 b(1/t)dtt <∞.

For each ρ ∈ C+ there are some constants Cs,Cr ,C′r > 0 such that for all t ∈R+we have the estimates

(i)∣∣∣s−1/a(ρ)(t)− eρts?(ρ)

∣∣∣ ≤ Cs,(ii)

∣∣∣r−1/a(ρ)(t)− eρtr?(ρ)∣∣∣ ≤ Cr a(1/t)/t,

(iii)∣∣∣r−1/a(ρ)(t)− eρtr?(ρ)

∣∣∣ ≤ C′r b(1/t)/t2.

These estimates are uniform in ρ on compact sets of C+.

Remark 5.2.5. a) The estimates from Lemma 5.2.4 remain true even onΣε0

with 0 < ε0 < α0, α0 > 0 sufficiently small.

b) Together with the fact that for each ρ ∈ C+ we have sρ(t) → s?(ρ) ,0, as t → ∞, see Lemma 5.2.2, it follows immediately that s−1/a(ρ) isunbounded on R+ for each ρ ∈C+.

90


Proof. Let K ⊂ C+ be a compact set. Then there are suitable constants δ0 ∈(0,π/2) and ρmax ≥ ρmin > 0 such that K ⊂ Σπ/2−δ0

∩Bρmax

(0) \Bρmin(0)

. We

set α0 = δ0/2 < π/4. We use the estimates and notations from Proposition5.2.3.

(i) For each ρ ∈ K we consider the holomorphic expression

sρ(λ)−s?(ρ)λ

,

on Σπ/2+α0− ρ. We want to apply the Analytic Representation Theorem

1.2.1 to this holomorphic expression on the shifted sector Σπ/2+α0− ρ.

We set ξ3 = maxξ1,2ρmax = ξ3(K) > 0. This implies for all ρ ∈ K andall λ ∈ Σπ/2+α0

− ρ with∣∣∣λ+ ρ

∣∣∣ ≥ ξ3 that∣∣∣∣∣sρ(λ)−s?(ρ)λ

∣∣∣∣∣ ≤ 1|λ+ ρ|

[∣∣∣∣∣1− a(λ+ ρ)a(ρ)

∣∣∣∣∣−1

+|λ+ ρ||λ||s?(ρ)|

]≤ M1

|λ+ ρ|.

with a constant M1 =M1(K) = 2 + 2maxρ∈K |s?(ρ)|.Now, we choose ν2 = minν1,ρmin/2 = ν2(K) > 0. Thus, it follows for allρ ∈ K and all λ ∈ Σπ/2+α0

− ρ with |λ+ ρ| ≤ ν2 that∣∣∣∣∣sρ(λ)−s?(ρ)λ

∣∣∣∣∣ ≤ 1|λ+ ρ|

[∣∣∣∣∣1− a(λ+ ρ)a(ρ)

∣∣∣∣∣−1

+|λ+ ρ||λ||s?(ρ)|

]≤ M2

|λ+ ρ|.

with a constant M2 =M2(K) = 1/2 + (1 + 2ρmax/ρmin)maxρ∈K |s?(ρ)|.Let ε0 ∈ (0,α0), ρ ∈ K and λ ∈ Σπ/2+ε0

−ρ such that we have ν2 ≤ |λ+ρ| ≤ξ3. We set G = Σπ/2+ε0

∩Bξ3

(0) \Bν2(0)

. For fixed ρ ∈ C+ the function

sρ(z − ρ)− s?(ρ)/(z − ρ) is holomorphic on Σπ/2+α0. Furthermore, there is

a constant M0 =M0(K,ε0,δ0,ν2,ξ3) =M0(K,ε0) > 0 such that

supρ∈K

supz∈G

∣∣∣sρ(z − ρ)− s?(ρ)/(z − ρ)∣∣∣ ≤ sup

(ρ,z)∈K×G

∣∣∣sρ(z − ρ)− s?(ρ)/(z − ρ)∣∣∣

= sup(ρ,z)∈K×∂G

∣∣∣sρ(z − ρ)− s?(ρ)/(z − ρ)∣∣∣

=M0 <∞,

since sρ(z − ρ) − s?(ρ)/(z − ρ) is continuous on the compact set ∂G ×K ,where K ∩ ∂G = ∅. Observe here that ν2 ≤ ρmin/2, ξ3 ≥ 2ρmax as wellas K ⊂ Σπ/2−δ0

. Thus, we obtain for all λ ∈ Σπ/2+ε0∩

Bξ3

(0) \Bν2(0)

− ρ

that ∣∣∣∣∣sρ(λ)−s?(ρ)λ

∣∣∣∣∣ |λ+ ρ||λ+ ρ|

≤ M0ξ2

|λ+ ρ|=

M3

|λ+ ρ|,

91


with M3 = M3(K,ε0) > 0. All together, we have for each ρ ∈ K and allλ ∈ Σπ/2+ε0

− ρ that∣∣∣∣∣sρ(λ)−s?(ρ)λ

∣∣∣∣∣ ≤ maxM1,M2,M3|λ+ ρ|

=Ms

|λ+ ρ|,

where Ms =Ms(K,ε0) > 0 a uniform constant for all ρ ∈ K . The AnalyticRepresentation Theorem 1.2.1 yields with g(s) = 1/s for all ε0 ∈ (0,α0)and all z ∈ Σε0

the estimate∣∣∣eρz [sρ(z)− s?(ρ)]∣∣∣ ≤ Cs,

and consequently ∣∣∣s−1/a(ρ)(z)− eρzs?(ρ)∣∣∣ ≤ Cs,

where Cs = Cs(K,ε0,α0) > 0 is a uniform2 constant for all ρ ∈ K .

(ii) For each ρ ∈ K we consider the holomorphic expression

rρ(λ)−r?(ρ)λ

,



We set ξ4 = maxξ1,ξ2,2ρmax = ξ4(K) > 0. This implies for all ρ ∈ K andall λ ∈ Σπ/2+α0

− ρ with |λ+ ρ| ≥ ξ4 that∣∣∣∣∣rρ(λ)−r?(ρ)λ

∣∣∣∣∣ ≤ |a(λ+ ρ)|[∣∣∣∣∣1− a(λ+ ρ)

a(ρ)

∣∣∣∣∣−1

+|r?(ρ)||λa(λ+ ρ)|

]≤M4|a(λ+ ρ)|.

with M4 =M4(K) = 2 + 2maxρ∈K |r?(ρ)|. Now, we choose the parameterν3 = minν1,ν2,ρmin/2 = ν3(K) > 0. Thus, it follows for all ρ ∈ K and allλ ∈ Σπ/2+α0

− ρ with |λ+ ρ| ≤ ν3 that∣∣∣∣∣rρ(λ)−r?(ρ)λ

∣∣∣∣∣ ≤ |a(λ+ ρ)|[∣∣∣∣∣1− a(λ+ ρ)

a(ρ)

∣∣∣∣∣−1

+|r?(ρ)||λa(λ+ ρ)|

]≤M5|a(λ+ ρ)|,

with a constant M5 =M5(K) = 1/2 + 2maxρ∈K∣∣∣r?(ρ)

∣∣∣.Let ε0 ∈ (0,α0), ρ ∈ K and λ ∈ Σπ/2+ε0

−ρ such that we have ν3 ≤∣∣∣λ+ ρ

∣∣∣ ≤ξ4. We set G = Σπ/2+ε0

∩Bξ4

(0) \Bν3(0)


2The constant Cs is given by Cs =Ms(K,ε0)max

e−sinε0sinε0

,2πe.

92


rρ(z − ρ)− r?(ρ)/(z − ρ) is holomorphic on Σπ/2+α0. Furthermore there is

some constant M6 =M6(K,ε0,δ0,ν3,ξ4) =M0(K,ε0) > 0 such that

supρ∈K

supz∈G

∣∣∣rρ(z − ρ)− r?(ρ)/(z − ρ)∣∣∣ ≤ sup

(ρ,z)∈K×G

∣∣∣rρ(z − ρ)− r?(ρ)/(z − ρ)∣∣∣


∣∣∣rρ(z − ρ)− r?(ρ)/(z − ρ)∣∣∣

=M6 <∞,

since rρ(z − ρ) − r?(ρ)/(z − ρ) is continuous on the compact set ∂G ×K , because ∂G ∩ K = ∅. Thus, we obtain that for all λ ∈ Σπ/2+ε0

∩Bξ4

(0) \Bν3(0)

− ρ the estimate∣∣∣∣∣rρ(λ)−r?(ρ)λ

∣∣∣∣∣ |a(λ+ ρ)||a(λ+ ρ)|

≤M6|a(λ+ ρ)|cα0a(|λ+ ρ|)

=M7|a(λ+ ρ)|,

with M7 = M7(K,ε0) = M6/[cα0a(ξ3)] > 0. All together, we have for all

ρ ∈ K and all λ ∈ Σπ/2+ε0− ρ that∣∣∣∣∣rρ(λ)−

r?(ρ)λ

∣∣∣∣∣ ≤maxM4,M5,M7|a(λ+ ρ)| ≤Mr a(|λ+ ρ|),

where Mr =Mr(K,ε0) = maxM4,M5,M7Cα0> 0 is a uniform constant

for all ρ ∈ K . Note that a : (0,∞)→R+ is non-negative, non-increasingwith limt→∞ a(t) = 0, since it is the Laplace transform of a completelymonotonic kernel, and supt∈(0,1] ta(t) = supt∈(0,1] 1/b(t) = 1/b(1) <∞ as

well as∫ 1

0 a(1/t)dtt < ∞ by assumption. The Analytic Representation

Theorem 1.2.1 yields with g = a for all ε0 ∈ (0,α0) and all z ∈ Σε0that∣∣∣eρz [rρ(z)− r?(ρ)]

∣∣∣ ≤ Cr a(1/ |z|)/ |z| ,and consequently∣∣∣r−1/a(ρ)(z)− eρzr?(ρ)

∣∣∣ ≤ Cr a(1/ |z|)/ |z| ,where Cr = Cr(K,ε0,α0) > 0 is a uniform3 constant for all ρ ∈ K .

(iii) For each ρ ∈ K we consider the derivative of rρ(λ)− r?(ρ)/λ, the holomor-phic expression

rρ′(λ) +

r?(ρ)λ2

3The constant Cr is given by Cr =Mr (K,ε0,α0)max

e−sinε0sinε0

,2πe.

93




Easy calculations yield the identity

rρ′(λ) =

(1

a(λ+ ρ)− 1a(ρ)

)−2 a′(λ+ ρ)a(λ+ ρ)2 = rρ(λ)2 a

′(λ+ ρ)a(λ+ ρ)2 .

With the relation a(λ+ ρ) = 1/[(λ+ ρ)b(λ+ ρ)] it follows that

rρ′(λ) +

r?(ρ)λ2 = b(λ+ ρ)

[rρ(λ)2a′(λ+ ρ)

b(λ+ ρ)a(λ+ ρ)2+

r?(ρ)

b(λ+ ρ)λ2

]= b(λ+ ρ)

[rρ(λ)2 a

′(λ+ ρ)(λ+ ρ)a(λ+ ρ)

+ r?(ρ)(λ+ ρ)a(λ+ ρ)

λ2

].

By Lemma 2.3.16 we know there is constant k = k(δ0) > 0 such that|λa′(λ)| ≤ k |a(λ)| for all λ ∈ Σπ/2+α0

.

Now, we choose ξ5 = maxξ? ,2ρmax = ξ5(K) > 0 and obtain for eachρ ∈ K and all λ ∈ Σπ/2+α0

− ρ with |λ+ ρ| ≥ ξ5 that∣∣∣∣∣rρ′(λ) +r?(ρ)λ2

∣∣∣∣∣ ≤ |b(λ+ ρ)|[|rρ(λ)|2

∣∣∣∣∣ a′(λ+ ρ)(λ+ ρ)a(λ+ ρ)

∣∣∣∣∣+|r?(ρ)|

∣∣∣∣∣a(λ+ ρ)(λ+ ρ)λ2

∣∣∣∣∣] ≤M8|b(λ+ ρ)|,

with a constant M8 = [2Cα0a(ρmin)]2k + maxρ∈K |r?(ρ)|Cα0

a(ξ5)2/ρmin =M8(K) > 0. We set ν4 = minν? ,ρmin/2 = ν4(K) > 0. Thus, we have foreach ρ ∈ K and all λ ∈ Σπ/2+α0

− ρ and with |λ+ ρ| ≤ ν4 that∣∣∣∣∣rρ′(λ) +r?(ρ)λ2

∣∣∣∣∣ ≤ |b(λ+ ρ)|[|rρ(λ)|2

∣∣∣∣∣ a′(λ+ ρ)(λ+ ρ)a(λ+ ρ)

∣∣∣∣∣+|r?(ρ)|

∣∣∣∣∣ a(λ+ ρ)(λ+ ρ)λ2

∣∣∣∣∣] ≤M9|b(λ+ ρ)|,

where the constant M9 = M9(K) > 0 is given by the expression M9 =[2cα0

a(ρmax)]2k + maxρ∈K |r?(ρ)|[Cα0/b(ν4)][2/ρmin]2. Keep in mind, that

ta(t) = 1/b(t) is non-decreasing on (0,∞).

Let ε0 ∈ (0,α0), ρ ∈ K ⊂ Σπ/2−δ0and λ ∈ Σπ/2+ε0

− ρ with ν4 ≤ |λ+ ρ| ≤ ξ5.We set G = Σπ/2+ε0

∩Bξ5

(0) \Bν4(0)


rρ′(z − ρ) + r?(ρ)/(z − ρ)2 is holomorphic on Σπ/2+α0

. Furthermore, thereis some constant M10 =M10(K,ε0,δ0,ν4,ξ5) =M10(K,ε0) > 0 such that

94


supρ∈K

supz∈G

∣∣∣∣rρ′(z − ρ) + r?(ρ)/(z − ρ)2∣∣∣∣ ≤ sup

(ρ,z)∈K×G

∣∣∣∣rρ′(z − ρ) + r?(ρ)/(z − ρ)2∣∣∣∣ ,


∣∣∣∣rρ′(z − ρ) + r?(ρ)/(z − ρ)2∣∣∣∣ ,

=M10 <∞,

because of the continuity of rρ′(z − ρ) + r?(ρ)/(z − ρ)2 on the compact

set ∂G × K , where ∂G ∩ K = ∅. Thus, we obtain for all ρ ∈ K and allλ ∈ Σπ/2+ε0

∩Bξ5

(0) \Bν4(0)

− ρ that

∣∣∣∣∣rρ′(λ) +r?(ρ)λ2

∣∣∣∣∣ |b(λ+ ρ)||b(λ+ ρ)|

≤M10|b(λ+ ρ)|cα0b(|λ+ ρ|)

=M11|b(λ+ ρ)|,

with M11 = M11(K,ε0) = M10Cα0/[cα0

b(ξ5)] > 0. All together, we havefor all ρ ∈ K and all λ ∈ Σπ/2+ε0

− ρ that

∣∣∣∣∣rρ(λ)−r?(ρ)λ

∣∣∣∣∣ ≤maxM8,M9,M11|b(λ+ ρ)| ≤M ′r b(|λ+ ρ|),

where Mr = M ′r(K,ε0,α0) = maxM8,M9,M11/cα0> 0 is a uniform con-

stant for all ρ ∈ K . Analogous to part (ii), the function b : (0,∞)→ R+satisfies the conditions for the Analytic Representation Theorem 1.2.1.So, it follows, together with the uniqueness of the Laplace transform,Theorem 1.1.4, that for all ε0 ∈ (0,α0) and all z ∈ Σε0

we have the esti-mate ∣∣∣∣eρz [zr−1/a(ρ)(z)− zr?(ρ)

]∣∣∣∣ ≤ C′r b(1/ |z|)/ |z| ,

and consequently∣∣∣zr−1/a(ρ)(z)− eρzzr?(ρ)∣∣∣ ≤ C′r b(1/ |z|)/ |z| ,

where C′r = C′r(K,ε0,α0) > 0 is a uniform4 constant for all ρ ∈ K .

4The constant C is given by C′r =M′r (K,ε0,α0)max

e−sinε0sinε0

,2πe.

95


5.3 Instability Theorem

Theorem 5.3.1 (Instability Theorem). Let X be a complex Banach space anda ∈ L1,loc(R+) \L1(R+) be a unbounded, completely monotonic kernel. We denoteby b ∈ L1,loc(R+) the corresponding completely monotonic Sonine kernel. Moreover,let the kernel a be θa-sectorial with θa ∈ (0,π/2]. Furthermore, we suppose that∫ 1

0 a(1/t)dtt <∞,

∫ 10 b(1/t)dtt <∞ and

[t 7→mina(1/t)/t, b(1/t)/t2

]∈ L1(R+).

Moreover, let A be a closed linear operator in X with dense domain such thatσ (−A)∩∂Σθa = ∅ and σ+ = σ (A)∩ −1/a(C+) ⊂ −Σθa is a non-empty compactset and σ− = σ (A) \ σ+ is a closed subset of C.

We denote by P+ the spectral projection of the operator A associated with thecompact spectral set σ+. Let A− be the part of A in Rg(Id−P+) with σ (A−) = σ−.We assume for some ϕ > θa that ρ(−A−) ⊃ Σϕ ∪0 and ‖(µ+A−)−1‖B(X−) ≤M/ |µ|for all µ ∈ Σϕ. Suppose that h ∈ C1(X;X) with h(0) = 0, h′(0) = 0.

Then, the equilibrium w∗ = 0 is unstable for the Volterra equation

w(t) + (a ∗Aw)(t) = w0 + (a ∗ h(w))(t), t ∈ [0,T ], (5.6)

where u0 ∈ X.

Proof. We use the same notation as introduced in Section 5.1. By the spectralprojection P+ of the operator A associated with the compact spectral set σ+ weget a decomposition of the space X = X+ ⊕X−, see Remark 5.1.2. The operatorA+ ∈ B(X+) denotes the part of A in RgP+.

We consider for J = [0,T ] and f± ∈ C(J ;X±) the Volterra equations on X±

w±(t) + (a ∗A±w±)(t) = f±(t), t ∈ J.

The boundedness of the operator A+ ∈ B(X+) yields the existence of theresolvent S+(t)t∈R+

and the integral resolvent R+(t)t∈R+on X+ given by

Dunford’s integral representation, see Corollary 5.1.7.The assumptions on the operator A− yield the parabolicity of the corre-

sponding Volterra equation on X−. Together with the completely monotonicityof the kernel a ∈ L1,loc(R+) this implies by Theorem 3.3.1 that there exists abounded resolvent S−(t)t∈R+

∈ B(X−) on X−, thus ‖S−(t)‖B(X−) ≤M−, t ∈ R+,

withM− ≥ 1. Moreover, the integrability conditions a and b yield the existenceof an integrable integral resolvent R−(t)t∈R+

on X−, so ‖R−‖L1(R+;B(X−)) = N−with some N− > 0.

In particular, the resolvent S(t)t∈R+and integral resolvent R(t)t∈R+

forthe linear Volterra equation

w(t) ∗ (a ∗Aw)(t) = f (t), t ∈ J,

96

5.3. Instability Theorem

where f ∈ C(J ;X), exist and they are given by

S(t) = S+(t) + S−(t), t ∈R+,

R(t) = R+(t) +R−(t), t ∈R+.

We assume, that the equilibrium w∗ = 0 is stable of the Volterra equation(5.6), i. e. for each ε > 0 there is some δ = δ(ε) > 0 such that for all w0 ∈ Bδ(0)there exists a mild solution w = w(·;w0) of the Volterra equation (5.6) on R+and ‖w‖C(R+;X) ≤ ε.

Let ε > 0 be arbitrary, but fixed. We consider for the correspondingparameter δ = δ(ε) > 0 some w0 ∈ Bδ(0). The mild solution w = w(·;w0) of theVolterra equation (5.6) is given as solution of

w(t) = S(t)w0 + (R ∗ h(w))(t), t ∈R+.

The spectral projection P+ yields the decomposition w(t) = P+w(t) + P−w(t) =v(t) +u(t). Whereupon u and v solve the equations

v(t) = S+(t)v0 + (R+ ∗ h(u + v))(t), t ∈R+ on X+,

u(t) = S−(t)u0 + (R− ∗ h(u + v))(t), t ∈R+ on X−,

with w0 = P+w0 + P−w0 = v0 +u0.We know there is some bounded set Ω ⊂ −1/a(C+) ⊂ −Σθa such that

σ+ ⊂Ω. We denote by γ the boundary of Ω and assume that it is sufficientlysmooth, cf. Remark 5.1.2, as well as dist(0,γ) > 0.

Firstly, we consider the expression

x =1

2πi

∫γ

(µ−A+)−1[∫ ∞

0

(−1µ

)a−1(−1/µ)e−a

−1(−1/µ)τh(w(τ))dτ]dµ

=1

2πi

∫γ

(µ−A+)−1g(µ)dµ;

where

g(µ) =∫ ∞

0

(−1µ

)a−1(−1/µ)e−a

−1(−1/µ)τh(w(τ))dτ.

By definition of x it follows that x ∈ X+. Note that we can change the orderof integration, since the path γ is compact and thus the continuous maph(w) is bounded along the path γ with dist(0,γ) > 0. One convince oneselfeasily, that g : − 1/a(C+)→ X is a holomorphic map, because a−1 : a(C+)→C+ is holomorphic as inverse function of the bijective holomorphic mapa : C+ → a(C+) with a′(λ) , 0 on C+. In particular, we have some bound‖x‖X ≤ 1

2π supµ∈γ∥∥∥(µ−A+)−1

∥∥∥B(X)‖g(µ)‖XL(γ) < ∞, where L(γ) < ∞ denotes

the length of the closed path γ .

97


Now, we take another bounded set Ω′ ⊂ −1/a(C+) ⊂ −Σθa such that Ω′ ⊃Ω.We denote by γ ′ the boundary of Ω′ and assume that it is sufficiently smoothwith dist(0,γ ′) > 0, too. Using Dunford’s integral representation, for eacht ∈R+ the resolvent S+(t) is given by

S+(t) =1

2πi

∫γ ′sµ′ (t)(µ

′ −A+)−1dµ′ ,

see Corollary 5.1.7. Due to the choice of the paths γ and γ ′ as well as theresolvent identity and Cauchy’s integral formula it follows that

S+(t)x =1

2πi

∫γ ′sµ′ (t)(µ

′ −A+)−1[

12πi

∫γ

(µ−A+)−1g(µ)dµ]dµ′ (5.7)

=1

2πi

∫γsµ(t)(µ−A+)−1g(µ)dµ. (5.8)

Using Dunford’s integral representation for the integral resolvent R+(t), t ∈R+,yields

(R+ ∗ h(w))(t) =∫ t

0

[∫γrµ(t − τ)(µ−A+)−1dµ

]h(w(τ))dτ

=∫γ

(µ−A+)−1[∫ t

0rµ(t − τ)h(w(τ))dτ

]dµ;

note that we can change the order of integration, since γ is a compact pathand therewith h(w) is bounded along this path, and rµ ∈ L1,loc(R+).

Now, we consider the component v of the mild solution on X+. For eacht ∈R+ we insert a zero in terms of S+(t)x, given by (5.7), and −S+(t)x, given by(5.8), where we split the integral in the definition of g at the point t. Hence,we obtain for all t ∈R+ that

(2πi)v(t)

=∫γ ′sµ′ (t)(µ

′ −A+)−1[v0 +

12πi

∫γ

(µ−A+)−1g(µ)dµ]dµ′

+∫γ

(µ−A+)−1[∫ t

0

rµ(t − τ)− sµ(t)

(−1µ

)a−1(−1/µ)e−a

−1(−1/µ)τh(w(τ))dτ

−∫ ∞tsµ(t)

(−1µ

)a−1(−1/µ)e−a

−1(−1/µ)τh(w(τ))dτ]dµ.

Now, we substitute ρ = a−1(−1/µ) in the integral along the contour γ . Thisyields a unique path Γ = a−1(−1/γ) ⊂ C+ with dist(0,Γ ) > 0. Thus, we have forall t ∈R+ that

98


(2πi)v(t)

=∫γ ′sµ′ (t)(µ

′ −A+)−1[v0 +

12πi

∫γ


−∫Γ

(1/a(ρ) +A+)−1[∫ t

0

r−1/a(ρ)(t − τ)− s−1/a(ρ)(t)ρa(ρ)e−ρτ

h(w(τ))dτ

−∫ ∞ts−1/a(ρ)(t)ρa(ρ)e−ρτh(w(τ))dτ

]a′(ρ)a(ρ)2 dρ.

In our next step we will show that the integral over Γ is uniformly boundedfor all t ∈ R+. Since the map h is continuously differentiable with h(0) = 0and h′(0) = 0, we have for all σ > 0 that there exists η = η(σ ) > 0 such thatfor all z ∈ Bη(0) we have ‖h(z)‖X ≤ σ ‖z‖X . For sufficiently small σ > 0 wechoose ε = η(σ ) and δ = δ(ε) > 0 according to the stability property of theequilibrium w∗ = 0. Hence, we have for all w0 ∈ Bδ(0) that ‖w‖C(R+;X) ≤ ε, thus‖h(w)‖C(R+;X) ≤ σ ‖w‖C(R+;X) ≤ ση. Consequently, we deduce for all ρ ∈ Γ andall t ∈R+ that∥∥∥∥∥∥

∫ t

0

[r−1/a(ρ)(t − τ)− ρa(ρ)e−ρτs−1/a(ρ)(t)

]h(w(τ))dτ

∥∥∥∥∥∥X

≤ ση∫ t

0

∣∣∣r−1/a(ρ)(t − τ)− ρa(ρ)e−ρτs−1/a(ρ)(t)∣∣∣dτ.

Now, we insert for all τ ∈ [0, t] the zero eρ(t−τ) [r?(ρ)− ρa(ρ)s?(ρ)] = 0, wheres?(ρ) and r?(ρ) the limits from Lemma 5.2.2. We obtain∣∣∣r−1/a(ρ)(t − τ)− ρa(ρ)e−ρτs−1/a(ρ)(t)

∣∣∣≤

∣∣∣r−1/a(ρ)(t − τ)− eρ(t−τ)r?(ρ)∣∣∣+

∣∣∣ρa(ρ)∣∣∣ ∣∣∣∣[s?(ρ)eρt − s−1/a(ρ)(t)

]e−ρτ

∣∣∣∣ .In combination with an easy substitution we get the estimate∥∥∥∥∥∥∫ t

0


]h(w(τ))dτ

∥∥∥∥∥∥X

≤ ση[∫ t

0

∣∣∣r−1/a(ρ)(τ)− eρτr?(ρ)∣∣∣dτ +

∣∣∣ρa(ρ)∣∣∣∫ t

0

∣∣∣∣[s?(ρ)eρt − s−1/a(ρ)(t)]e−ρτ

∣∣∣∣dτ] .Since Γ is a compact set in C+, by Lemma 5.2.4, there are uniform constantsC+,M+ > 0 such that for all ρ ∈ Γ and all t ∈R+ we have the estimates∣∣∣s−1/a(ρ)(t)− eρts?(ρ)

∣∣∣ ≤M+,∣∣∣r−1/a(ρ)(t)− eρtr?(ρ)∣∣∣ ≤ C+ min

a(1/t)t

,b(1/t)t2

,

99


we define N+ =∥∥∥t 7→ C+ mina(1/t)/t, b(1/t)/t2

∥∥∥L1(R+)

. Thus, we have for allt ∈R+ that ∥∥∥∥∥∥

∫ t

0


]h(w(τ))dτ

∥∥∥∥∥∥X

≤ ση[N+ +

∣∣∣ρa(ρ)∣∣∣ M+

Reρ

]≤ σηM0,

with some uniform constant M0 =N+ + supρ∈Γ∣∣∣∣ρa(ρ) M+

Reρ

∣∣∣∣, for all ρ ∈ Γ and all

t ∈R+; note that dist(Γ ,0) > 0.Furthermore, we have by the differentiability of the map h the following

estimate for each t ∈R+∥∥∥∥∥∫ ∞ts−1/a(ρ)(t)ρa(ρ)e−ρτh(w(τ))dτ

∥∥∥∥∥X

≤ ση∣∣∣ρa(ρ)

∣∣∣∫ ∞t

∣∣∣∣e−ρ(τ−t)[e−ρts−1/a(ρ)(t)

]∣∣∣∣dτ.Using the relation

∣∣∣e−ρts−1/a(ρ)(t)∣∣∣ ≤ ∣∣∣∣e−ρt [s−1/a(ρ)(t)− eρts?(ρ)

]∣∣∣∣+∣∣∣s?(ρ)

∣∣∣ ≤M+ + supρ∈Γ

∣∣∣s?(ρ)∣∣∣ =M1,

which is uniform in ρ ∈ Γ ⊂ C+ and valid for all t ∈ R+. Hence we concludefor each t ∈R+ that∣∣∣∣∣∫ ∞

ts−1/a(ρ)(t)ρa(ρ)e−ρτh(w(τ))dτ

∣∣∣∣∣≤ σηM1

∣∣∣ρa(ρ)∣∣∣∫ ∞t

∣∣∣e−ρ(τ−t)∣∣∣dτ = ση

∣∣∣ρa(ρ)∣∣∣ M1

Reρ≤ σηM2,

with some uniform constant M2 = supρ∈Γ∣∣∣ρa(ρ)

∣∣∣ M1Reρ , since dist(Γ ,0) > 0.

This shows for all t ∈R+ that the expression

∫Γ

(1/a(ρ) +A+)−1[∫ t

0


h(w(τ))dτ

−∫ ∞ts−1/a(ρ)(t)ρa(ρ)e−ρτh((u + v)(τ))dτ

]a′(ρ)a(ρ)2 dρ

is uniformly bounded by

100


∥∥∥∥∥∥∫Γ

(1/a(ρ) +A+)−1[∫ t

0


h(w(τ))dτ

−∫ ∞ts−1/a(ρ)(t)ρa(ρ)e−ρτh((u + v)(τ))dτ

]a′(ρ)a(ρ)2 dρ

∥∥∥∥∥∥X

≤ ση [M0 +M2]K <∞,

with K = supρ∈Γ∥∥∥(1/a(ρ) +A+)−1

∥∥∥B(X)

∣∣∣∣ a′(ρ)a(ρ)2

∣∣∣∣L(Γ ), where L(Γ ) <∞ denotes thelength of the closed contour Γ .

Since the mild solution w is assumed to be bounded, its components u andv have to be bounded for all t ∈R+ as well. Thus the expression

S+(t) [v0 + x] =1

2πi

∫γ ′sµ′ (t)(µ

′ −A+)−1[v0 +

12πi

∫γ


has to be bounded in X+. Indeed, assuming that there is some constant M > 0such that for all t ∈ R+ we have ‖S+(t) [v0 + x]‖X+

≤ M. We take a bounded

set Ω′′ ⊂ −1/a(C+) ⊂ −Σθa such that Ω′′ ⊃Ω′ and denote by γ ′′ the boundaryof Ω′′. For each µ ∈ Ω′′ there is some ρ ∈ K ⊂ Σπ/2+δ0

∩ Bρmax(0) \ Bρmin

(0),where δ0 ∈ (0,π/2) and 0 < ρmin ≤ ρmax are suitable parameters, such thatµ = −1/a(ρ). By Lemma 5.2.4 there is some uniform constant C > 0 such thatfor each µ ∈Ω′′ and all t ≥ t0 = ln(2C/M)/ρmin we have the estimate∣∣∣sµ(t)

∣∣∣ ≥ eρmintM2> 0,

where M = minρ∈K |s?(ρ)| > 0.Hence, for t ≥ t0 the expression

F(t) =1

2πi

∫γ ′′

1sµ′′ (t)

(µ′′ −A+)−1dµ′′ ∈ B(X+),

is well-defined and we have the estimate

‖F(t)‖B(X+) ≤e−ρmint

Mπsupµ′′∈γ ′′

∥∥∥(µ′′ −A+)−1∥∥∥B(X+)

L(γ ′′) <∞,

where L(γ ′′) <∞ denotes the length of the closes path γ ′′. In particular, wehave ‖F(t)‖B(X+) → 0 as t →∞. Due to the choice of the paths γ ′ and γ ′′ aswell as the resolvent identity and Cauchy’s integral formula it follows that

F(t)S+(t) =1

2πi

∫γ ′′

1sµ′′ (t)

(µ′′ −A+)−1[

12πi

∫γ ′sµ′ (t)(µ

′ −A+)−1dµ′]dµ′′

=1

2πi

∫γ ′

(µ′ −A+)−1dµ′ = P+.

101


This implies on one hand that

‖F(t)S+(t) [v0 + x]‖X+≤ ‖F(t)‖B(X+) ‖S+(t) [v0 + x]‖X+

≤ ‖F(t)‖B(X+)M→ 0

as t→∞, and on the other hand that

F(t)S+(t) [v0 + x] = P+ [v0 + x] = v0 + x,

since v0 + x ∈ X+. So, we conclude v0 + x = 0.Consequently, we have

−(2πi)v(t) =∫Γ

(1/a(ρ) +A+)−1[∫ t

0

r−1/a(ρ)(t − τ)− ρa(ρ)e−ρτs−1/a(ρ)(t)

h((u + v)(τ))dτ

−∫ ∞tρa(ρ)e−ρτs−1/a(ρ)(t)h((u + v)(τ))dτ

]a′(ρ)a(ρ)2 dρ.

This implies for all t ∈R+ the estimate

‖v(t)‖X ≤σK2π

supρ∈Γ

[∫ t

0

∣∣∣r−1/a(ρ)(t − τ)− s−1/a(ρ)(t)ρa(ρ)e−ρτ∣∣∣‖(u + v)(τ)‖X dτ

+∣∣∣ρa(ρ)

∣∣∣∫ ∞t

∣∣∣eρ(t−τ)∣∣∣ ∣∣∣e−ρts−1/a(ρ)(t)

∣∣∣‖(u + v)(τ)‖X dτ]

≤σK2π

[M0 +M2] supτ∈R+

[‖u(τ)‖X + ‖v(τ)‖X

]. (5.9)

Now, we consider the component u of the mild solution on X−. The bound-edness of the resolvent S−(t)t∈R+

, the integrability of the integral resolventR−(t)t∈R+

on X− and the differentiability of the map h yield for all t ∈R+ that

‖u(t)‖X ≤ ‖S−(t)‖B(X−) ‖u0‖X +∫ t

0‖R−(t − τ)‖B(X−) ‖h((u + v)(τ))‖X dτ

≤M− ‖u0‖X + σN− supτ∈R+

[‖u(τ)‖X + ‖v(τ)‖X

]. (5.10)

Combining the estimates for the component v from (5.9) and for the compo-nent u from (5.10), we get for each t ∈R+ the uniform estimate

‖u(t)‖X + ‖v(t)‖X ≤M− ‖u0‖X + σ[N− +

K2π

[M0 +M2]]

supτ∈R+

[‖u(τ)‖X + ‖v(τ)‖X

].

We set C =N− + K2π [M0 +M2] and we choose σ > 0 sufficiently small, such that

σC < 1/2, hence we have for all t ∈R+ the estimate

supτ∈R+

[‖u(τ)‖X + ‖v(τ)‖X] ≤ 2M− ‖u0‖X ,

102


in particular we have ‖u(0)‖X + ‖v(0)‖X ≤ 2M− ‖u0‖ and therewith ‖v(0)‖X ≤(2M− − 1)‖u0‖X . This is a contradiction to the stability of the equilibriumw∗ = 0, since we have a restriction to the initial values inside the ball Bδ(0). Itthe claim follows.

In case of the standard kernel the previous theorem reads as follows.

Corollary 5.3.2. Let X be a complex Banach space, α ∈ (0,1) and A be a closedlinear operator in X with dense domain such that σ (−A)∩∂Σα π

2= ∅ and −σ+ =

σ (−A)∩Σα π2

is a non-empty compact set and σ− = σ (A) \ σ+ is a closed set in C.We denote by P+ the spectral projection of the operator A associated with the

compact spectral set σ+. Let A− be the part of A in Rg(Id−P+) with σ (A−) = σ−. Weassume that A− is an invertible sectorial operator with spectral angle ϕA− < π−α

π2 .

Moreover, suppose that f ∈ C1(X;X) with f (0) = 0, f ′(0) = 0.Then, the equilibrium u∗ = 0 is unstable of the Volterra equation with standard

kernel


where u0 ∈ X.

The following corollary is an easy consequence for Volterra equations withstandard kernel in the finite dimensional case.

Corollary 5.3.3. Let α ∈ (0,1). Suppose that the spectrum of the matrix A ∈Rn×nbe such that σ (−A)∩∂Σα π

2= ∅ and there is at least one eigenvalue λ0 ∈ σ (A) with

|argλ0| > π −απ2 . Furthermore, let f ∈ C1(Rn;Rn) with f (0) = 0 and f ′(0) = 0.Then, the equilibrium u∗ = 0 is unstable for the Volterra equation


where u0 ∈Rn.

103

Part III

Quasilinear ParabolicFractional Evolution Equations

105

6Maximal Lp-RegularityThe goal of the last part of this thesis are stability studies of quasilinear frac-tional evolution equations. A possible approach for quasilinear problemsis the use of the property of maximal Lp-regularity for the linearised prob-lem. For this reason we explain in this chapter the necessary underlyingobjects, see Section 6.1, and introduce the concept of maximal Lp-regularityfor linear fractional evolution equations, see Section 6.2. We refer to Zacher’swork [Zac03] that all these considerations keep true if one replace the time-fractional evolution equation with the standard kernel by a wider class ofkernels for which also the concept of maximal Lp-regularity is known.

6.1 Preliminaries

As in the classical case certain geometrical properties of the underlying Banachspaces and operators are relevant for the considerations about the maximalLp-regularity of the equation (6.1). We introduce these notions.

Definition 6.1.1 (Banach space of classHT ). A Banach space X is said tobelong to the class HT , if the Hilbert transform is bounded on Lp(R;X) forsome (and then all) p ∈ (1,∞). Here the Hilbert transform Hf of a functionf ∈ S(R;X), the Schwartz space of rapidly decreasing X-valued functions, isdefined by

(Hf )(t) = limR→∞

∫R−1≤|s|≤R

f (t − s)dsπs, t ∈R.

Remark 6.1.2. a) We refer to Burkholder [Bur83] and Bourgain [Bou83] forthe proof that the class of Banach spaces of class HT coincides with theclass of UMD spaces, where UMD is the abbreviation for unconditionalmartingale difference property.

b) Each Banach space of class HT is super-reflexive and all Hilbert spacesbelong to the class of HT . If (Ω,A,µ) is a σ -finite measure space, p ∈(1,∞), then Lp(Ω;X) is of class HT if X has this property. Moreover,closed subspaces, quotients, duals and finite products are preserved,and complex interpolation spaces (X,Y )θ as well as real interpolationspaces (X,Y )θ,p are of class HT , provided X and Y have this propertyand θ ∈ (0,1), p ∈ (1,∞). For proofs of these results, we refer to thesurvey article by Burkholder [Bur86] and Amann [Ama95].

107

6. Maximal Lp-Regularity

Moreover, let Ω , ∅ be an open subset of Rn. Then for all m ∈ N0

the Sobolev spaces Hmp (Ω) are of class HT if and only if p ∈ (1,∞),

see [Are04, Section 6.1.3 Example d)].

But of course not each Banach space is class HT . A counterexample isgiven by the well-known space of continuous functions C(Ω), whereΩ ⊂R

n is a non-empty open set, cf. [Are04, Section 6.1.3 Example e)].

Furthermore, we are interested in the class of operators which are R-sectorial. We start with the definition of R-bounded families of boundedlinear operators.

Definition 6.1.3 (R-Boundedness). Let X and Y be Banach spaces. A familyof operators T ⊂ B(X,Y ) is called R-bounded, if there is a constant C > 0 andp ∈ [1,∞) such that for each N ∈N, Tj ∈ T , xj ∈ X and for all independent,symmetric, −1,1-valued random variables εj on a probability space (Ω,A,µ)the inequality ∥∥∥∥∥∥∥∥

N∑j=1

εjTjxj

∥∥∥∥∥∥∥∥Lp(Ω;Y )

≤ C

∥∥∥∥∥∥∥∥N∑j=1

εjxj

∥∥∥∥∥∥∥∥Lp(Ω;X)

is valid. The smallest such C is called R-bound of T , we denote it by R(T ).

Remark 6.1.4. 1. Consider Ω = [0,1], A the Borel sets in [0,1] and µ theLebesgue measure. Then the Rademacher functions rk(t) = sgn(sin(2kπt)),k ∈N, are the prototype for the random variables εk , cf. [PS16, Example4.1.2].

By means of the notion of R-boundedness stochastic analysis is intro-duced in operator theory.

2. The definition of R-boundedness is independent of p ∈ [1,∞). If theoperator family T ⊂ B(X,Y ) isR-bounded, then it is uniformly bounded,sup‖T ‖B(X,Y ) : T ∈ T ≤ R(T ). If X and Y are Hilbert spaces, then theR-boundedness of the operator family T ⊂ B(X,Y ) is equivalent to itsuniform boundedness. For a proof of these results, we refer to Prüss andSimonett [PS16, Remark 4.1.3].

With the aid of the definition of R-boundedness we are able to defineR-sectorial operators by replacing bounded with R-bounded in the definitionof sectorial operators, cf. Definition 3.2.3.

Definition 6.1.5 (R-Sectorial Operator). Let X be a complex Banach space,and assume that A is a sectorial operator in X. Then the operator A is calledR-sectorial if

RA(0) =Rt(t +A)−1 : t > 0 <∞.

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6.1. Preliminaries

The R-angle ϕRA of the operator A is defined by means of

ϕRA = infθ ∈ (0,π) : RA(π −θ) <∞,

where

RA(θ) =Rλ(λ+A)−1 : |argλ| ≤ θ.

The class of R-sectorial operators in X will be denoted by RS(X).

Remark 6.1.6. a) TheR-angle of anR-sectorial operator A is well-definedand it is always not smaller than the spectral angle of A, see Denk,Hieber and Prüss [DHP03, p. 43].

b) A sufficient condition for R-sectoriality is proved by Clément and Prüssin [CP01]. Let X be a Banach space of class HT and let A ∈ BIP (X)with power angle θA. Then A is R-sectorial with ϕRA ≤ θA.

A sectorial operator A is said to admit bounded imaginary powers (ab-breviation: BIP ) if Ais ∈ B(X) for each s ∈ R, and there is a constantC > 0 such that ‖Ais‖B(X) ≤ C for all |s| ≤ 1. The class of such oper-ators is denoted by BIP (X). The power angle θA of A is given byθA = limsup|s|→∞ log‖Ais‖B(X)/ |s|.

c) Denk, Hieber and Prüss proved in [DHP03, Chapter II] that differentialoperators which are elliptic in a special sense and satisfy the Lopatinskii-Shapiro condition on domainsG ⊂R

n with sufficiently smooth boundaryare R-sectorial operators on Lp(G).

The Vector-Valued Bessel-Potential Space Hαp . For an arbitrary Banach

space X of class HT we define the derivation operator of second order B0in Lp(R;X), p ∈ (1,∞), with domain D(B0) =H2

p (R;X) by means of (B0u)(t) =−∂2

t u(t), t ∈ R, u ∈ D(B0). We refer to Prüss [Prü12, p. 226] for the fact thatB0 ∈ BIP (Lp(R;X)) with power angle θB0

= 0.For α ∈ R+ we define Hα

p (R;X) = D(Bα/20 ). Then it is known, that for m ∈N and α ∈ (0,1) we have H2mα

p (R;X) = [Lp(R;X);H2mp (R;X)]α, the complex

interpolation space of order α, cf. [Prü12, p. 226]. For J = [0,T ] or J =R+ we follow Zacher [Zac05, p. 85] and put Hα

p (J ;X) =f |J : f ∈Hα

p (R;X),

endowed with the norm ‖f ‖Hαp (J ;X) = inf‖g‖Hα

p (R;X) : g |J = f . Furthermore,we set 0H

αp (J ;X) = f |J : f ∈ Hα

p (R;X) and suppf ⊂ R+. These spaces arevector-valued Bessel-potential spaces.

For α ∈ (1/p,1) one can prove that these spaces coincide with the definitionof the vector-valued Bessel-potential spaces Hα

p (J ;X) = [Lp(J ;X);H1p (J ;X)]α

and 0Hαp (J ;X) = [Lp(J ;X); 0H

1p (J ;X)]α, respectively; the domains of the α-th

power of the time-derivative ∂t on Lp(J ;X) (with trace zero).

109


Moreover, for α ∈ (0,1) the operator ∂αt with domain D(∂αt ) = 0Hαp (J ;X)

given by

∂αt u = ∂t (g1−α ∗u) , u ∈ 0Hαp (J ;X),

coincides with (∂t)α, the derivation operator of (fractional) order α; ∂αt u is calledthe fractional derivative of u of order α, cf. [Zac05, Example 2.1].

6.2 Maximal Lp-Regularity for Fractional EvolutionEquations

Now, we are in the situation to introduce the concept of maximal Lp-regularityfor linear fractional evolution equation.

Let X be a Banach space of class HT , p ∈ (1,∞), α ∈ (1/p,1) and letf : R+ → X. We assume that the operator A ∈ RS(X) with R-angle ϕRA <π −απ2 . We consider the fractional evolution equation

∂αt u +Au = f , t ∈R+, u(0) = 0, (6.1)

in the space Lp(R+;X), for p ∈ (1,∞). For this purpose, we extend the operatorA in the canonical way to Lp(R+;X) with natural domain Lp(R+;XA).

We denote by J either a compact interval [0,T ], T > 0, or R+. The definitionof maximal Lp-regularity for the equation (6.1) is as follows.

Definition 6.2.1 (Maximal Lp-Regularity). Suppose the operator A : D(A) ⊂X→ X is linear, closed and densely defined. Then the operator A has maximalLp-regularity on J if for each f ∈ Lp(J ;X) there exists a unique solution u ∈0H

αp (J ;X)∩Lp(J ;XA) satisfying the fractional evolution equation (6.1) almost

everywhere in J .

Remark 6.2.2. a) By the closed graph theorem one can show that thereexists some constant CMR > 0 such that for each f ∈ Lp(J ;X0) and theunique solution u of (6.1) we have

‖u‖0H

αp (J ;X)∩Lp(J ;XA) ≤ CMR ‖f ‖Lp(J ;X) .

Note that the constant CMR can be chosen independently of J .

b) Via Sobolev embedding one can prove that the space 0Hαp (J ;X)∩Lp(J ;XA)

is continuously embedded in the space C(J ;Xγ ) with the real interpo-lation space Xγ = (X,XA)1− 1

αp ,p. In particular, there is some constant

Cemb > 0 such that for u ∈ 0Hαp (J ;X)∩Lp(J ;XA) we have

‖u‖C(J ;Xγ ) ≤ Cemb ‖u‖0Hαp (J ;X)∩Lp(J ;XA) ,

where the embedding constant Cemb does not depend of J (here it iscrucial that u is vanishing at t = 0).

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6.2. Maximal Lp-Regularity for Fractional Evolution Equations

We have the following theorem about maximal Lp-regularity for frac-tional evolution equations. This theorem is a special cases of Theorem 3.1.1considering Remark 3.1.1 in [Zac10].

Theorem 6.2.3. Let X be a Banach space of class HT , p ∈ (1,∞) and α ∈ (1/p,1).Let the operator A be an invertible and R-sectorial operator on X with R-angleϕRA < π −α

π2 .

Then the fractional evolution equation (6.1) has a unique solution in the space

0Hαp (R+;X)∩Lp(R+;XA) if and only if f ∈ Lp(R+;X).

Remark 6.2.4. The theorem reads in the same way, if one replaces R+ bysome compact interval J = [0,T ], with T > 0. In this situation the assumptionson the operator A can be weakened. It suffices that the operator µ + A isR-sectorial on X with R-angle ϕRµ+A < π −α

π2 , for some µ ≥ 0.

Now, we consider nontrivial initial values

∂αt (u −u0) +Au = f , t ∈R+, u(0) = u0. (6.2)

Using the notation gα(t) = tα−1/Γ (α), t ∈R+, we are able to rewrite (6.2) as anevolutionary integral equation

u + gα ∗Au = gα ∗ f +u0, t ∈R+. (6.3)

We denote by S(t)t∈R+the resolvent of the Volterra equation (6.3). For the

definition of the resolvent we refer to Section 3.1.For a proof of the following result about the trace space, we refer to Prüss

and Simonett [PS16, Proposition 4.5.14, p.190].

Proposition 6.2.5 (Trace Space). Let X be a Banach space of class HT , p ∈(1,∞), α ∈ (1/p,1), and suppose A ∈ RS(X) is invertible with ϕRA < π −α

π2 . Let

S(t)t∈R+denote the resolvent of (6.3).

Then we have S(·)x ∈Hαp (R+;X)∩Lp(R+;XA) if and only if x ∈ (X,XA)1− 1

αp ,p.

The corresponding map x 7→ S(·)x is continuous between the relevant spaces. Inparticular, we have a constant C1 > 0 such that for all x ∈ (X,XA)1− 1

αp ,pwe have

‖S(·)x‖Hαp (R+;X)∩Lp(R+;XA) ≤ C1 ‖x‖(X,XA)1− 1

αp ,p.

From the preceding results we obtain the theorem about maximal Lp-regularity of fractional evolution equations with non-trivial initial values, cf.Theorem 4.5.15 in [PS16].

Theorem 6.2.6 (Maximal Lp-Regularity). Let p ∈ (1,∞), α ∈ (1/p,1) and X bea Banach space of class HT . Suppose that A ∈ RS(X) is invertible, ϕRA +απ2 < π.Then (6.2) admits a unique solution u ∈ Hα

p (R+;X)∩ Lp(R+;XA) if and only iff ∈ Lp(R+;X) and u0 ∈ (X,XA)1− 1

αp ,p.

111


Remark 6.2.7. a) The maximal Lp-regularity theorem takes the same formif one replaces R+ by an interval [0,T ] with T > 0. In this case one canweaken the assumptions on the operator A; it suffices to assume thatfor some µ ≥ 0 we have µ+A ∈ RS(X) with ϕRµ+A < π −α

π2 , see [Zac03,

Remark 3.1.3].

b) We introduce the following notation

E1(R+) =Hαp (R+;X)∩Lp(R+;XA),

0E1(R+) = 0Hαp (R+;X)∩Lp(R+;XA),

E0(R+) = Lp(R+;X).

In case of a finite interval [0,T ], we write

E1(T ) =Hαp ([0,T ];X)∩Lp([0,T ];XA),

0E1(T ) = 0Hαp ([0,T ];X)∩Lp([0,T ];XA),

E0(T ) = Lp([0,T ];X).

c) Let f ∈ Lp(R+;X) be arbitrary, but fixed. We can decompose the solutionu of (6.2) as u = v+w, where v solves the problem with zero initial value

∂αt v +Av = f , t ∈R+, v(0) = 0,

and w solves (6.2) with zero right hand side

∂αt (w −u0) +Aw = 0, t ∈R+, w(0) = u0.

In particular, the solution w is given as w = S(·)u0, where, as usual,S(t)t∈R+

denotes the resolvent of (6.3). Consequently, ‖w‖E1(R+) can by

estimated with the aid of Proposition 6.2.5. Due to v(0) = 0 we have‖v‖

E1(R+) = ‖v‖0E1(R+) and we can use the maximal Lp-regularity estimate.

Hence,

‖u‖E1(R+) ≤ ‖v‖0E1(R+) + ‖w‖

E1(R+)

≤ CMR ‖f ‖E0(R+) +C1 ‖u0‖(X,XA)1− 1αp ,p

.

These considerations are the same in the case where one replaces R+by a finite interval J = [0,T ], T > 0. Here, all constants can be chosenindependently of J .

For α = 1 we have the known results for classical evolution equations.

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6.2. Maximal Lp-Regularity for Fractional Evolution Equations

Remark 6.2.8. Note that it is possible to consider the property of maximalLp-regularity for a wider class of equations, cf. [Zac03]. But in contrast to theclassical case there are still some basic questions about maximal Lp-regularityopen. For example it is not known what properties of the operator A, suchas sectoriality, are implied by maximal Lp-regularity. Moreover, the relationbetween maximal Lp-regularity on a compact interval [0,T ] and on R+ is notknown. For this thesis these questions are not immediately relevant, since herethe property that a certain operator has maximal Lp-regularity is paramount.But independently of these considerations it is certainly worthwhile to thinkabout the above raised questions.

113

7Local Well-Posedness and MaximalSolutions of Quasilinear ProblemsThis section concerned with the strong solvability of quasilinear fractionalevolution equations. In order to establish local well-posedness, the basic ideais to use a fixed point argument which relies on the maximal Lp-regularity foran associated linear problem, see Theorem 7.1.1.

Afterwards, we explain how to continue a local solution and thus we areable to define the maximal solution of the quasilinear problem. Moreover, wegive a characterisation of the maximal interval of existence, see Section 7.2.


Assumptions. The spaces X0 and X1 are Banach spaces of class HT suchthat X1 → X0 with dense embedding. For p ∈ (1,∞) and α ∈ (1/p,1) let theV be an open subset of the real interpolation space Xγ = (X0,X1)1− 1

αp ,p, and

u0 ∈ V . Moreover, we have maps (A,F) : V →B(X1,X0)×X0. We denote by Jeither a compact interval [0,T ], T > 0, or R+.

We consider abstract quasilinear parabolic problems of the form

∂αt (u −u0) +A(u)u = F(u), t ∈ J, u(0) = u0. (7.1)

We are interested in (local) solutions u of (7.1) with maximal Lp-regularity, i. e.

u ∈Hαp ([0,T ];X0)∩Lp([0,T ];X1).

The trace space of this class of functions is given by Xγ .

7.1 Local Well-Posedness

First of all we prove the local well-posedness of the quasilinear problem (7.1).The following theorem including its proof is analogous to Prüss and Simonett,Theorem 5.1.1, [PS16, Theorem 5.1.1, pp. 195-200].

Theorem 7.1.1 (Local Well-Posedness). Let p ∈ (1,∞), α ∈ (1/p,1) and u0 ∈ Vbe given and suppose that (A,F) ∈ C1−(V ;B(X1,X0) ×X0). Assume in additionthat for some µ ≥ 0 the operator µ+A(u0) ∈ RS(X0) with ϕRµ+A(u0) < π −α

π2 , i. e.

the operator A(u0) has maximal Lp-regularity on a compact interval [0,T∗].

Then there exist T = T (u0) ∈ (0,T∗] and ε = ε(u0) > 0 with BXγε (u0) ⊂ V such

that (7.1) has a unique solution

u(·;u1) ∈Hαp ([0,T ];X0)∩Lp([0,T ];X1)∩C([0,T ];V ),

115

7. Local Well-Posedness and Maximal Solutions of Quasilinear Problems

on [0,T ], for any initial value u1 ∈ BXγε (u0).

Moreover, there exists a constant c = c(u0) > 0 such that for all u1,u2 ∈ BXγε (u0)

the estimate

‖u(·;u1)−u(·;u2)‖Hαp ([0,T ];X)∩Lp([0,T ];X1) ≤ c ‖u1 −u2‖Xγ

is valid.

Proof. This proof is analogous to the proof of [PS16, Thm. 5.1.1, pp. 195-199].Since u0 ∈ V and by Lipschitz continuity of (A,F), there exists some ε0 > 0

and a constant L > 0 such that BXγε0

(u0) ⊂ V and

‖A(w1)v −A(w2)v‖X0≤ L‖w1 −w2‖Xγ ‖v‖X1

, (7.2)

as well as

‖F(w1)−F(w2)‖X0≤ L‖w1 −w2‖Xγ , (7.3)

is valid for all w1,w2 ∈ BXγε0

(u0), v ∈ X1.For T ∈ (0,T∗] we introduce a reference function u∗0 ∈ E1(T ) =Hα

p ([0,T ];X)∩Lp([0,T ];X1) as the solution of the linear problem

∂αt (w −u0) +A(u0)w = 0, t ∈ [0,T ], w(0) = u0. (7.4)

Observe that by the assumption, the operator µ+A(u0) ∈ RS(X0) withR-angleϕRµ+A(u0) < π −α

π2 , the corresponding Volterra equation with standard kernel

u + (gα ∗ [µ+A(u0)]u) = u0, t ∈ [0,T ],

is parabolic and therefore admits a bounded resolvent family Sµ+A(u0)(t)t∈R+

on X0, cf. Theorem 3.3.1. Using a perturbation result for parabolic Volterraequation, see [Prü12, Theorem 3.2], with the term −µ(gα ∗ u), we know thatthe linear Volterra equation

u + gα ∗A(u0)u = u0, t ∈ [0,T ],

also admits a resolvent family S(t)t∈R+= SA(u0)(t)t∈R+

⊂ B(X0). Thus,thesolution of the linear reference problem (7.4) is given by u∗0 = S(·)u0.

The strong continuity on R+, together with the uniform boundedness prin-ciple, implies that there is a constant M0 > 0 such that supt∈[0,T∗] ‖S(t)‖B(X0) ≤M0 <∞. Moreover, the resolvent family S(t)t∈R+

considered as operator fam-ily on X1 is also strongly continuous on R+, and hence there is some constantM1 > 0 such that supt∈[0,T∗] ‖S(t)‖B(X1) ≤M1 <∞. Via the properties of the realinterpolation space Xγ , we deduce that there is some constant Mγ > 0 suchthat supt∈[0,T∗] ‖S(t)‖B(Xγ ) ≤Mγ <∞.

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7.1. Local Well-Posedness

Let u1 ∈ BXγε (u0) with ε ∈ (0, ε0] be given. For r ∈ (0,1] we define the set

Br,T ,u1⊂ E1(T ) by

Br,T ,u1= v ∈ E1(T ) : v(0) = u1 and

∥∥∥v −u∗0∥∥∥E1(T )≤ r.

Note that Br,T ,u1is a non-empty complete metric space if it is endowed with

the E1(T )-norm. We will show that for all v ∈Br,T ,u1we have v(t) ∈ BXγε0

(u0) forall t ∈ [0,T ], provided that r, T , ε > 0 are sufficiently small. For this purposewe define u∗1 ∈ E1(T ) as the unique solution of

∂αt (w −u1) +A(u0)w = 0, t ∈ [0,T ], w(0) = u1.

Note that u∗1 = S(·)u1. Given v ∈Br,T ,u1we estimate as follows

‖v −u0‖C([0,T ];Xγ ) ≤∥∥∥v −u∗1∥∥∥C([0,T ];Xγ )

+∥∥∥u∗1 −u∗0∥∥∥C([0,T ];Xγ )

+∥∥∥u∗0 −u0

∥∥∥C([0,T ];Xγ )

.

Since u0 is fixed, there exists a number T0 = T0(u0) ∈ (0,T∗] such that∥∥∥u∗0 −u0

∥∥∥C([0,T0];Xγ )

= supt∈[0,T0]

∥∥∥u∗0(t)−u0

∥∥∥Xγ≤ ε0/3,

since the resolvent family is strongly continuous on R+, even considered asoperator family on the space Xγ . Observe that v(0)−u∗1(0) = u1 −u1 = 0, andvia the continuous embedding of 0E1(T ) = 0H

αp ([0,T ];X0)∩Lp([0,T ];X1) →

C([0,T ];Xγ ) we have∥∥∥v −u∗1∥∥∥C([0,T ];Xγ )≤ Cemb

∥∥∥v −u∗1∥∥∥0E1(T )

,

with the constant Cemb > 0 being independent of T > 0. Therefore, we obtainvia the triangle inequality∥∥∥v −u∗1∥∥∥C([0,T ];Xγ )

≤ Cemb(∥∥∥v −u∗0∥∥∥E1(T )

+∥∥∥u∗0 −u∗1∥∥∥E1(T )

),

and with the definition of v ∈Br,T ,u1it follows that∥∥∥v −u∗1∥∥∥C([0,T ];Xγ )

≤ Cemb(r +

∥∥∥u∗0 −u∗1∥∥∥E1(T )

).

As mentioned above, the assumptions on the operator µ+A(u0) ensure thatsupt∈[0,T∗] ‖S(t)‖B(Xγ ) ≤ Mγ , with some constant Mγ > 0. So, we get withu∗0 = S(·)u0 and u∗1 = S(·)u1 that∥∥∥u∗0 −u∗1∥∥∥C([0,T ];Xγ )

= supt∈[0,T ]

‖S(t)(u0 −u1)‖Xγ ≤Mγ ‖u0 −u1‖Xγ ,

117


since T ∈ (0,T∗]. Via the trace space result from Proposition 6.2.5 we knowthat there is some constant C1 > 0 which is independent of T > 0 such that∥∥∥u∗0 −u∗1∥∥∥E1(T )

≤ C1 ‖u0 −u1‖Xγ .

Hence, we have with Cγ = maxMγ ,CembC1 that∥∥∥u∗0 −u∗1∥∥∥C([0,T ];Xγ )+Cemb

∥∥∥u∗0 −u∗1∥∥∥E1(T )≤ Cγ ‖u0 −u1‖Xγ . (7.5)

For ε ≤ ε0/(3Cγ ), r ≤ ε0/(3Cemb) and T ∈ [0,T0], we obtain

‖v −u0‖C([0,T ];Xγ ) ≤ Cembr +Cγε+∥∥∥u∗0 −u0

∥∥∥C([0,T ];Xγ )

≤ ε0. (7.6)

Throughout the remaining part of this proof we will assume that u1 ∈ BXγε (u0)

with ε ≤ ε0/(3Cγ ), T ∈ (0,T0], and r ≤ ε0/(3Cemb). Under these assumptions,we may define a mapping

Tu1: Br,T ,u1

→ E1(T ), v 7→ Tu1v = u,

where u is the unique solution of the linear problem

∂αt (u −u1) +A(u0)u = F(v) + (A(u0)−A(v))v, t ∈ [0,T ], u(0) = u1.

In order to apply the contraction mapping principle, firstly, we show thatTu1

(Br,T ,u1) ⊂Br,T ,u1

, and then that Tu1defines a strict contraction on Br,T ,u1

,i. e. there is some κ ∈ (0,1) such that for all v, v ∈Br,T ,u1

we have∥∥∥Tu1v −Tu1

v∥∥∥E1(T )

≤ κ ‖v − v‖E1(T ) .

Self-mapping. At first we take care of the self-mapping property. Foreach v ∈Br,T ,u1

we know that Tu1v −u∗1 solves the equation

∂αt w+A(u0)w = F(v) + (A(u0)−A(v))v, t ∈R+, w(0) = 0.

The property of maximal Lp-regularity on the compact interval [0,T∗] of theoperator A(u0) implies the estimate∥∥∥Tu1

v −u∗1∥∥∥E1(T )

≤ CMR ‖F(v) + (A(u0)−A(v))v‖E0(T ) ,

and the constant CMR > 0 is independent of T ∈ (0,T∗]. Now, we estimate(A(u0)−A(v))v in E0(T ). Using the Lipschitz continuity (7.2) we obtain that

118


‖(A(u0)−A(v))v‖E0(T ) ≤ L

(∫ T

0‖u0 − v(t)‖pXγ ‖v(t)‖pX1

dt

)1/p

≤ L‖u0 − v‖C([0,T ];Xγ ) ‖v‖E1(T ) .

In view of the definition of v ∈ Br,T ,u1we have ‖v‖

E1(T ) ≤ r +∥∥∥u∗0∥∥∥E1(T )

, andhence

‖(A(u0)−A(v))v‖E0(T ) ≤ L‖u0 − v‖C([0,T ];Xγ )

(r +

∥∥∥u∗0∥∥∥E1(T )

).

Next we use the assumed Lipschitz continuity of the map F from (7.3) toestimate

‖F(v)‖E0(T ) ≤ ‖F(v)−F(u0)‖

E0(T ) + ‖F(u0)‖E0(T )

≤ T 1/p(L‖v −u0‖C([0,T ];Xγ ) + ‖F(u0)‖X0

).

Combining these estimates we have∥∥∥Tu1v −u∗0

∥∥∥E1(T )

≤∥∥∥u∗1 −u∗0∥∥∥E1(T )

+∥∥∥Tu1

v −u∗1∥∥∥E1(T )

≤∥∥∥u∗1 −u∗0∥∥∥E1(T )

+CMR(‖F(v)‖

E0(T ) + ‖(A(u0)−A(v))v‖E0(T )

)≤

∥∥∥u∗1 −u∗0∥∥∥E1(T )+CMRT

1/p(L‖v −u0‖C([0,T ];Xγ ) + ‖F(u0)‖X0

)+CMRL‖u0 − v‖C([0,T ];Xγ )

(r +

∥∥∥u∗0∥∥∥E1(T )

).

One easily verifies that∥∥∥u∗0 −u0

∥∥∥C([0,T ];Xγ )

,∥∥∥u∗0∥∥∥E1(T )

tend to zero as T → 0.

Therefore we get for sufficiently small r > 0, T > 0 and ε > 0 that∥∥∥Tu1v −u∗0

∥∥∥E1(T )

≤∥∥∥u∗1 −u∗0∥∥∥E1(T )

+ r/2,

here we employ the estimate ‖v −u0‖C([0,T ];Xγ ) ≤ Cembr+Cγε+∥∥∥u∗0 −u0

∥∥∥C([0,T ];Xγ )

from (7.6). With the estimate (7.5) we get∥∥∥u∗0 −u∗1∥∥∥E1(T )≤ (Cγ /Cemb)‖u0 −u1‖Xγ .

The preceding estimates imply that∥∥∥Tu1v −u∗0

∥∥∥E1(T )

≤ (Cγ /Cemb)‖u1 −u0‖Xγ + r/2 ≤ r/2 + r/2 = r,

with a possible smaller ε > 0. This proves the self-mapping property of Tu1on

Br,T ,u1.

119


Contraction. Now, we turn to the contraction property. Let u1,u2 ∈BXγε0

(u0) be arbitrary but fixed as well as v1 ∈ Br,T ,u1and v2 ∈ Br,T ,u2

. Thismeans, that Tuivi is the solution of the equation

∂αt (u −ui) +A(u0)u = F(vi) + (A(u0)−A(vi))vi , t ∈ [0,T ], u(0) = ui ,

for i ∈ 1,2. One convinces oneself that Tuivi − S(·)ui solves the equation

∂αt u +A(u0)u = F(vi) + (A(u0)−A(vi))vi , t ∈ [0,T ], u(0) = 0,

for i ∈ 1,2. Due to the property of maximal Lp-regularity of the operatorA(u0) on the compact interval [0,T∗] we have∥∥∥Tu1

v1 −Tu2v2

∥∥∥E1(T )

≤‖S(·)(u1 −u2)‖E1(T ) +CMR ‖F(v1)−F(v2)‖

E0(T ) (7.7)

+CMR ‖(A(u0)−A(v1))v1 − (A(u0)−A(v2))v2‖E0(T ) .

The first term on the right-hand side of equation (7.7) can be estimated byProposition 6.2.5, thereby we obtain that

‖S(·)(u1 −u2)‖E1(T ) ≤ C1 ‖u1 −u2‖Xγ , (7.8)

with some constant C1 > 0 independent of T > 0. For the second term on theright-hand side of (7.7) we use the Lipschitz continuity of the map F from(7.3) and get

‖F(v1)−F(v2)‖E0(T ) ≤ L

(∫ T

0‖v1(t)− v1(t)‖pXγ dt

)1/p

≤ LT 1/p ‖v1 − v2‖C([0,T ];Xγ ) . (7.9)

Moreover, since v1(0)−u1 = 0 as well as v2(0)−u2 = 0 and by the boundednessof the resolvent family S(t)t∈R+

on Xγ we deduce that

‖v1 − v2‖C([0,T ];Xγ ) ≤ ‖v1 − v2 − S(·)(u1 −u2)‖C([0,T ];Xγ ) + ‖S(·)(u1 −u2)‖C([0,T ];Xγ )

≤ Cemb ‖v1 − v2 − S(·)(u1 −u2)‖0E1(T ) +Mγ ‖u1 −u2‖Xγ ,

with T -independent constants Mγ and Cemb. By the triangle inequality andthe estimate (7.8) it follows that

‖v1 − v2‖C([0,T ];Xγ ) ≤ Cemb(‖v1 − v2‖E1(T ) +C1 ‖u1 −u2‖Xγ ) +Mγ ‖u1 −u2‖Xγ= Cemb ‖v1 − v2‖E1(T ) + 2Cγ ‖u1 −u2‖Xγ , (7.10)

with Cγ = maxMγ ,CembC1 like above. Together with (7.9) we have

‖F(v1)−F(v2)‖E0(T ) ≤ LT

1/p(Cemb ‖v1 − v2‖E1(T ) + 2Cγ ‖u1 −u2‖Xγ

).

120


For the remaining term in (7.7) we use the Lipschitz continuity of the map Afrom (7.2):

‖(A(u0)−A(v1))v1 − (A(u0)−A(v2))v2‖E0(T )

≤ ‖(A(v1)−A(u0)) (v1 − v2)‖E0(T ) + ‖(A(v1)−A(v2))v2‖E0(T )

≤ L(‖v1 −u0‖C([0,T ];Xγ ) ‖v1 − v2‖E1(T ) + ‖v1 − v2‖C([0,T ];Xγ ) ‖v2‖E1(T )

).

As in (7.6), we can made the term ‖v1 −u0‖C([0,T ];Xγ ) as small as we wish bychoosing r > 0, T ∈ (0,T∗] and ε > 0 sufficiently small. By the definition ofv2 ∈Br,T ,u2

we have

‖v2‖E1(T ) ≤∥∥∥v2 −u∗0

∥∥∥E1(T )

+∥∥∥u∗0∥∥∥E1(T )

≤ r +∥∥∥u∗0∥∥∥E1(T )

,

hence ‖v2‖E1(T ) is as small as we want, provided r > 0 and T ∈ (0,T∗] are smallenough. Lastly, the term ‖v1 − v2‖C([0,T ];Xγ ) can be estimated like in (7.10). Insummary, if we choose r > 0, T > 0 and ε > 0 sufficiently small, we obtain

∥∥∥Tu1v1 −Tu2

v2

∥∥∥E1(T )

≤ C1 ‖u1 −u2‖Xγ+CMRLT

1/p(C1 ‖v1 − v2‖E1(T ) + 2Cγ ‖u1 −u2‖Xγ

)+CMRL

(‖v1 −u0‖C([0,T ];Xγ ) ‖v1 − v2‖E1(T )

+(Cemb ‖v1 − v2‖E1(T ) + 2Cγ ‖u1 −u2‖Xγ )‖v2‖E1(T )

)≤ 1

2‖v1 − v2‖E1(T ) + c ‖u1 −u2‖Xγ ,

with some constant c = c(u0) > 0, for all u1,u2 ∈ BXγε (u0) and all v1 ∈ Br,T ,u1

and all v2 ∈ Br,T ,u2. In the very special case u1 = u2 we get the contraction

mapping property of Tu1on Br,T ,u1

.

Now we are in position to apply Banach’s fixed point theorem to obtaina unique fixed point u ∈ Br,T ,u1

of Tu1, i. e. Tu1

u = u. Therefore u ∈ Br,T ,u1is

the unique local solution of (7.1). Furthermore, if u(·;u1) and u(·;u2) denote

the solutions of (7.1) with initial values u1,u2 ∈ BXγε (u0), respectively, the last

assertion of the theorem follows from the last estimate above. This completesthe proof.

121


7.2 Maximal Solutions

In this section we show that a local solution u ∈ E1(τ)∩C([0, τ];Xγ ) of thequasilinear parabolic fractional evolution equation (7.1) can be continued toa solution u ∈ E1(τ + δ)∩C([0, τ + δ];Xγ ) with some sufficiently small δ > 0,where u restricted to the interval [0, τ] is equal to u. On the basis of thiscontinuation we are able to define a maximal solution for the quasilinearproblem (7.1).

Lemma 7.2.1. Let p ∈ (1,∞) and α ∈ (1/p,1) be given and suppose that (A,F) ∈C1−(V ;B(X1,X0)×X0). Assume that for all v ∈ V there is some µ = µ(v) ≥ 0 suchthat µ+A(v) ∈ RS(X0) with ϕRµ+A(v) < π−α

π2 , i. e. the operator A(v) has maximal

Lp-regularity on each compact interval [0,T∗]. Let u ∈ E1(τ)∩C([0, τ] V ) be alocal unique solution of the quasilinear problem (7.1) with u0 ∈ V .

Then there exists some δ > 0 such that there is a unique solution of maximalLp-regularity u ∈ E1(τ + δ)∩C([0, τ + δ];V ) of the quasilinear problem (7.1) andu(t) = u(t) for all t ∈ [0, τ].

Proof. Let u0 ∈ V be arbitrary, but fixed. By the Local Well-Posedness Theorem7.1.1 there is a unique solution u ∈ E1(τ)∩C([0, τ],V ) on the interval [0, τ],for some τ > 0. This solution can be continued to some larger interval [0, τ +δ] ⊂ [0, τ + 1]. Indeed, for uτ = u(τ) ∈ V we introduce a reference functionw ∈ E1(τ + 1)∩C([0, τ + 1];Xγ ) as the solution of the linear problem

∂αt (w −u0) +A(uτ )w = (A(uτ )−A(u))u +F(u)χ[0,τ], t ∈ [0, τ + 1],

withw(0) = u0. Obviously, we havew = u on [0, τ] by uniqueness. In particular,w(τ) = uτ . By the continuity of w in Xγ it follows that for all η > 0 there issome r = r(η) ∈ (0,1] such that for all δ ∈ (0, r] we have

‖w −uτ‖C([τ,τ+δ];Xγ ) = supt∈[τ,τ+δ]

‖w(t)−uτ‖Xγ ≤ η. (7.11)

Since V is an open set with uτ ∈ V there is some ε0 > 0 such BXγε0

(uτ ) ⊂ V . Wechoose the parameter η = ε0/2 according to the relation (7.11) and fix thecorresponding parameter r = r(ε0/2) ∈ (0,1]. We denote by Cemb the uniformembedding constant for 0E1(T ) → C([0,T ];Xγ ), with T > 0. We consider forδ ∈ (0, r] and ρ ∈ (0, ε0/(2Cemb)] the set

B(τ,δ,ρ) = v ∈ E1(τ + δ) : v(t) = u(t) for all t ∈ [0, τ] and ‖v −w‖0E1(τ+δ) ≤ ρ.

Since w = u on [0, τ] by uniqueness, the setB(τ,δ,ρ) is not empty and itbecomes a complete metric space when endowed with the metric induced bythe norm of E1(τ + δ).

122

7.2. Maximal Solutions

Firstly, we will show that for all δ ∈ (0, r] and ρ ∈ (0, ε0/(2Cemb)] we havefor v ∈B(τ,δ,ρ) that ‖v −uτ‖C([τ,τ+δ];Xγ ) ≤ ε0. In fact,

‖v −uτ‖C([τ,τ+δ];Xγ ) ≤ ‖v −w‖C([0,τ+δ];Xγ ) + ‖w −uτ‖C([τ,τ+δ];Xγ )

≤ Cemb ‖v −w‖0E1(τ+δ) + ε0/2

≤ Cemb ρ+ ε0/2

≤ ε0.

In particular, v(t) ∈ BXγε0(uτ ) ⊂ V for all t ∈ [τ,τ + δ] and hence, A(v(t)) and

F(v(t)) are well-defined for all t ∈ [0, τ + δ].Next, we define the mapping G : B(τ,δ,ρ)→ E1(τ + δ), which assigns to

v ∈B(τ,δ,ρ) the solution u = G(v) of the linear problem

∂αt (u −u0) +A(uτ )u = (A(uτ )−A(v))v +F(v), t ∈ [0, τ + δ], (7.12)

with u(0) = u0. Since v = u on [0, τ], we have also u = u on [0, τ], by uniqueness.We will prove that G is a self-mapping of the set B(τ,δ,ρ) and a contractionfor sufficiently small parameters.

Self-mapping. To prove the self-mapping property we have to estimate‖u −w‖

0E1(τ+δ). The difference u −w satisfies on [0, τ + δ] the equation

∂αt (u −w) +A(uτ )(u −w) = (A(uτ )−A(v))v +F(v)χ[τ,τ+δ],

with (u −w)(0) = 0. Via the maximal Lp-regularity property of the operatorA(uτ ) on the interval [0, τ + 1] it follows that

‖u −w‖0E1(τ+δ) ≤ C

A(uτ )MR

‖(A(uτ )−A(v))v‖Lp([τ,τ+δ];X0) + ‖F(v)‖Lp([τ,τ+δ];X0)

.

Using the Lipschitz continuity of the maps A and F on the ball BXγε0

(uτ ) weobtain that

‖(A(uτ )−A(v))v‖Lp([τ,τ+δ];X0)

≤ L(∫ τ+δ

τ‖uτ − v(t)‖pXγ ‖v(t)‖pX1

dt

)1/p

≤ L‖uτ − v‖C([τ,τ+δ];Xγ )

(∫ τ+δ

τ‖v(t)‖pX1

dt

)1/p

≤ L[Cembρ+ ‖w −uτ‖C([τ,τ+δ];Xγ )

] (ρ+ ‖w‖Lp([τ,τ+δ];X1)

),

as well as

‖F(v)‖Lp([τ,τ+δ];X0) ≤ δ1/p

(L‖uτ − v‖C([τ,τ+δ];Xγ ) + ‖F(uτ )‖X0

)≤ δ1/p

(L[Cembρ+ ‖w −uτ‖C([τ,τ+δ];Xγ )

]+ ‖F(uτ )‖X0

).

Since ‖uτ −w‖C([τ,τ+δ];Xγ ), as well as ‖w‖Lp([τ,τ+δ];X1) tends to zero as δ→ 0 it

123


follows that for sufficiently small ρ > 0 we have

‖u −w‖0E1(τ+δ) ≤ ρ.

This proves the self-mapping property of the map G.

Contraction. To prove the contraction property of the map G, we definefor v1,v2 ∈ B(τ,δ,ρ) the corresponding solutions of the equation (7.12) de-noted by u1 = G(v1), u2 = G(v2), like above. This implies for the differenceu1 − u2 on [0, τ + δ] the equation

∂αt (u1 − u2) +A(uτ )(u1 − u2) = (A(uτ )−A(v1)v1 − (A(uτ )−A(v2))v2

+F(v1)−F(v2)χ[τ,τ+δ],

with (u1 − u2)(0) = 0. By means of the maximal Lp-regularity property on theinterval [0, τ + 1] of the operator A(uτ ) we get the estimate

‖u1 − u2‖0E1(δ+τ) ≤CA(uτ )MR

‖(A(uτ )−A(v1))v1 + (A(uτ )−A(v2))v2‖Lp([τ,τ+δ];X0)

+‖F(v1)−F(v2)‖Lp([τ,τ+δ];X0)

.

Using the Lipschitz estimate from (7.2) we obtain

‖(A(uτ )−A(v1))v1 + (A(uτ )−A(v2))v2‖Lp([τ,τ+δ];X0)

≤ ‖(A(v1)−A(uτ ))(v1 − v2)‖Lp([τ,τ+δ];X0) + ‖(A(v1)−A(v2))v2‖Lp([τ,τ+δ];X0)

≤ L(‖v1 −uτ‖C([τ,τ+δ];Xγ ) ‖v1 − v2‖Lp([τ,τ+δ];X1)

+‖v1 − v2‖C([τ,τ+δ];Xγ ) ‖v2‖Lp([τ,τ+δ];X1)

)≤ L

([Cembρ+ ‖w −uτ‖C([τ,τ+δ];Xγ )

]‖v1 − v2‖Lp([τ,τ+δ];X1)

+‖v1 − v2‖C([τ,τ+δ];Xγ )

[ρ+ ‖w‖Lp([τ,τ+δ];X1)

]).

Furthermore, with the Lipschitz estimate from (7.3) we get

‖F(v1)−F(v2)‖Lp([τ,τ+δ];X0) ≤ Lδ1/p ‖v1 − v2‖C([τ,τ+δ];Xγ ) .

We use the continuous embedding 0E1(τ + δ) → C([0, τ + δ];Xγ ) to estimatethe term ‖v1 − v2‖C([τ,τ+δ];Xγ ) ≤ Cemb ‖v1 − v2‖0E1(τ+δ). This yields the estimate

‖u1 − u2‖0E1(τ+δ)

≤ CA(uτ )MR L

[Cembρ+ ‖w −uτ‖C([τ,τ+δ];Xγ )

]+Cemb

([ρ+ ‖w‖Lp([τ,τ+δ];X1)

]+ δ1/p

)‖v1 − v2‖0E1(τ+δ) .

124

7.2. Maximal Solutions

Since ‖w −uτ‖C([τ,τ+δ];Xγ ) and ‖w‖Lp([τ,τ+δ];X1) tend to zero as δ→ 0, the con-traction property follows.

Thus, we can use the contraction mapping principle to show that there isa unique fixed point u ∈ E1(τ + δ)∩C([0, τ + δ];V ) on [0, τ + δ]. This solutioncoincides with u on [0, τ]. Thus, u is the desired continuation of the localsolution u.

We call u defined on an interval J maximal solution if there does not exista solution v on an interval J ′ strictly containing J such that v restricted to Jequals u. If u is a maximal solution on J , then J is called the maximal intervalof existence, and J = [0, τ(u0)) where

τ(u0) = supT > 0: u ∈ E1(T )∩C([0,T ];V ) is solution of (7.1).

Lemma 7.2.2. Let p ∈ (1,∞) and α ∈ (1/p,1) be given and suppose that (A,F) ∈C1−(V ;B(X1,X0)×X0). Assume that for all v ∈ V there is some µ = µ(v) ≥ 0 suchthat µ+A(v) ∈ RS(X0) with ϕRµ+A(v) < π−α

π2 , i. e. the operator A(v) has maximal

Lp-regularity on any compact interval [0,T∗].Then the quasilinear problem (7.1) with initial value u0 ∈ Xγ has a unique

solution u on a maximal interval of existence [0, τ(u0)), which is characterised byone of the following options.

(i) τ(u0) =∞, i. e. global existence;

(ii) τ(u0) <∞ and liminft→τ(u0) distXγ (u(t),∂V ) = 0;

(iii) τ(u0) <∞ and limt→τ(u0)−u(t) does not exists in Xγ .

Proof. Suppose τ(u0) < ∞, distXγ (u(t),∂V ) ≥ η, for some η > 0 and all t ∈[0, τ(u0)) and assume that the solution u converges in Xγ to some u1 ∈ V ,u1 = limt→τ(u0)u(t). Thus, there is a continuous extension u on [0, τ(u0)] withu = u on [0, τ(u0)) and u(τ(u0)) = u1.

Let t ∈ [0, τ(u0)). Analogous to the preceding proof there is some δ =δ(u(t)) > 0 such that we can continue the solution u from [0, t] to [0, t + δ].By uniqueness, on [t, τ(u0)) this is of course the solution we already have.The parameter δ can be chosen independently of u(t). In fact, the set K =u([0, τ(u0)]) is a compact set in Xγ . Thus, for sufficiently small ε > 0 there is

some N =N (ε) ∈N and x1, . . . ,xN ∈ K such that K ⊂⋃Ni=1B

Xγε (xi) ⊂ V . We fix

i ∈ 1, . . . ,N (ε) with ‖u(t)− xi‖Xγ ≤ ε. We follow the arguments in the proof ofthe above Lemma 7.2.1 and replace uτ by xi . By a perturbation argument, allestimates are valid for sufficiently small parameters ε0, ρ and δ, which can bechosen independently of u(t). The solution may be continued past τ(u0) (taket sufficiently close to τ(u0)) and it follows a contradiction.

125

8Stability for Quasilinear FractionalEvolution EquationsAn essential step in our proof of the stability result for quasilinear fractionalevolution equations, Theorem 8.2.3, is the time-weighting and rewriting ofthe equation in a way similar to the procedure for the semilinear Volterraequation in Part II. To achieve this we will first introduce a product rule forfractional derivatives. Here special attention will be paid to the occurringcommutator term. Subsequently, for two different time-weighting functionswe prove Lp-estimates on R+ for the commutator term, see Lemma 8.1.2 andLemma 8.1.6.

On the basis of these considerations we are able to prove with the aid ofmaximal Lp-regularity of the linearised equation a stability result for quasilin-ear fractional evolution equations, Theorem 8.2.3. We can prove asymptoticstability with the same regularity assumptions as in the classical case. Butto prove the optimal asymptotic decay behaviour we need an additionalregularity assumption on the non-linearities.

Finally, we apply our stability result to an example, see Section 8.3.

8.1 Commutator Term Estimates

We start with a product rule for fractional derivatives. Afterwards, we intro-duce the so-called commutator term which is motivated by the second termin the product rule. For two special choices of the time-weighting functionwe prove Lp-estimates on R+, see Lemma 8.1.2 and Lemma 8.1.6.

8.1.1 Product Rule

Like our approach for the semilinear problem in Part II, we want to rewriteour equation after multiplication with some function. The following helpfullemma concerning a product rule for fractional derivatives is taken from[KZ15, Lemma 2.3].

Lemma 8.1.1 (Product Rule). Let J = [0,T ] with T > 0 be a compact intervaland p ∈ (1,∞), α ∈ (1/p,1) and ε ∈ (0,1 − α) be given. Furthermore, let X be aBanach space of class HT , v ∈ 0H

αp (J ;X) and ϕ ∈ Cα+ε(J ;B(X)).

Then, we have ϕv ∈ 0Hαp (J ;X) and for all t ∈ J that

∂αt (ϕv)(t) = ϕ(t)∂αt v(t) +∫ t

0[−g1−α(t − τ)] (ϕ(t)−ϕ(τ))v(τ)dτ.

127

8. Stability for Quasilinear Fractional Evolution Equations

8.1.2 Lp-Estimates for the Commutator Term

Let p ∈ (0,∞) and α ∈ (1/p,1) be arbitrary, but fixed. We are interested inLp-estimates for the so-called commutator term for t ∈R+ given by∫ t

0[−g1−α(t − τ)]

ϕ(t)−ϕ(τ)ϕ(τ)

dτ,

with two special choices for ϕ.

Lemma 8.1.2. Let ξ ∈ (0,α −1/p) and ε0 > 0. We define the map ϕε0: R+→R+

by

ϕε0(t) = 1 + ε0

t

1 + t1−ξ, t ∈R+.

Then, there are constants C(α,p,ξ) > 0 and M(α,p,ξ) > 0 which are independentof ε0 such that ∥∥∥t 7→ g1−α(t)[ϕε0

(t)− 1]∥∥∥Lp(R+)

≤ ε0M(α,p,ξ),

as well as∥∥∥∥∥∥t 7→∫ t

0[−g1−α(t − τ)]

ϕε0(t)−ϕε0

(τ)ϕε0

(τ)dτ

∥∥∥∥∥∥Lp(R+)

≤ ε0C(α,p,ξ).

Proof. Obviously, the map ϕε0is continuously differentiable on R+. Consider-

ing the derivative of ϕε0it is easy to see that ϕε0

is strictly increasing on R+,since

ϕε0(t) = ε0

1 + ξt1−ξ

(1 + t1−ξ )2> 0.

We have by the definition of the standard kernel g1−α that(∫ ∞0

∣∣∣g1−α(t)[ϕε0(t)− 1]

∣∣∣p dt)1/p

=ε0

Γ (1−α)

(∫ ∞0

∣∣∣∣∣ 1tα

t

1 + t1−ξ

∣∣∣∣∣p dt)1/p

≤ ε0

Γ (1−α)

(∫ 1

0

∣∣∣t1−α∣∣∣p dt)1/p

+(∫ ∞

1

∣∣∣t−α+ξ∣∣∣p dt)1/p

=

ε0

Γ (1−α)

[((1−α)p+ 1)−1/p + ((α − ξ)p − 1)−1/p

]= ε0M(α,p,ξ),

128

8.1. Commutator Term Estimates

and hence[t 7→ g1−α(t)[ϕε0

(t)− 1]]∈ Lp(R+). Next, we consider the map

t 7→∫ t

0[−g1−α(t − τ)]

ϕε0(t)−ϕε0

(τ)ϕε0

(τ)dτ.

At first, we estimate this integral for t ≤ 1. We have ϕε0(t) ≥ 1, t ∈R+, hence∫ t

0[−g1−α(t − τ)]

ϕε0(t)−ϕε0

(τ)ϕε0

(τ)dτ ≤

∫ t

0[−g1−α(t − τ)]

[ϕε0

(t)−ϕε0(τ)

]dτ.

For each t ∈R+ we have the trivial estimate ϕε0(t) ≤ ε0(1 + ξt1−ξ), and hence

for t ∈ [0,1] we obtain ϕε0(t) ≤ 2ε0. By the mean value theorem there is some

ν ∈ (τ, t) ⊂ [0,1] such that ϕε0(t) −ϕε0

(τ) = ϕε0(ν)(t − τ), and so we have the

uniform estimate ϕε0(t)−ϕε0

(τ) ≤ 2ε0(t − τ). Hence,∫ t

0[−g1−α(t − τ)]

ϕε0(t)−ϕε0

(τ)ϕε0

(τ)dτ ≤ 2ε0

∫ t

0[−g1−α(t − τ)] (t − τ)dτ

= 2ε0

∫ t

0

αs−α

Γ (1−α)ds

= ε0C1(α)t1−α . (8.1)

To estimate for t ≥ 1 the integral∫ t

0[−g1−α(t − τ)]

ϕε0(t)−ϕε0

(τ)ϕε0

(τ)dτ, (8.2)

we split the integral at t/2 and estimate the summands separately.For the integral (8.2) from 0 to t/2 we use on one hand the completely

monotonicity of the standard kernel and get for all τ ∈ (0, t/2) that −g1−α(t −τ) ≤ −g1−α(t/2) = (t/2)−1−αα/Γ (1−α). On the other hand we use monotonicityof ϕε0

(t) ≥ 1, t ∈R+, to obtain that

ϕε0(t)−ϕε0

(τ)ϕε0

(τ)≤ ϕε0

(t)− 1 = ε0t

1 + t1−ξ≤ ε0t

ξ .

So, we get∫ t/2

0[−g1−α(t − τ)]

ϕε0(t)−ϕε0

(τ)ϕε0

(τ)dτ ≤

∫ t/2

0

( t2

)−1−α αΓ (1−α)

ε0tξ dτ

= ε0C2(α)t−(α−ξ).

To estimate the integral (8.2) from t/2 to t we use again the mean valuetheorem. Thus, there is some ν ∈ (τ, t) ⊂ (t/2, t) such that ϕε0

(t) −ϕε0(τ) ≤

129


ϕε0(ν)(t − τ). Due to the monotonicity of the enumerator and denominator we

have for ν ∈ (t/2, t) that

ϕε0(ν) = ε0

1 + ξν1−ξ

(1 + ν1−ξ )2≤ ε0

1 + ξt1−ξ

(1 + (t/2)1−ξ )2≤ ε023−2ξt−(1−ξ).

Using this estimate we deduce that∫ t

t/2[−g1−α(t − τ)]

ϕε0(t)−ϕε0

(τ)ϕε0

(τ)dτ

≤ ε023−2ξt−(1−ξ)∫ t

t/2[−g1−α(t − τ)] (t − τ)dτ

= ε023−2ξt−(1−ξ)∫ t

0

αs−α

Γ (1−α)ds

= ε0C3(α,ξ)t−(α−ξ).

Combining these both estimates we obtain for t ≥ 1 that∫ t

0[−g1−α(t − τ)]

ϕε0(t)−ϕε0

(τ)ϕε0

(τ)dτ ≤ ε0C4(α,ξ)t−(α−ξ),

with C4(α,ξ) = maxC2(α),C3(α,ξ). Together with the estimate (8.1) for smallt ≤ 1 we thus have∫ t

0[−g1−α(t − τ)]

ϕε0(t)−ϕε0

(τ)ϕε0

(τ)dτ ≤ ε0C0(α,ξ)min

t1−α , t−(α−ξ)

,

with C0(α,ξ) = max C4(α,ξ),C1(α). In particular, we have∥∥∥∥∥∥t 7→∫ t

0[−g1−α(t − τ)]

ϕε0(t)−ϕε0

(τ)ϕε0

(τ)dτ

∥∥∥∥∥∥Lp(R+)

≤ ε0C(α,p,ξ),

with C(α,p,ξ) = C0(α,ξ)[(p(1−α) + 1)−1/p + (p(α − ξ)− 1)−1/p

].

Now, we are interested in Lp-estimates for the commutator term in the caseϕε = 1/sε, ε > 0, where sε(t) = Eα(−εtα), t ∈R+ denotes the scalar resolvent forthe scalar Volterra equation with standard kernel, see Section 2.3.3. In thissituation Lemma 3.5.1 yields for all t ∈ (0,∞) that∫ t

0[−g1−α(t − τ)]

ϕε(t)−ϕε(τ)ϕε(τ)

dτ =∫ t

0[−g1−α(t − τ)]

[1sε(t)

− 1sε(τ)

]sε(τ)dτ

= ε+ g1−α(t)[1− 1

sε(t)

]≥ 0.

We need the following auxiliary lemma about the scaling-invariance ofthe scalar resolvent sµ(t) = Eα(−µtα).

130


Lemma 8.1.3. The function sµ(t) is scaling-invariant in the sense that sµ(λt) =sµλα (t), for all λ ∈ Σπ, µ ∈C and t ∈R+.

Proof. It is obvious, that sµ(λt) = Eα (−µ(λt)α) = Eα (−(µλα)tα) = sµλα (t), for allλ ∈ Σπ, µ ∈C and t ∈R+.

Lemma 8.1.4. There is a constant C = C(α) > 0 such that for ε > 0

ε −g1−α(t)sε(t)

+ g1−α(t) ≤ Cg1−α(t),

is valid for all t > 0.

Proof. Due to the scaling-invariance of the standard kernel g1−α and the scalarresolvent sε, it is enough to show this relation for ε = 1 and all t > 0. In fact,suppose that we have for all t > 0 the inequality

1−g1−α(t)s1(t)

+ g1−α(t) ≤ Cg1−α(t).

With the scaling t = λs (λ > 0) and the scaling behaviour of the standardkernel g1−α(λt) = λ−αg1−α(t) this inequality is equivalent to the relation

1−λ−αg1−α(s)s1(λs)

+λ−αg1−α(s) ≤ Cλ−αg1−α(s).

From Lemma 8.1.3 we know that s1(λs) = sλα (s). Consequently, we obtain forall λ > 0 and s > 0 that

λα −g1−α(s)sλα (s)

+ g1−α(s) ≤ Cg1−α(s),

the claim follows with ε = λα.Now, we will prove for t > 0 that

|s1(t)− g1−α(t)| ≤ C0g1−α(t)s1(t).

For all t ∈ (0,1] we have via the triangle inequality that

|s1(t)− g1−α(t)| ≤ s1(t)g1−α(t)g1−α(t)

+ g1−α(t)s1(t)s1(t)

.

Since s1(t) and g1−α(t) are monotonically decreasing, we can estimate as fol-lows

|s1(t)− g1−α(t)| ≤[

1g1−α(1)

+1

s1(1)

]g1−α(t)s1(t).

131


We know from Lemma 2.2.6 that for each t ∈ R+ we have the pointwiseestimate s1(t) ≥ (1 + 1/g1−α(t))−1, in particular s1(1) ≥ (1 + 1/g1−α(1))−1 =(1 + Γ (1−α))−1. Together with g1−α(1) = 1/Γ (1−α) we obtain for all t ∈ (0,1]

|s1(t)− g1−α(t)| ≤ C1g1−α(t)s1(t),

where C1 = C1(α) = [1 + 2Γ (1−α)] > 0.To prove the corresponding inequality for t ≥ 1 we use Lemma 8.1.5.

There is some constant C = C(α) > 0 such that for all t ∈ (0,∞) we have theestimate |s1(t)− g1−α(t)| ≤ Ct−2α. Moreover, we have for each t ∈ (0,∞) thelower estimate s1(t) ≥ (1+1/g1−α(t))−1. Since g1−α is monotonically decreasingon (0,∞) we obtain for t ≥ 1 the lower estimate s1(t) ≥ g1−α(t)(1 + g1−α(1))−1 =g1−α(t)(1 + 1/Γ (1−α))−1. Together with the definition of the standard kernelwe then have that

|s1(t)− g1−α(t)| ≤ C Γ (1−α)2g1−α(t)2

≤ C2 g1−α(t)s1(t),

where C2 = C2(α) = C Γ (1−α)2[1 + 1/Γ (1−α)] > 0. All in all, we have for allt ∈ (0,∞) the estimate

|s1(t)− g1−α(t)| ≤ C0g1−α(t)s1(t),

with the constant C0 = C0(α) = maxC1,C2.This implies for all t ∈ (0,∞) that∣∣∣∣∣1− g1−α(t)

s1(t)

∣∣∣∣∣ ≤ C0g1−α(t),

as well as

0 ≤ 1−g1−α(t)s1(t)

+ g1−α(t) ≤∣∣∣∣∣1− g1−α(t)

s1(t)

∣∣∣∣∣+ g1−α(t) ≤ (C0 + 1)g1−α(t).

Hence, the claim follows with C = C(α) = C0 + 1.

In the above proof we used the following lemma about the decay behaviourof the difference s1 − g1−α.

Lemma 8.1.5. There is some constant C > 0 such that for all t ∈ (0,∞) we havethe estimate

|s1(t)− g1−α(t)| ≤ Ct−2α .

Proof. For each λ ∈C+ we know that the Laplace transforms of the functionss1 and g1−α are given by

s1(λ) =1λ

11 +λ−α

, g1−α(λ) = λα−1.

132


For this reason we have for each λ ∈C+ that

s1(λ)− g1−α(λ) = − λ2α−1

λα + 1,

as well as

ddλ

[s1(λ)− g1−α(λ)] = −λ2α−2 (α − 1)λα + 2α − 1(λα + 1)2 := λ2α−2h(λ).

We want to apply the Analytic Representation Theorem 1.2.1 to the expres-sions s1(λ)− g1−α(λ), for α ∈ (0,1/2), and d/ dλ [s1(λ)− g1−α(λ)], for α ∈ [1/2,1).

Case α ∈ (0,1/2). The term λ2α−1/(λα + 1) possesses an holomorphicextension to Σπ. By Lemma 2.3.10 we have for each η ∈ (0,π/2) some constantc = c(η) > 0 such that for all z ∈ Σπ/2+η we have |1 + z| ≥ c(1 + |z|). We fix someη ∈ (0, π2 ) and consider the expression s1(λ)− g1−α(λ) on λ ∈ Σ π

2 +γ with γ = η/2.We have λα ∈ Σα( π2 +γ) ⊂ Σ π

2 +γ and obtain that∣∣∣∣∣∣ λ2α−1

λα + 1

∣∣∣∣∣∣ ≤ |λ|2α−1

c(|λ|α + 1)≤ |λ|

2α−1

c.

The map g : (0,∞)→ R+ with g(s) = s2α−1, s ∈ (0,∞), satisfies all conditionsfor the Analytic Representation Theorem 1.2.1, thus we have that there is aconstant C0 > 0 such that for all t ∈ (0,∞) we have |s1(t)− g1−α(t)| ≤ C0t

−2α.

Case α ∈ [1/2,1). Obviously, the term λ2α−2(λ)h(λ) possesses a holomor-phic extension to Σπ. We fix some η ∈ (0,π/2) and consider the expressionλ2α−2h(λ) for all λ ∈ Σπ/2+γ with γ = η/2, as above. Since λα ∈ Σπ/2+η we have∣∣∣∣∣λ2α−2 (α − 1)λα + 2a− 1

(λα + 1)2

∣∣∣∣∣ ≤ |λ|2α−2 (1−α) |λ|α + |2α − 1|c2(|λ|α + 1)2 .

In our next step we will show that h(λ) is uniformly bounded on Σπ/2+γ . Forall λ ∈ Σπ/2+γ with |λ| ≤ 1 we know that

|h(λ)| ≤(1−α) + |2α − 1|

c2 =M1.

For all λ ∈ Σπ/2+γ with |λ| ≥ 1 we have

|h(λ)| ≤(1−α) + |2α − 1| |λ|−α

c2(|λ|α + 2 + |λ|−α)≤ (1−α) + |2α − 1|

2c2 =M2.

All together we have for all λ ∈ Σπ/2+γ the estimate |h(λ)| ≤maxM1, M2 =:M.In particular, we obtain for all λ ∈ Σπ/2+γ the estimate∣∣∣∣∣ ddλ [s1(λ)− g1−α(λ)]

∣∣∣∣∣ ≤M |λ|2α−2 .

133


The Analytic Representation Theorem 1.2.1 with g(s) = s2α−2, s ∈ (0,∞) to-gether with the uniqueness of the Laplace transform, Theorem 1.1.4, yieldthat there is a constant C > 0 such that for all t ∈ (0,∞) we have the estimate

|t(s1(t)− g1−α(t))| ≤ Ct−2α+1.

The claim follows with C = maxC0,C.

Finally, we are able to prove the following.

Lemma 8.1.6. For each η > 0 there is some ε = ε(η) > 0 such that∥∥∥∥∥∥t 7→∫ t

0[−g1−α(t − τ)]]

ϕε(t)−ϕε(τ)ϕε(τ)

dτ

∥∥∥∥∥∥Lp(R+)

=(∫ ∞

0

∣∣∣∣∣ε − g1−α(t)sε(t)

+ g1−α(t)∣∣∣∣∣p dt)1/p

≤ η;

where ϕε = 1/sε and sε denotes the scalar resolvent for the scalar Volterra equationwith standard kernel.

Proof. Let ε > 0. We define ψε : (0,∞)→R+ via

ψε(t) = ε+ g1−α(t)[1− 1

sε(t)

].

For each t ∈ (0,∞) we have on one hand that 0 ≤ ψε(t) ≤ ε, and on the otherhand there is some constant C = C(α) > 0 (independent of ε) such that we havethe estimate ψε(t) ≤ Cg1−α(t), cf. Lemma 8.1.4. Now, let η > 0 be arbitrary, butfixed. Then there is some t∗ = t∗(η) > 0 such that(∫ ∞

t∗

Cpgp1−α(t)dt

)1/p

=Ct

1/p−α∗

Γ (1−α)(1−αp)1/p≤ η/2.

Now, we choose ε = ε(η) > 0 so small that εpt∗ ≤ (η/2)p. Then we have thefollowing estimate for the Lp-norm of ψε:∥∥∥ψε∥∥∥Lp(R+)

=(∫ ∞

0

∣∣∣ψε(t)∣∣∣p dt)1/p

≤(∫ t∗

0

∣∣∣ψε(t)∣∣∣p dt)1/p

+(∫ ∞

t∗

∣∣∣ψε(t)∣∣∣p dt)1/p

≤(∫ t∗

0εp dt

)1/p

+(∫ ∞

t∗

Cpgp1−α(t)dt

)1/p

≤ η/2 + η/2 = η.

134


8.2 Stability Theorem

After these preparations we are in the situation to examine quasilinear frac-tional evolution equations in regards to stability of equilibria. On the basisof time-weighting and rewriting of the equation, together with the maximalLp-regularity assumption on the linearised problem, we can prove asymptoticstability. The optimal decay rate of t−α, like the scalar resolvent, we can onlyprove under an additional regularity assumption on the non-linearities.

We consider again the setting from Chapter 7.

Assumptions. The spaces X1 and X0 are Banach spaces for class HT suchthat X1 → X0 with dense embedding. For p ∈ (1,∞) and α ∈ (1/p,1) let Vbe an open subset of the real interpolation space Xγ = (X0,X1)1− 1

αp ,p, and

u0 ∈ V . For simplicity we choose the norms in the involved spaces in sucha way that ‖v‖X0

≤ ‖v‖Xγ ≤ ‖v‖X1for all v ∈ X1. Moreover, we have maps

(A,F) : V →B(X1,X0)×X0.

We consider abstract quasilinear parabolic problems of the form

∂αt (u −u0) +A(u)u = F(u), t ∈R+, u(0) = u0. (8.3)

The trace space of this class of functions is given by Xγ . We assume for theopen set V ⊂ Xγ that

(A,F) ∈ C1(V ;B(X1,X0)×X0).

Let u∗ ∈ V ∩X1 be an equilibrium of equation (8.3), i. e. A(u∗)u∗ = F(u∗).Introducing the deviation v = u−u∗ from the equilibrium u∗ and linearising

around u∗, the equation for v then reads as follows

∂αt (v − v0) +A0v = G(v), t ∈R+, v(0) = v0. (8.4)

where v0 = u0 −u∗ and for all v ∈ X1 we have

A0v = A(u∗)v + (A′(u∗)v)u∗ −F′(u∗)v.

We set V∗ = V − u∗, then the map G : V∗ ∩X1 → X0 can be written as G(v) =G1(v) +G2(v,v), where for all w ∈ X1 and all v ∈ V∗ := V −u∗ ⊂ Xγ we have

G1(v) =(F(u∗ + v)−F(u∗)−F′(u∗)v)− (A(u∗ + v)−A(u∗)−A′(u∗)v)u∗, (8.5)

G2(v,w) =− (A(u∗ + v)−A(u∗))w. (8.6)

Before we turn our attention to the asymptotic stability of the equilibrium,we prove the next lemma concerning the growth behaviour of the map G in aneighbourhood of zero on the basis of different regularity assumptions on themaps A and F.

135


Lemma 8.2.1. Let V ⊂ Xγ be an open subset, (A,F) ∈ C1(V ;B(X1;X0)×X0) besuch that for some u∗ ∈ V ∩ X1 we have A(u∗)u∗ = F(u∗). We set V∗ = V − u∗.Furthermore, let the map G : V∗ ∩ X1 → X0, given by G(v) = G1(v) +G2(v,v),where G1 and G2 are like in (8.5) and (8.6).

Then, for some sufficiently small r0 > 0 we have BXγr0 (0) ⊂ V∗ and the following

is true.

(i) There is some constant C0 = C0(r0) > 0 such that for all η > 0 there is some

r = r(η) ∈ (0, r0] such that for all v1,v2 ∈ BXγr (0)∩X1 we have

‖G(v1)−G(v2)‖X0≤ C0

(η + r + ‖v2‖X1

)‖v1 − v2‖X1

.

In particular, we have for all v ∈ BXγr (0)∩X1 that

‖G(v)‖X0≤ C0(η + r)‖v‖X1

.

(ii) For (A,F) ∈ C2(V ;B(X1;X0) × X0) we have that there is some constant

M0 = M0(r0) > 0 such that for all δ ∈ (0, r0] and v ∈ BXγδ (0)∩X1 we have

the estimate

‖G(v)‖X0≤M0 ‖v‖Xγ ‖v‖X1

.

Proof. From the regularity assumption we obtain that G1 ∈ C1(V∗;X0) andG2 ∈ C1(V∗ ×X1;X0). Moreover, we have

G1(0) = G2(0,0) = 0 ∈ X0,

G′1(0) = 0 ∈ B(Xγ ;X0),

G′2(0,0) = 0 ∈ B(Xγ ×X1;X0),

where G′1 and G′2 denote the Fréchet derivatives of G1 and G2, respectively.The set V is open in Xγ with u∗ ∈ V , this implies that there is some r0 > 0

such that BXγr0 (0) ⊂ V∗ = V −u∗.

(i) The assumptions imply that A : V →B(X1;X0) and F : V → X0 are con-tinuously Frechét-differentiable at u∗ ∈ V . Hence, there is an boundedoperator A′(u∗) ∈ B(Xγ ;B(X1;X0)) such that for all η > 0 there is an

r = r(η) ∈ (0, r0] such that for all v ∈ BXγr (0) we have the estimate∥∥∥A(u∗ + v)−A(u∗)−A′(u∗)v

∥∥∥B(X1;X0)≤ η ‖v‖Xγ .

The analogous statement holds for the map F. There is an operatorF′(u∗) ∈ B(Xγ ;X0) such that for all η > 0 there is an r = r(η) ∈ (0, r0] such

that for all v ∈ BXγr (0) we have∥∥∥F(u∗ + v)−F(u∗)−F′(u∗)v

∥∥∥X0≤ η ‖v‖Xγ .

136


A continuously Frechét-differentiable map is locally Lipschitz contin-uous, for this reason there is a constant L > 0 (depending only on a

appropriately small r0) such that for all r ∈ (0, r0] and all v ∈ BXγr (0) it

follows that

‖A(u∗ + v)−A(u∗)‖B(X1;X0) ≤ L‖v‖Xγ .

With these properties it is clear, that we have the following estimates forthe map G = G1 +G2 : V∗→ X0. For each η > 0 there is an r = r(η) ∈ (0, r0]

such that for all v1,v2 ∈ BXγr (0) we obtain

‖G1(v1)−G1(v2)‖X0≤ η ‖v1 − v2‖Xγ ,

for all w ∈ X1 and v1,v2 ∈ BXγr (0) we have

‖G2(v1,w)−G2(v2,w)‖X0≤ L‖w‖X1

‖v1 − v2‖Xγ ,

and for w1,w2 ∈ X1 and v ∈ BXγr (0) we get

‖G2(v,w1)−G2(v,w2)‖X0≤ Lr ‖w1 −w2‖X1

.

Note that the constant L does not depend on r ∈ (0, r0] with r0 > 0appropriately chosen. Combining these estimates, we have for all v1,v2 ∈BXγr (0)∩X1 the estimate

‖G(v1)−G(v2)‖X0≤ (η +L‖v2‖X1

)‖v1 − v2‖Xγ +Lr ‖v1 − v2‖X1

≤ C0(η + r + ‖v2‖X1)‖v1 − v2‖X1

,

where the constant C0 = max1,L is independent of r ∈ (0, r0]. Now,we choose v2 = 0 and remember that G(0) = 0 ∈ X0. Hence, for all

v ∈ BXγr (0)∩X1 it follows that

‖G(v)‖X0≤ C0(η + r)‖v‖X1

.

(ii) By the assumption on the maps A and F we deduce G1 ∈ C2(V∗;X0) andG2 ∈ C2(V∗ ×X1;X0). Now, we consider the Taylor expansion of G at thepoint 0. For all v ∈ V∗ ∩X1 we have

G(v) = G(0) +G′(0)v +∫ 1

0(1− t)G′′(tv)(v,v)dt.

Due to the linearity of the Fréchet derivative we have to consider theTaylor expansion of G1 and G2 at the point 0, respectively.

137


Taylor expansion of G1 at the point 0. For all v ∈ V∗ we have

G1(v) = G1(0) +G′1(0)v +∫ 1

0(1− t)G′′1 (tv)(v,v)dt

=∫ 1

0(1− t)G′′1 (tv)(v,v)dt,

since G1(0) = 0 and G′1(0)v = 0. The first derivative G′1 : V∗→B(Xγ ;X0)and second derivative G′′1 : V∗→B(Xγ ;B(Xγ ;X0)) are given by

G′1(v)h =[F′(u∗ + v)−F′(u∗)

]h−

[(A′(u∗ + v)−A′(u∗))h

]u∗,

G′′1 (v)(h, h) = F′′(u∗ + v)(h, h)−(A′′(u∗ + v)(h, h)

)u∗,

for all v ∈ V∗ and h, h ∈ Xγ . For each v ∈ V∗ we have

‖G1(v)‖X0≤ supt∈[0,1]

∥∥∥G′′1 (tv)(v,v)∥∥∥X0

≤ supt∈[0,1]

[∥∥∥F′′(u∗ + tv)(v,v)∥∥∥X0

+∥∥∥(A′′(u∗ + tv)(v,v))u∗

∥∥∥X0

]≤ supt∈[0,1]

∥∥∥F′′(u∗ + tv)∥∥∥B2(Xγ ;X0) ‖v‖

2Xγ

+ supt∈[0,1]

∥∥∥A′′(u∗ + tv)∥∥∥B2(Xγ ;B(X1;X0)) ‖v‖

2Xγ‖u∗‖X1

.

Taylor expansion of G2 at the point (0,0). We have for all v ∈ V∗and w ∈ X1 that

G2(v,w) = G2(0,0) +G′2(0,0)(v,w) +∫ 1

0(1− t)G′′2 (tv, tw)((v,w), (v,w))dt

=∫ 1

0(1− t)G′′2 (tv, tw)((v,w), (v,w))dt,

since G2(0,0) = 0 and G′2(0,0) = 0. Furthermore, the first derivativeG′2 : V∗×X1→B(Xγ ×X1;X0) as well as second derivative G′′2 : V∗×X1→B(Xγ ×X1;B(Xγ ×X1;X0)) are given by

G′2(v,w)(h1,h2) = − (A′(u∗ + v)h1)w − (A(u∗ + v)−A(u∗))h2,

G′′2 (v,w)((h1,h2), (h1, h2)) = −(A′′(u∗ + v)(h1, h1)

)w

− (A′(u∗ + v)h1) h2 −(A′(u∗ + v)h1

)h2,

for all v ∈ V∗ and h1, h1 ∈ Xγ , and all w ∈ X1 and h2, h2 ∈ X1. For allv ∈ V∗ and w ∈ X1 we have

‖G2(v,w)‖X0≤ supt∈[0,1]

∥∥∥G′′2 (tv, tw)((v,w), (v,w))∥∥∥X0

138


≤ supt∈[0,1]

[∥∥∥(A′′(u∗ + tv)(v,v)) tw∥∥∥X0

+ 2∥∥∥(A′(u∗ + tv)v)w

∥∥∥X0

]≤ supt∈[0,1]

∥∥∥A′′(u∗ + tv)∥∥∥B2(Xγ ;B(X1;X0)) ‖v‖

2Xγ‖w‖X1

+ 2 supt∈[0,1]

∥∥∥A′(u∗ + tv)∥∥∥B(Xγ ;B(X1;X0)) ‖v‖Xγ ‖w‖X1

.

It follows for all v ∈ V∗ ∩X1 that

‖G(v)‖X0≤ ‖G1(v)‖X0

+ ‖G2(v,v)‖X0

≤ supt∈[0,1]

∥∥∥F′′(u∗ + tv)∥∥∥B2(Xγ ;X0) ‖v‖

2Xγ

+ supt∈[0,1]

∥∥∥A′′(u∗ + tv)∥∥∥B2(Xγ ;B(X1;X0)) ‖v‖

2Xγ‖u∗‖X1

+ supt∈[0,1]

∥∥∥A′′(u∗ + tv)∥∥∥B2(Xγ ;B(X1;X0)) ‖v‖

2Xγ‖v‖X1

+ 2 supt∈[0,1]

∥∥∥A′(u∗ + tv)∥∥∥B(Xγ ;B(X1;X0)) ‖v‖Xγ ‖v‖X1

.

Due to the continuity of the first and second derivative of A and F byassumption we know that for some appropriates small r0 (if necessary

r0 has to be chosen smaller) we have for all w ∈ BXγr0 (0) that∥∥∥A′(u∗ +w)

∥∥∥B(Xγ ;B(X1;X0))≤ 1 +

∥∥∥A′(u∗)∥∥∥B(Xγ ;B(X1;X0))=MA1

,∥∥∥A′′(u∗ +w)∥∥∥B2(Xγ ;B(X1;X0))

≤ 1 +∥∥∥A′′(u∗)∥∥∥B2(Xγ ;B(X1;X0))

=MA2,∥∥∥F′′(u∗ +w)

∥∥∥B2(Xγ ;B(X1;X0))≤ 1 +

∥∥∥F′′(u∗)∥∥∥B2(Xγ ;B(X1;X0))=MF .

This yields for all δ ∈ (0, r0] and all v ∈ BXγδ ∩X1 the estimate

‖G(v)‖X0≤ ‖G1(v)‖X0

+ ‖G2(v,v)‖X0

≤MF ‖v‖Xγ ‖v‖X1

+MA2

(‖u∗‖X1

+ ‖v‖Xγ)‖v‖Xγ ‖v‖X1

+ 2MA1‖v‖Xγ ‖v‖X1

.

and hence

‖G(v)‖X0≤M0 ‖v‖Xγ ‖v‖X1

,

with M0(r0) =MF +MA2

(‖u∗‖X1

+ r0)

+ 2MA1.

139


We have the following terms of stability for the equilibrium u∗ of thequasilinear equation (8.3). Basically, these notions are similar to the semilinearcase, cf. Definition 4.2.1, but we recall them for the sake of completeness.

Definition 8.2.2 (Stability). (i) We call the equilibrium u∗ stable in Xγ forthe quasilinear equation (8.3) if for each ε > 0 there is some δ = δ(ε) > 0

such that for all u0 ∈ BXγδ (u∗) a solution u = u(·;u0) ∈ E1(R+)∩C(R+;Xγ )

of the equation (8.3) exists and ‖u −u∗‖C(R+;Xγ ) < ε.

(ii) We call the equilibrium u∗ asymptotically stable in Xγ with an algebraicdecay rate with exponent β, if u∗ is stable in Xγ for the quasilinearequation (8.3) and there is some δ0 > 0 and a constant C > 0 such

that for all u0 ∈ BXγδ0

(u∗) the solution u = u(·;u0) satisfies the estimate

‖u(t)−u∗‖Xγ ≤ C ‖u0‖Xγ t−β , t ∈ (0,∞).

Now, we formulate our stability result for the abstract quasilinear parabolicproblem (8.3). The peculiarity of this result is the different decay rates accord-ing to the regularity assumptions on the maps A and F.

Theorem 8.2.3 (Stability Theorem). Let X0 and X1 be Banach spaces of classHT such that X1 → X0 is a dense embedding. Let p ∈ (1,∞) and α ∈ (1/p,1). Fur-thermore, let V be an open subset of the real interpolation spaceXγ = (X0,X1)1− 1

αp ,p.

Suppose that (A,F) ∈ C1(V ;B(X1;X0)×X0) and assume that u∗ ∈ V ∩X1 is anequilibrium of (8.3), i. e. A(u∗)u∗ = F(u∗).

For the linearisation of (8.3) given by A0v = A(u∗)v + (A′(u∗)v)u∗ − F′(u∗)v,v ∈ X1, we assume that the operator A0 ∈ RS(X0) is invertible with ϕRA0

< π−απ2 ,i. e. the operator A0 has the property of maximal Lp-regularity on R+.

(i) Then the equilibrium u∗ is stable in Xγ for the quasilinear equations (8.3).

(ii) Moreover, the equilibrium u∗ is asymptotically stable in Xγ with an algebraicdecay rate with exponent between 0 and α − 1/p .

Under the additional assumption (A,F) ∈ C2(V ;B(X1,X0)×X0) we have that

(iii) The decay rate in Xγ is algebraic with exponent α.

Remark 8.2.4. Obviously, the asymptotic stability property of the equilibriumu∗ in part (ii) is stronger than the stability statement in part (i) of Theorem8.2.3. But in regard to the following proof and the used arguments it seems tobe useful to split these both statements.

Proof of Theorem 8.2.3. Firstly, we show that the assumptions on the operatorA0 imply that there is some µ ≥ 0 such that µ+A(u∗) ∈ RS(X0) with ϕµ+A(u∗) <π −απ2 , i. e. the operator A(u∗) has maximal Lp-regularity on each compact

140


interval [0,T∗]. Indeed, we have A(u∗) = A0 − S, where Sv = (A′(u∗)v)u∗ +F′(u∗)v for all v ∈ Xγ . The assumption on the maps A and F imply thatS ∈ B(Xγ ;X0). Moreover, observe that due to the invertibility of the operatorA0 and the definition of the real interpolation space Xγ we have for all v ∈ X1

that ‖Sv‖X0≤ C ‖v‖1−θX0

‖A0v‖θX0, where θ = 1− 1

αp ∈ (0,1) is the interpolationparameter. Hence, by means for Young’s inequality we deduce for all ε > 0and all v ∈ X1 that

‖Sv‖X0≤ C

[εθ ‖A0v‖X0

+ (1−θ)ε−θ/(1−θ) ‖v‖X0

],

cf. [DHP03, p. 12]. Thus, S is a relatively bounded perturbation of A0with small relative bound. Now, we choose ε > 0 sufficiently small, suchthat ε < CA0

/(1 + a), where a = Rλ(λ +A0)−1 : λ ∈ Σθ, θ > ϕRA0, and CA0

=supλ>0‖A0(λ + A0)−1‖B(X0). Then, we choose a sufficiently large parameterµ > βMA0

(1 + a)/(1 − εCA0(1 + a)), where MA0

= supλ>0‖λ(λ +A)−1‖B(X0).Theperturbation result for relatively bounded perturbations [DHP03, Proposition4.3] implies that µ+A(u∗) ∈ RS(X0) and ϕRµ+A(u∗)

= ϕRA0< π −απ2 . Hence A(u∗)

has maximal Lp-regularity on each compact interval [0,T∗].Thus, the quasilinear problem (8.3) is locally well-posed for all initial

values u0 ∈ Xγ in a small ball around u∗, see Theorem 7.1.1.

(i) We first choose r0 > 0 sufficiently small and C0 > 0 as in Lemma 8.2.1(i); next, we choose η ∈ (0,1/(8CMRC0)]. By Lemma 8.2.1 we may chooser = r(η) ∈ (0,minr0,1/(8CMRC0)] such that the estimates in Lemma8.2.1 (i) hold true.1 Here, CMR denotes the constant of maximal Lp-regularity, cf. Remark 6.2.2 a).

If r0 > 0 is sufficiently small, we know by the local well-posedness of theequation that there is some T > 0 such that for all v0 ∈ B

Xγr0 (0) ⊂ V∗ there

is a unique local strong solution v of the equation (8.4) in the class

v ∈ E1(T )∩C([0,T ];Xγ ).

The solution v with initial value v0 ∈ BXγr (0) can be extended to a maxi-

mal interval of existence [0, t∗). If t∗ is finite, then either v leaves the ball

BXγr (0) at time t∗, or the limit limt→t∗ v(t) does not exist in Xγ . We show

that this cannot happen for initial values v0 ∈ BXγδ (0), with sufficiently

small δ ≤ r to be chosen later. Now, we define the exit time for the ballBXγr (0), that is

t0 := supt ∈ (0, t∗) : ‖v(τ)‖Xγ ≤ r for all τ ∈ [0, t],1For the proof of part (i) and (ii) it would be enough to claim CMRC0 (η + r) ≤ 1/2, but in

part (iii) it is a technical necessity to have CMRC0 (η + r) ≤ 1/4. For uniform estimates in allthree parts of the proof we require the parameters with the stronger property.

141


and suppose that t0 < t∗. We know that v ∈ E1(t0) ∩ C([0, t0];Xγ ) isthe unique local strong solution of (8.4), hence we have for almost all

t ∈ [0, t0] that v(t) ∈ X1 ∩BXγr (0). The properties of the map G yield for

almost all t ∈ [0, t0] that ‖G(v(t))‖X0≤ C0(η + r)‖v(t)‖X1

, and hence

‖G(v)‖E0(t0) ≤ C0(η + r)‖v‖

E1(t0) . (8.7)

To prove the stability of the equilibrium v∗ = 0 of the equation (8.4), wedecompose our solution into the part which comes from the initial valuev0 and the remainder v(t) = v(t) − S(t)v0. The remainder v solves theproblem

∂αt v +A0v(t) = G(v), t ∈ [0, t∗), v(0) = 0.

As usual S(·)v0 solves the corresponding resolvent equation for the oper-ator A0. We estimate the E1(t0)-norm of v via ‖v‖

E1(t0) ≤ ‖S(·)v0‖E1(t0) +‖v‖

0E1(t0). By Proposition 6.2.5 there is some constant C1 > 0 indepen-dent of t0 such that ‖S(·)v0‖E1(t0) ≤ ‖S(·)v0‖E1(R+) ≤ C1 ‖v0‖Xγ . On theother hand we have by maximal Lp-regularity of the operator A0 for theremainder v the estimate

‖v‖0E1(t0) ≤ CMR ‖G(v)‖

E0(t0) ;

observe the independence of the constant CMR > 0 of the time t0. By theestimate (8.7) we obtain with the above choice of η and r that

‖v‖0E1(t0) ≤ C0CMR(η + r)‖v‖

E1(t0) ≤12‖v‖

E1(t0) .

The decomposition of the solution v yields

‖v‖0E1(t0) ≤

12

(‖S(·)v0‖E1(t0) + ‖v‖

0E1(t0)

),

and hence ‖v‖0E1(t0) ≤ C1 ‖v0‖Xγ . On the whole, we have with the uni-

form constant C1 > 0 the estimate ‖v‖E1(t0) ≤ 2C1 ‖v0‖Xγ .

Additionally, we consider the Xγ-norm of v(t) for t ∈ [0, t0]: ‖v(t)‖Xγ ≤‖S(t)v0‖Xγ + ‖v(t)‖Xγ . By Corollary 3.3.6, we have for all t ∈ R+ that‖S(t)v0‖Xγ ≤ Cγ ‖v0‖Xγ . The remainder v has trace zero, so we can use theembedding 0E1(t0) → C([0, t0],Xγ ) to obtain with a uniform constantCemb > 0 that ‖v‖C([0,t0];Xγ ) ≤ Cemb ‖v‖0E1(t0). Together with the aboveestimate on the 0E1(t0)-norm of v we have

‖v‖C([0,t0];Xγ ) ≤ CembC1 ‖v0‖Xγ .

142


Combining the estimates for the initial-value part and the remainderyields

‖v‖C([0,t0];Xγ ) ≤ ‖S(·)v0‖C([0,t0];Xγ ) + ‖v‖C([0,t0];Xγ )

≤ Cγ ‖v0‖Xγ +C1Cemb ‖v0‖Xγ ,

i. e. we have the uniform estimate ‖v(t)‖Xγ ≤ K ‖v0‖Xγ for all t ∈ [0, t0],

where K = Cγ + C1Cemb > 0. In particular, for all v0 ∈ BXγδ (0) with

0 < δ ≤ δ1 = minr/(2K), r we have for t = t0 that

‖v(t0)‖Xγ ≤ r/2.

This is a contradiction to the definition of t0, and we conclude thatt0 = t∗.

The above arguments yield for each δ ∈ (0,δ1] and all v0 ∈ BXγδ (0) the

uniform estimates ‖v‖E1(T ) ≤ 2C1 ‖v0‖Xγ as well as ‖v(t)‖C([0,T ];Xγ ) ≤ r/2

for all T < t∗. This implies t∗ =∞, the global existence of the solution

v ∈ E1(R+) with ‖v‖E1(R+) ≤ 2C1 ‖v0‖Xγ and v(t) ∈ B

Xγr (0) ⊂ V∗ for all

t ∈R+, and the stability of the equilibrium v∗ = 0 in Xγ . Indeed, givenε > 0 we choose η and r as before and δ < minδ1, ε/K, thus we have

‖v‖C(R+;Xγ ) < ε for all v0 ∈ BXγδ (0).

(ii) We choose r0 > 0, C0 > 0 as well as η ∈ (0,1/(8CMRC0)] and r = r(η) ∈(0,minr0,1/(8CMRC0)] as in the first part (i). We use the stabilityproperty of the equilibrium v∗ = 0 and fix for ε = r the corresponding

parameter δ0 = δ0(ε) > 0 such that for all v0 ∈ BXγδ0

(0) there is a solution

v ∈ E1(R+)∩C(R+;Xγ ) and v(t) ∈ BXγε (0) for all t ∈R+. We use again the

decomposition v = S(·)v0 + v where v solves the problem

∂αt v +A0v = G(v), t ∈R+, v(0) = 0. (8.8)

To study the decay behaviour of the solution v we examine every part ofthe decomposition by its own.

a) Decay behaviour of the initial-value part S(·)v0 in Xγ . Thedecay behaviour of the linear part is given by Corollary 3.3.6, sayingthat for all t ∈ R+ we have a constant Cγ > 0 such that ‖S(t)v0‖Xγ ≤Cγ min1, t−α‖v0‖Xγ .

b) Decay behaviour of the remainder term v in Xγ . To examinethe decay behaviour of the remainder term v, we consider for some

143


appropriate function ϕε0the product w = ϕε0

v and show that this ex-pression is uniformly bounded in Xγ . This yields the decay behaviourof 1/ϕε0

for the remainder v.

b) (i) Introducing the time-weighted remainder w. We fix someξ ∈ (0,α − 1/p) and set for all t ∈R+

ϕε0(t) = 1 + ε0

t

1 + t1−ξ,

with some arbitrary ε0 > 0 which will be chosen later. Now, we multiplythe equation (8.8) for the remainder term v by ϕε0

. For w = ϕε0v, this

yields the equation

∂αt w(t) +A0w(t) =ϕε0(t)G

(w(t)/ϕε0

(t) + S(t)v0

)(8.9)

+∫ t

0[−g1−α(t − τ)]

ϕε0(t)−ϕε0

(τ)ϕε0

(τ)w(τ)dτ, t ∈R+,

with w(0) = 0, using Lemma 8.1.1. Our aim is to show that ‖w(t)‖Xγ isuniformly bounded for all t ∈R+. To achieve this, we estimate the solu-tion w by using the property of maximal Lp-regularity of the operatorA0. We will show that every term on the right hand side of (8.9) is anelement of Lp(R+;X0) = E0(R+).

b) (ii) Estimates for the commutator term of the time-weightedremainder w. Firstly, we consider the map

t 7→∫ t

0[−g1−α(t − τ)]

ϕε0(t)−ϕε0

(τ)ϕε0

(τ)w(τ)dτ.

Since v = v − S(·)v0 ∈ 0E1(R+)∩C(R+;Xγ ) and ϕε0is continuous on R+

it is obvious that w is continuous on R+ with values in Xγ and for eachT > 0 we have w ∈ 0E1(T )∩C([0,T ];Xγ ).

Let w. l. o. g. T ≥ 1 be arbitrary, but fixed.We have by Lemma 8.1.2 that∥∥∥∥∥∥t 7→∫ t

0[−g1−α(t − τ)]

ϕε0(t)−ϕε0

(τ)ϕε0

(τ)w(τ)dτ

∥∥∥∥∥∥Lp([0,T ];X0)

≤ ‖w‖C([0,T ];Xγ )

∥∥∥∥∥∥t 7→∫ t

0[−g1−α(t − τ)]

ϕε0(t)−ϕε0

(τ)ϕε0

(τ)dτ

∥∥∥∥∥∥Lp(R+)

≤ ε0C ‖w‖C([0,T ];Xγ )

≤ ε0CCemb ‖w‖0E1(T ) ,

144


where the constant C = C(α,p,ξ) is independent of T and ε0, the em-bedding constant Cemb > 0 enjoys the same property since w has tracezero. Thus, we know that

t 7→∫ t

0[−g1−α(t − τ)]

ϕε0(t)−ϕε0

(τ)ϕε0

(τ)w(τ)dτ ∈ Lp([0,T ];X0).

b) (iii) Estimates for the map t 7→ ϕε0(t)G(v(t)). Now, we turn

to the expression

t 7→ ϕε0(t)G(v(t)) = ϕε0

(t)G(w(t)/ϕε0

(t) + S(t)v0

)and the question, whether this map belongs to Lp([0,T ];X0).

Due to the choice of ε = r ∈ (0, r0] and δ0 = δ0(ε), we have for all ini-

tial values v0 ∈ BXγδ0

(0) that v(t) ∈ BXγr (0), t ∈ R+. Moreover, the previ-

ous proof of the stability part (i) implies v ∈ E1(R+) and the estimate‖v‖

E1(R+) ≤ 2C1 ‖v0‖Xγ . In particular, we have v(t) ∈ X1 for almost allt ∈R+. Together with the properties of the map G from Lemma 8.2.1 (i)we obtain the estimate

‖G(v)‖Lp([0,1];X0) = ‖G(v)‖E0(1)

≤ C0(η + ε)‖v‖E1(1)

≤ C0(η + ε)2C1 ‖v0‖Xγ≤ C1 ‖v0‖Xγ ,

since C0(η + ε) ≤ 1/2. Moreover, for all t ∈ [0,1] and ε0 ∈ [0,1] we have1 ≤ ϕε0

(t) ≤ 2, it follows that∥∥∥ϕε0

G(v)∥∥∥Lp([0,1];X0)

≤ 2C1 ‖v0‖Xγ .

Using the properties of the map G from Lemma 8.2.1 (i) as well as the

fact that for almost all t ∈ R+ we have that v(t) ∈ BXγε ∩X1, the same

considerations as mentioned above imply∥∥∥ϕε0G(v)

∥∥∥Lp([1,T ];X0)

≤ C0(η + ε)(∫ T

1

∥∥∥ϕε0(t)v(t)

∥∥∥pX1dt

)1/p

≤ C0(η + ε)

(∫ T

1‖w‖pX1

dt

)1/p

+(∫ ∞

1

∥∥∥ϕε0(t)S(t)v0

∥∥∥pX1dt

)1/p .Via Corollary 3.3.7 (iii) there is a constant M > 0 such that we have theestimate ‖S(t)‖B(X0;X1) ≤Mt−α for all t > 0, and hence

145


∥∥∥ϕε0G(v)

∥∥∥Lp([1,T ];X0)

≤ C0(η + ε)

‖w‖0E1(T ) +M ‖v0‖X0

(∫ ∞1

[ϕε0

(t)t−α]pdt

)1/p .On can easily show that for ε0 ∈ [0,1] we have the uniform estimate∥∥∥t 7→ ϕε0

(t)t−α∥∥∥Lp([1,∞))

≤ (αp−1)−1/p+(p(α−ξ)−1)−1/p =M1(α,p,ξ) =M1;

observe the independence of T and ε0. Consequently, we have∥∥∥ϕε0G(v)

∥∥∥Lp([0,T ];X0)

≤ 2C1 ‖v0‖Xγ

+C0(η + ε)[‖w‖

E1(T ) +M ‖v0‖X0M1

]<∞,

thus t 7→ ϕε0(t)G(v(t)) ∈ Lp([0,T ];X0).

b) (iv) Estimates for the time-weighted remainderw. By the max-imal Lp-regularity property of the operator A0 in equation (8.9) we havethe estimate

‖w‖0E1(T ) ≤ CMR

∥∥∥ϕε0G(v)

∥∥∥E0(T )

+

∥∥∥∥∥∥t 7→∫ t

0[−g1−α(t − τ)]

ϕε0(t)−ϕε0

(τ)ϕε0

(τ)w(τ)dτ

∥∥∥∥∥∥E0(T )

≤ CMR

2C1 ‖v0‖Xγ +C0(η + ε)‖w‖

0E1(T )

+C0(η + ε)M ‖v0‖X0M1 + ε0CCemb ‖w‖0E1(T )

.

Since CMRC0(η + ε) ≤ 1/2 by the choice of the parameters above, wehave

‖w‖0E1(T ) ≤ CMR

[4C1 +MM1]‖v0‖Xγ + 2ε0CCemb ‖w‖0E1(T )

.

Now, we fix ε0 = min(4CembCMRC)−1,1, and it follows

‖w‖0E1(T ) ≤ 2CMR [4C1 +MM1]‖v0‖Xγ .

Using again the embedding theorem yields the uniform estimate

‖w‖C([0,T ];Xγ ) ≤ Cemb ‖w‖0E1(T )

≤ 2CembCMR [4C1 +MM1]‖v0‖Xγ .

Hence, we have for each t ∈ [0,T ] and all v0 ∈ BXγδ0

(0) that

‖w(t)‖Xγ ≤ ‖w‖C([0,T ];Xγ ) ≤M2 ‖v0‖Xγ <∞,

146


with M2 = 2CembCMR [4C1 +MM1] independent of T > 0. This uni-form estimate implies for each t ∈ R+ that ‖w(t)‖Xγ ≤M2 ‖v0‖Xγ for all

v0 ∈ BXγδ0

(0). Recalling the definition w = ϕε0v, together with the lower

estimate ϕε0(t) ≥ (ε0/2)tξ , t ∈R+, it follows that

‖v(t)‖Xγ ≤M2 ‖v0‖Xγϕε0

(t)≤

2M2 ‖v0‖Xγε0

t−ξ → 0,

as t→∞, with an algebraic decay rate of exponent ξ.

c) Conclusion. Together with the decay rate of the initial-valuepart S(·)v0 we have for t > 0 that

‖v(t)‖Xγ ≤ ‖v(t)‖Xγ + ‖S(t)v0‖Xγ ≤ M ‖v0‖Xγ t−ξ ,

for all v0 ∈ BXγδ0

(0), with M = max2M2/ε0,Cγ .

(iii) We choose r0 > 0, C0 > 0 as well as η ∈ (0,1/(8CMRC0)] and r = r(η) ∈(0,minr0,1/(8CMRC0)] as above in part (i). We use the stability prop-erty of the equilibrium v∗ = 0 and fix for ε = r the corresponding pa-

rameter δ0 = δ0(ε) > 0 such that for all v0 ∈ BXγδ (0) with δ ≤ δ0 there is

a solution v ∈ E1(R+)∩C(R+;Xγ ) and v(t) ∈ BXγε (0) for all t ∈ R+. We

decompose the global solution into the initial-value part S(·)v0 and tworemainder terms v = S(·)v0 + v + v, where v and v solve the subsequentequations

∂αt v +A0v = G(S(·)v0), t ∈R+, v(0) = 0, (8.10)

∂αt v +A0v = G(v)−G(S(·)v0), t ∈R+, v(0) = 0, (8.11)

We study the decay behaviour of each part of the decomposition.

a) Decay behaviour of the initial-value part S(·)v0 in Xγ . Thedecay behaviour of the initial-value part is given by Corollary 3.3.6, wehave for all t ∈R+ that ‖S(t)v0‖Xγ ≤ Cγ min1, t−α‖v0‖Xγ .

b) Introducing the time-weighted remainders w and w. To provethe decay behaviour of the terms v and v we consider the functionϕε0

= 1/sε0, where sε0

denotes the scalar resolvent, the parameter ε0 > 0will be fixed later. The aim is to show that w = ϕε0

v and w = ϕε0v are

uniformly bounded in Xγ . This shows that v and v decay like sε0as

t→∞.

147


We multiply the equations (8.10) and (8.11) by ϕε0and obtain with the

notation w = ϕε0v and w = ϕε0

v, respectively, the equations

∂αt w(t) +A0w(t) = ϕε0(t)G(S(t)v0) (8.12)

+∫ t

0[−g1−α(t − τ)]

ϕε0(t)−ϕε0

(τ)ϕε0


with w(0) = 0, and

∂αt w(t) +A0w(t) = ϕε0(t) [G(v(t))−G(S(t)v0)] (8.13)

+∫ t

0[−g1−α(t − τ)]

ϕε0(t)−ϕε0

(τ)ϕε0


with w(0) = 0, using Lemma 8.1.1. To reach our goal we make use of themaximal Lp-regularity property of the operator A0. Thus, we have toshow that every term on the right-hand side of the equation (8.12) and(8.13), respectively, are elements of Lp([0,T ];X0) = E0(T ) for each T > 0.

Trivially, we have for each t ∈ [0,1] and all ε0 ∈ [0,1] that there is someconstant Mϕ > 1 such that ϕε0

(t) ∈ [1,Mϕ]. It is well-known that for allt ≥ 1 there is some constant Cϕ > 0 (independent of ε0 ∈ [0,1]) such thatϕε0

(t) ≤ Cϕtα.

c) Decay behaviour of the term v in Xγ . At first, we show thatv ∈ 0E1(R+)∩ C(R+;Xγ ). For this end, we show that t 7→ G(S(t)v0) ∈Lp(R+;X0) for each v0 ∈ B

Xγδ (0) with sufficiently small δ > 0.

c) (i) Estimates for the map t 7→ ϕε0(t)G(S(t)v0). By Corollary

3.3.6 we have for all t ∈ R+ that ‖S(t)v0‖Xγ ≤ Mγ min1, t−α‖v0‖Xγ .

Hence, for δ ≤ δ2 = minδ0, ε/Mγ ,1 we have for all v0 ∈ BXγδ (0) that

S(t)v0 ∈ BXγε (0) as well as for almost all t ∈R+ that S(t)v0 ∈ B

Xγε (0)∩X1.

Using the properties of the map G it follows for almost all t ∈ R+ that‖G(S(t)v0)‖X0

≤ C0(η + ε)‖S(t)v0‖X1and therefore

‖G(S(·)v0)‖Lp([0,1];X0) ≤ C0(η + ε)‖S(·)v0‖Lp([0,1];X1) ,

and considering Proposition 6.2.5 we have ‖S(·)v0‖E1(R+) ≤ C1 ‖v0‖Xγ andhence

‖G(S(·)v0)‖Lp([0,1];X0) ≤ C0(η + ε)C1 ‖v0‖Xγ .

Next, using the estimate ‖S(t)v0‖X1≤Mt−α ‖v0‖X0

, cf. Corollary 3.3.7 (iii),we deduce with the same considerations as above that

148


‖G(S(·)v0)‖Lp([1,∞);X0) ≤ C0(η + ε)‖S(·)v0‖Lp([1,∞);X1)

≤ C0(η + ε)M(∫ ∞

1t−αp dt

)1/p

‖v0‖X0

≤ C0(η + ε)MM3 ‖v0‖X0,

with M3 =M3(α,p) = (αp − 1)−1. Altogether, we have

‖G(S(·)v0)‖Lp(R+;X0) ≤ C0(η + ε) [C1 +MM3]‖v0‖Xγ ,

and thus t 7→ G(S(t)v0) ∈ Lp(R+;X0). The maximal Lp-regularity prop-erty of the operator A0 applied to equation (8.10) yields v ∈ 0E1(R+)∩C(R+;Xγ ). Since ϕε0

is continuous it follows directly that w is contin-uous on R+ a function taking values in Xγ and for each T > 0 we havew ∈ 0E1(T )∩C([0,T ];Xγ ). Let w. l. o. g. T ≥ 1 be arbitrary, but fixed.

In our second step, we will show that the terms on the right-hand sideof equation (8.12) are elements of Lp([0,T ];X0). We begin by lookingat the map t 7→ ϕε0

(t)G(S(t)v0). With the aid of the the considerationsabove we have∥∥∥ϕε0

G(S(·)v0)∥∥∥Lp([0,1];X0)

≤ C0(η + ε)Mϕ ‖S(·)v0‖Lp([0,1];X1)

≤ C0(η + ε)MϕC1 ‖v0‖Xγ .

As mentioned before, for almost all t ∈ R+ we have S(t)v0 ∈ BXγε ∩X1.

Using Lemma 8.2.1 (ii) as well as Corollary 3.3.7 there are constantsM0(ε), Mγ , M > 0 such that for almost all t ≥ 1 we deduce

‖G(S(t)v0)‖X0≤M0 ‖S(t)v0‖Xγ ‖S(t)v0‖X1

≤M0Mγ t−α ‖v0‖XγMt−α ‖v0‖X0

.

Consequently, we have∥∥∥ϕε0G(S(·))v0)

∥∥∥Lp([1,T ];X0)

=(∫ T

1ϕpε0(t)‖G(S(t)v0)‖pX0

dt

)1/p

≤ CϕM0MγM

(∫ ∞1t−αp dt

)1/p

‖v0‖Xγ ‖v0‖X0

≤ CϕM0MγMM3 ‖v0‖2Xγ ,

with M3 independent of T > 0, see above. Combining the previousestimates we obtain∥∥∥ϕε0

G(S(·)v0)∥∥∥Lp([0,T ];X0)

≤C0(η + ε)MϕC1 ‖v0‖Xγ+CϕM0MγMM3 ‖v0‖2Xγ ,

hence we have t 7→ ϕε0(t)G(S(t)v0) ∈ Lp([0,T ];X0).

149


c) (ii) Estimates for the commutator term of the time-weightedremainder w. Next, we consider the map

t 7→∫ t

0[−g1−α(t − τ)]

ϕε0(t)−ϕε0

(τ)ϕε0

(τ)w(τ)dτ.

Lemma 8.1.6 implies for each ρ > 0 that there is an ε0 = ε0(ρ) > 0 suchthat we have the estimates∥∥∥∥∥∥t 7→

∫ t

0[−g1−α(t − τ)]

ϕε0(t)−ϕε0

(τ)ϕε0

(τ)w(τ)dτ

∥∥∥∥∥∥Lp([0,T ];X0)

≤ ‖w‖C([0,T ];Xγ )

∥∥∥∥∥∥t 7→∫ t

0−g1−α(t − τ)

ϕε0(t)−ϕε0

(τ)ϕε0

(τ)dτ

∥∥∥∥∥∥Lp(R+)

≤ ρ ‖w‖C([0,T ];Xγ )

≤ ρCemb ‖w‖0E1(T ) ,

using the embedding property with embedding constant Cemb > 0 inde-pendent of T > 0. Hence, this map is an element of Lp([0,T ];X0).

c) (iii) Estimates for the time-weighted remainder w. Now, weare in the situation to apply the maximal Lp-regularity property of theoperator A0, thereby getting that

‖w‖0E1(T ) ≤ CMR

[∥∥∥ϕε0G(S(·)v0)

∥∥∥E0(T )

+

∥∥∥∥∥∥t 7→∫ t

0−g1−α(t − τ)

ϕε0(t)−ϕε0

(τ)ϕε0

(τ)w(τ)dτ

∥∥∥∥∥∥E0(T )

≤ CMR

[C0(η + ε)MϕC1 ‖v0‖Xγ +CϕM0MγMM3 ‖v0‖2Xγ

+ ρCemb ‖w‖0E1(T )

].

Now, we choose ρ = (2CembCMR)−1 and fix the corresponding ε0 =ε0(ρ) > 0. Note that ρ and ε0 are independent of T > 0. This implieswith CMRC0(η + ε) ≤ 1/2 that

‖w‖0E1(T ) ≤MϕC1 ‖v0‖Xγ + 2CMRCϕM0MγMM3 ‖v0‖2Xγ .

In particular, we have for all v0 ∈ BXγδ (0) with δ ≤ δ2 the uniform estimate

‖w‖0E1(T ) ≤M4 ‖v0‖Xγ ,

150


with the T -independent constant M4 = MϕC1 + 2CMRCϕM0MγMM3.Finally, we use the continuous embedding 0E1(T ) → C([0,T ];Xγ ) withT -independent embedding constant Cemb > 0 to see that

‖w‖C([0,T ];Xγ ) ≤ Cemb ‖w‖0E1(T ) ≤ CembM4 ‖v0‖Xγ .

Since T ≥ 1 was arbitrary, this uniform estimate yields for all t ∈R+ that‖w(t)‖Xγ ≤M5 ‖v0‖Xγ , withM5 = CembM4 which is T -independent. With

the definition w = v/sε0it follows for all v0 ∈ B

Xγδ2

(0) that

‖v(t)‖Xγ ≤M5 ‖v0‖Xγ sε0(t)→ 0,

as t→∞, with the algebraic rate t−α.

d) Decay behaviour of the term v in Xγ . From the considera-tions above we know that v, S(·)v0, v ∈ E1(R+)∩C(R+;Xγ ), for all v0 ∈BXγδ (0), and δ ≤ δ2. This implies in particular v = v − S(·)v0 − v ∈

0E1(R+) ∩ C(R+;Xγ ). We know that ϕε0is continuous on R+, hence

w is continuous on R+ with values in Xγ and for all T > 0 we havew ∈ 0E1(T )∩C([0,T ];Xγ ).

Let w. l. o. g. T ≥ 1 be arbitrary, but fixed. In our next step, we will showthat every term on the right-hand side of equation (8.13) is an elementof Lp([0,T ];X0).

d) (i) Estimates for the map t 7→ ϕε0(t)[G(v(t))−G(S(t)v0)]. At

first we consider the map t 7→ ϕε0(t)[G(v(t))−G(S(t)v0)]. Analogous to

the examination above we have with a suitable choice of the parameters

for almost all t ∈R+ that v(t), S(t)v0 ∈ BXγε (0)∩X1 for all v0 ∈ B

Xγ (0)δ with

δ ≤ δ2. Thus, Lemma 8.2.1 (i) implies∥∥∥ϕε0[G(v)−G(S(·)v0)]

∥∥∥Lp([0,1];X0)

≤∥∥∥ϕε0

G(v)∥∥∥Lp([0,1];X0)

+∥∥∥ϕε0

G(S(·)v0)∥∥∥Lp([0,1];X0)

≤MϕC0(η + ε)[‖v‖

E1(R+) + ‖S(·)v0‖E1(R+)

].

The proof of part (i) yields, that ‖v‖E1(R+) ≤ 2C1 ‖v0‖Xγ and with Lemma

6.2.5 it follows that∥∥∥ϕε0[G(v)−G(S(·)v0)]

∥∥∥Lp([0,1];X0)

≤MϕC0(η + ε)3C1 ‖v0‖Xγ ;

with Mϕ = supt∈[0,1], ε0∈[0,1]|ϕε0(t)|. Using again Lemma 8.2.1 (i) we have

for almost all t ≥ 1 that

151


‖G(v(t))−G(S(t)v0)‖X0≤ C0(η + ε+ ‖S(t)v0‖X1

)‖v(t)− S(t)v0‖X1

≤ C0(η + ε+M ‖v0‖X0)‖v(t)− S(t)v0‖X1

.

Here we apply Corollary 3.3.7 (iii) which gives us a constant M > 0such that for all t ≥ 1 we have the estimate ‖S(t)v0‖X1

≤M ‖v0‖X0. The

decomposition of our solution v − S(·)v0 = v + v leads us to the estimate∥∥∥ϕε0[G(v)−G(S(·)v0)]

∥∥∥Lp([1,T ],X0)

≤ C0(η + ε+M ‖v0‖X0)

(∫ T

1

∥∥∥ϕε0(t)v

∥∥∥pX1dt

)1/p

+(∫ T

1

∥∥∥ϕε0(t)v

∥∥∥pX1dt

)1/p≤ C0(η + ε+M ‖v0‖X0

)[‖w‖

0E1(T ) + ‖w‖0E1(T )

].

On the whole we have∥∥∥ϕε0[G(v)−G(S(·)v0)]

∥∥∥Lp([0,T ];X0)

≤MϕC0(η + ε)3C1 ‖v0‖Xγ+C0(η + ε+M ‖v0‖X0

)[‖w‖

0E1(0,T ) + ‖w‖0E1(0,T )

],

and thus we have t 7→ ϕε0(t) [G(v(t))−G(S(t)v0)] ∈ Lp([0,T ];X0).

d) (ii) Estimates for the commutator term for the time-weightedremainder w. Next, we consider the map

t 7→∫ t

0[−g1−α(t − τ)]

ϕε0(t)−ϕε0

(τ)ϕε0

(τ)w(τ)dτ.

Lemma 8.1.6 yields that for all ρ > 0 there is an ε0 = ε0(ρ) > 0 such that∥∥∥∥∥∥t 7→∫ t

0[−g1−α(t − τ)]

ϕε0(t)−ϕε0

(τ)ϕε0

(τ)w(τ)dτ

∥∥∥∥∥∥Lp([0,T ];X0)

≤ ‖w‖C([0,T ];Xγ )

∥∥∥∥∥∥t 7→∫ t

0[−g1−α(t − τ)]

ϕε0(t)−ϕε0

(τ)ϕε0

(τ)dτ

∥∥∥∥∥∥Lp(R+)

≤ ρ ‖w‖C([0,T ];Xγ )

≤ ρCemb ‖w‖0E1(T ) ,

using the embedding property with embedding constant Cemb > 0 inde-pendent of T > 0. Hence, this map is an element of Lp([0,T ];X0).

d) (iii) Estimates for the time-weighted remainder w. Now, weare in the situation to apply the maximal Lp-regularity property of theoperator A0 to the equation (8.13). This gives

152


‖w‖0E1(T ) ≤ CMR

[∥∥∥ϕε0[G(v)−G(S(·)v0)]

∥∥∥E0(T )

+

∥∥∥∥∥∥t 7→∫ t

0[−g1−α(t − τ)]

ϕε0(t)−ϕε0

(τ)ϕε0

(τ)w(τ)dτ

∥∥∥∥∥∥E0(T )

≤ CMR

[C0(η + ε)Mϕ3C1 ‖v0‖Xγ

+C0

(η + ε+M ‖v0‖X0

) ‖w‖

0E1(T ) + ‖w‖0E1(T )

+ ρCemb ‖w‖0E1(T )

].

Now, we use the same choice as above for ρ = (2CembCMR)−1 and thecorresponding ε0 = ε0(ρ) > 0. This implies

‖w‖0E1(T ) ≤ 2CMR

[C0(η + ε)Mϕ3C1 ‖v0‖Xγ

+ C0

(η + ε+M ‖v0‖Xγ

) ‖w‖

0E1(T ) + ‖w‖0E1(T )

].

Now, we choose δ ≤ δ3 = minδ2, (8CMRC0M)−1 and consider v0 ∈BXγδ (0). This, together with CMRC0(η + ε) ≤ 1/4, implies the estimate

2CMRC0(η + ε+M ‖v0‖Xγ ) ≤ 3/4 and consequently

‖w‖0E1(T ) ≤ 12MϕC1 ‖v0‖Xγ + 3 ‖w‖

0E1(T )

≤[12MϕC1 + 3M4

]‖v0‖Xγ .

The map w has trace zero, so we can use the continuous embedding withT -independent embedding constant Cemb > 0 and get

‖w‖C([0,T ];Xγ ) ≤ Cemb ‖w‖0E1(T )

≤ Cemb[12MϕC1 + 3M4

]‖v0‖Xγ .

In particular, we have for all v0 ∈ BXγδ (0) with δ ≤ δ3 the uniform estimate

‖w‖C([0,T ];X0) ≤M6 ‖v0‖Xγ , (8.14)

with T -independent constant M6 = Cemb[12MϕC1 + 3M4

]. This uni-

form estimate yields for all t ∈ R+ that ‖w(t)‖Xγ ≤M6 ‖v0‖Xγ . With the

definition w = v/sε0it follows for all v0 ∈ B

Xγδ3

(0) that

‖v(t)‖Xγ ≤M6 ‖v0‖Xγ sε0(t)→ 0,

for t→∞.

153


d) Conclusion. With the well-known estimate sε0(t) ≤ K0t

−α, cf.Section 2.3.3 Standard Kernel, it follows, together with the decay be-haviour of S(t)v0, v and v, that

‖v(t)‖Xγ ≤ ‖S(t)v0‖Xγ + ‖v(t)‖Xγ + ‖v(t)‖Xγ≤

[Cγ t

−α +M5sε0(t) +M6sε0

(t)]‖v0‖Xγ

≤ Mt−α ‖v0‖Xγ ,

for all v0 ∈ BXγδ (0), where δ ≤ δ3 and M = Cγ +K0(M5 +M6) > 0. The

claim follows.

Remark 8.2.5. At this point should be mentioned that the above considera-tions can be adapted to the more general class of quasilinear equations withthe property of maximal Lp-regularity treated by Zacher [Zac03].

8.3 Example – Time-Fractional Diffusion Equation

We want to apply our theory to a concrete equation. This equation has alreadybeen studied in the classical case by Prüss [Prü02, Example 7.2]. Initially, wecollect important facts about the Neumann Laplacian on bounded domainsin Lp-setting for each p ∈ (1,∞). Afterwards, we turn to the question of localwell-posedness and then to the stability of certain equilibria.

The Neumann Laplacian on Bounded Domains

On L2(Ω). Firstly, we consider the Laplacian with homogeneous Neumannboundary conditions on a bounded domain Ω ⊂R

n with C2-boundary Γ = ∂Ωon the space L2(Ω). We collect important properties of its spectrum and thesemigroup generated by this operator. As usual, we denote by H1

2 (Ω) the firstSobolev space.

We define a : H12 (Ω) ×H1

2 (Ω) → C by a(u,v) =∫Ω∇u∇v dx, for all u,v ∈

H12 (Ω). The form a is closed and symmetric. We denote by −∆N the operator

associated with the form a. Then ∆N is a self-adjoint operator, called NeumannLaplacian on Ω, with D(∆N ) = u ∈ H1

2 (Ω) : ∆u ∈ L2(Ω) and ∂νu = 0 weakly,cf. [Are04, Example 5.3.3 c)]. Here, ∂νu = 0 in the weak sense can by definedby the condition

∫Ω

[∇u∇v +∆uv]dx = 0 for all v ∈ C1(Ω).Using regularity theory of elliptic operators, one can show that D(∆N ) =

u ∈H22 (Ω) : ∂νu = 0 on Γ , cf. [Yag10, Theorem 2.6].

154

8.3. Example – Time-Fractional Diffusion Equation

Properties of the C0-Semigroup. Since the Neumann Laplacian is in-duced by a form, it is the generator of a C0-semigroup

(et∆N

)t∈R+

which

admits a holomorphic continuation to some sector Σθ with angle θ ∈(0, π2

).

Additionally, we know that the semigroup(et∆N

)t∈R+

generated by the Neu-

mann Laplacian on L2(Ω) is contractive. A proof of this property is givenin [Vra03, Theorem 4.2.1].

Thus, the Neumann Laplacian is self-adjoint and generates a boundedholomorphic, positive and contractive C0-semigroup

(et∆N

)t∈R+

on L2(Ω),

cf. [Are04, Section 2.6].

The Spectrum. It is known that on L2(Ω) the Neumann Laplacian opera-tor has compact resolvent, its spectrum is discrete consisting only of eigenval-ues with finite multiplicity and zero is an isolated eigenvalue, cf. [BM16, Ap-pendix A]. Thus, σ (−∆N ) = λk ∈ [0,∞) : k ∈N0 with 0 = λ0 < λ1 ≤ λ2 . . . ≤λk ≤ . . ., k ∈N.

On Lp(Ω). Now, we consider the Neumann Laplacian on Lp(Ω), p ∈ [1,∞].

The semigroup(et∆N

)t∈R+

generated by the Neumann Laplacian on L2(Ω) is

symmetric submarkovian, see [Are04, Example 7.1.4], i. e. for each t ∈ R+et∆N is positivity preserving and

∥∥∥et∆Nu∥∥∥∞ ≤ ‖u‖∞ for all 0 ≤ u ∈ L2(Ω) ∩

L∞(Ω). Hence, there exist consistent extrapolation C0-semigroups(et∆

pN

)t∈R+

on Lp(Ω), p ∈ [1,∞]; the corresponding generators are denoted by −∆pN .

The Lp-theory of elliptic operators implies thatD(∆pN ) = u ∈H2p (Ω) : ∂νu =

0 on Γ , p ∈ (1,∞), see [Yag10, Theorem 2.15].

Properties of the C0-Semigroup. It is well-known that for p ∈ (1,∞) thegenerated semigroup

(et∆

pN

)t∈R+

on Lp(Ω) is bounded, holomorphic, positive

and contractive, see [Are04, The Heritage List 7.2.2].Indeed, the symmetric submarkovian property of

(et∆N

)t∈R+

implies, to-

gether with the fact that the Neumann Laplacian is self-adjoint, that its adjoint

operator is submarkovian, too. Thus, for p ∈ 1,∞ we have∥∥∥∥et∆

pN f

∥∥∥∥Lp(Ω)

≤

‖f ‖Lp(Ω), for all f ∈ Lp(Ω)∩ L2(Ω) and all t ∈ R+; and by interpolation alsofor each p ∈ (1,∞). In other words, the extrapolation semigroup on Lp(Ω) iscontractive, for all p ∈ (1,∞).

Furthermore, due to the holomorphy of the semigroup generated by theNeumann Laplacian on L2(Ω) and together with Gaussian estimates, see[Are04, Example 7.4.1 (b)], it follows that the extrapolation semigroups on

155


Lp(Ω) are holomorphic with the same angle for each p ∈ [1,∞], cf. [Are04,Paragraph 7.4.4].

All in all, we have that the Neumann Laplacian generates a boundedholomorphic, positive and contractive C0-semigroup on Lp(Ω) for all p ∈(1,∞). Now, it follows from Remark 4.9 c), together with Theorem 4.2 andCorollary 4.4, from Weis [Wei01], that for each µ > 0 the operator −∆pN +µ onLp(Ω) is R-sectorial with R-angle < π

2 .

The Spectrum. The Neumann Laplacian has compact resolvent, seeabove. Hence, the spectrum is p-independent, p ∈ (1,∞), cf. [Are04, TheHeritage List 7.2.2], i. e. σ (−∆pN ) = σ (−∆N ).

Time-Fractional Diffusion Equation on Bounded Domains

We want to illustrate our theory on a quasilinear time-fractional diffusionequation on a bounded domain.

Assumptions. Let Ω ⊂Rn be a bounded domain with C2-boundary Γ = ∂Ω.

Suppose that f ∈ C1(Rn+1;R) and a ∈ C1(Rn+1;R) such that a(u,v) > 0 for allu ∈R and v ∈Rn. Moreover, let p ∈ (1,∞) and α ∈ (1/p,1).

We consider the quasilinear time-fractional diffusion equation∂αt (u −u0)− a(u,∇u)∆u = f (u,∇u), t > 0, x ∈Ω,

∂νu = 0, t > 0, x ∈ Γ ,u = u0, t = 0, x ∈Ω.

(8.15)

Here, we denote by ∇u =(∂x1u, . . . ,∂xnu

)Tthe gradient of the map u with

respect to the spatial variable x. By ∆u we mean the Laplacian of the map uwith respect to the spatial variable x, that is ∆u =

∑ni=1∂

2xiu. Since we have a

C2-boundary Γ the outward pointing unit normal ν : Γ →Rn is well-defined

and the normal derivative of u along Γ is given by ∂νu(x) =∑ni=1νi(x)∂xiu(x),

x ∈ Γ .Firstly, we are looking for a unique local solution of the problem (8.15) in

the space of maximal Lp-regularity

Hαp

([0,T ];Lp(Ω)

)∩Lp

([0,T ];H2

p (Ω)),

for some T > 0.In the following we always consider the case p > n+ 2

α . We set X0 = Lp(Ω)and

X1 = v ∈H2p (Ω) : ∂νv = 0 on Γ .

156

8.3. Example – Time-Fractional Diffusion Equation

Note that X1 coincides with the domain of the Lp-realsation of the NeumannLaplacian. The space X1 is densely embedded in X0 and both spaces are Ba-

nach spaces of class HT . By the Besov space B2− 2

αppp (Ω) =

(Lp(Ω),H2

p (Ω))

1− 1αp ,p

the trace space is given by

Xγ = (X0,X1)1− 1αp ,p

= v ∈ B2− 2

αppp (Ω) : ∂νu = 0 on Γ ,

see [Ama09, Section 4.9]. Since we consider p > n+ 2α we have the embeddings

B2− 2

αqpp (Ω) → C(Ω) and also B

1− 2αq

pp (Ω) → C(Ω), see [AF03, Chapter 7].Together with the assumptions on a and f we have for

A : Xγ →B (X1;X0) , u 7→ A(u) = −a(u,∇u)∆,

that A ∈ C1(Xγ ;B (X1;X0)

)and for

F : Xγ → X0, u 7→ F(u) = f (u,∇u),

that F ∈ C1(Xγ ;X0

).

For u ∈Hαp

([0,T ];Lp(Ω)

)∩Lp

([0,T ];H2

p (Ω))

we have the embedding

Hαp

([0,T ];Lp(Ω)

)∩Lp

([0,T ];H2

p (Ω))→ C

([0,T ];B

2− 2αp

pp (Ω))→ C

([0,T ]×Ω

),

since we consider α > 1/p. Moreover, for

∂xiu ∈Hα2p

([0,T ];Lp(Ω)

)∩Lp

([0,T ];H1

p (Ω)),

i ∈ 1, . . . ,n, we have the embedding

Hα2p

([0,T ];Lp(Ω)

)∩Lp

([0,T ];H1

p (Ω))→ C

([0,T ];B

1− 2αp

pp (Ω))→ C

([0,T ]×Ω

),

due to the condition p > n+ 2α , cf. [Zac03, p. 96].

Let u0 ∈ Xγ be arbitrary, but fixed. We consider the operator A(u0) =−a(u0,∇u0)∆; a differential operator of second order with variable coeffi-cients. By the above embeddings it is obvious that the coefficient a(u0,∇u0)is continuous on Ω. Moreover, the principal symbol of A(u0) is given bya(u0(x),∇u0(x))|ξ |2, ξ ∈ Rn, it is parameter elliptic with angle zero for eachx ∈Ω, and the Lopatinskii-Shapiro Condition is satisfied (essentially, one hasto realise this condition for the Neumann Laplacian). Thus, we conclude withTheorem 8.2 from Denk, Hieber and Prüss [DHP03] that up to some shift theoperator A(u0) is R-sectorial with R-angle < π

2 . Thus, the problem is locally

157


well-posed for p > n+ 2α , see Theorem 7.1.1.

Now, we want to study the stability of equilibria for equation (8.15). TheNeumann boundary condition guarantees that each constant function u∗ withf (u∗,0) = 0 defines an equilibrium, i. e. A(u∗)u∗ = F(u∗).

The linearisation of A(u)u −F(u) around the constant solution u∗ is givenfor all w ∈ X1 by

A0w = A(u∗)w+ (A′(u∗)w)u∗ −F′(u∗)w= −a(u∗,0)∆w − [au(u∗,0)w+ av(u∗,0)∇w]∆u∗ − f ′(u∗,0)w

= −a(u∗,0)∆w − f ′(u∗,0)w.

The eigenvalues of the operator A0 are given by (µk)k∈N0with

µk = a(u∗,0)λk − f ′(u∗,0), k ∈N0,

where (λk)k∈N0are the eigenvalues of the Neumann Laplacian, see above.

Thus, for f ′(u∗,0) < 0 we have (µk)k∈N0⊂ (0,∞). Hence, the operator A0 is

invertible. Furthermore, this operator is R-sectorial with R-angle<π2 , since

the multiplication with a positive number a(u∗,0) do not modify the propertyofR-sectoriality, by the permanence properties ofR-sectorial operators whichare similar to those for sectorial operators, cf. [DHP03, p. 43 and Proposition1.3]. In particular, the equilibrium is asymptotically stable, see Theorem 8.2.3.

Moreover, if we consider f ∈ C2(Rn+1;R) and a ∈ C2(Rn+1;R) then we haveA ∈ C2

(Xγ ;B (X1;X0)

)as well as F ∈ C2

(Xγ ;X0

). By Theorem 8.2.3 we know

that there is a sufficiently small δ0 > 0 and a constant C > 0 such that for all

u0 ∈ BXγδ0

(u∗) we have for the corresponding solution u(·;u0) of (8.15) for eacht ∈ (0,∞) the estimate

‖u(t;u0)−u∗‖Xγ ≤ Ct−α ‖u0‖Xγ .

We summarise our results concerning the stability of the equilibrium u∗.

Theorem 8.3.1. Let Ω ⊂ Rn be a bounded domain with C2-boundary Γ = ∂Ω.

Suppose that f ∈ C1(Rn+1;R) and a ∈ C1(Rn+1;R) such that a(u,v) > 0 for allu ∈R and v ∈Rn. Moreover, let p ∈ (1,∞) and α ∈ (1/p,1) with p > n+ α

2 .Suppose that u∗ is a constant function with f (u∗,0) = 0 satisfies the stability

condition

f ′(u∗,0) < 0.

(i) Then the equilibrium u∗ is asymptotically stable in Xγ for the quasilineartime-fractional diffusion equation (8.15).

(ii) If we assume additionally that f ∈ C2(Rn+1;R) and a ∈ C2(Rn+1;R), thenu∗ is asymptotically stable in Xγ with an algebraic decay rate of exponent α.

158

Appendices

159

AOperators and their SpectrumIn the following we always denote by X a complex Banach space with norm‖·‖X . We refer to Arendt, Batty, Hieber and Neubrander [ABHN01, AppendixB] and Lunardi [Lun95, Appendix A.0] for the following definitions andresults as well as their proofs.

A.1 Closed Operators

Definition A.1.1 (Linear, Densely Defined and Closed Operator). LetD(A)be a linear subspace of the complex Banach space X.

(i) A linear map A : D(A)→ X is called linear operator on X with domainD(A). We denote by RgA = Ax : x ∈ D(A) the range of the operator Aand by KerA = x ∈ D(A) : Ax = 0 the kernel of the operator A.

(ii) The linear operator A is densely defined, if D(A) is dense in X.

(iii) The linear operator A is closed if its graph G(A) = (x,Ax) : x ∈ D(A) isclosed in X ×X.

(iv) The linear operator A is invertible if there is bounded operator A−1 onX such that A−1Ax = x for all x ∈ D(A) as well as A−1y ∈ D(A) andAA−1y = y for all y ∈ X.

Remark A.1.2. The definition of closed linear operators is equivalent to thecondition that for all (xn)n∈N ⊂ D(A) such that limn→∞ xn = x and limn→∞Axn =y in X we have x ∈ D(A) and Ax = y.

Proposition A.1.3. Let A be a linear operator on X with domainD(A). We denoteby XA = (D(A),‖·‖A) the domain of the operator equipped with the graph normdefined for all x ∈ D(A) by ‖x‖A = ‖x‖X + ‖Ax‖X .

Then the operator A : D(A)→ X is always bounded with respect to the graphnorm. Furthermore, the linear operator A is closed if and only if XA is a Banachspace.

Definition A.1.4 (Part of an Operator in an Subspace). Let A be a linearoperator on X, and let Y be a closed subspace of X. The part of the operator Ain the subspace Y is the operator AY : D(AY )→ Y on Y with domain D(AY ) =y ∈ D(A)∩Y : Ay ∈ Y given by AY y = Ay.

161

A. Operators and their Spectrum

A.2 Spectrum and Resolvent Operator

Definition A.2.1. Let A : D(A) ⊂ X → X be a linear operator . The resolventset ρ(A) and the spectrum σ (A) of A are defined by

ρ(A) = λ ∈C : λ−A : D(A)→ X is invertible, σ (A) = C \ ρ(A).

For λ ∈ ρ(A) the operator (λ−A)−1 : X→ X is called the resolvent operator (ofA at λ).

Remark A.2.2. If ρ(A) , ∅, then the linear operator A is closed. Let A be aclosed linear operator and λ ∈ ρ(A). The resolvent operator (λ−A)−1 has therange D(A), it is closed and belongs to B(X).

Proposition A.2.3. Let A be a linear operator on X. Then the resolvent set ρ(A)is open and the spectrum σ (A) is closed in C.

Let µ ∈ ρ(A) and λ ∈C with |λ−µ| ≤∥∥∥(µ−A)−1

∥∥∥−1B(X)

, then λ ∈ ρ(A), and

(λ−A)−1 =∞∑n=0

(µ−λ)n(µ−A)−(n+1),

where the series in norm-convergent. Hence,

∥∥∥(λ−A)−1∥∥∥B(X)

≤

∥∥∥(µ−A)−1∥∥∥B(X)

1−∣∣∣λ−µ∣∣∣∥∥∥(µ−A)−1

∥∥∥B(X)

.

Moreover, the map λ 7→ (λ−A)−1 is holomorphic on ρ(A) with values in B(X)and for all λ ∈ ρ(A) and all n ∈N we have(

ddλ

)n(λ−A)−1 = (−1)nn! (λ−A)−(n+1).

The resolvent set ρ(A) is the biggest domain of holomorphy of the map λ 7→(λ−A)−1.

Corollary A.2.4. Let A be a linear operator on X with domain D(A). For eachµ,λ ∈ ρ(A) we have

(i) (λ−A)−1− (µ−A)−1 = (µ−λ)(λ−A)−1(µ−A)−1 = (µ−λ)(µ−A)−1(λ−A)−1,i. e. the so-called resolvent identity is satisfied and the resolvent operatorscommute;

(ii) A(λ−A)−1 = λ(λ−A)−1 − Id;

(iii) for all x ∈ D(A) we have A(λ − A)−1x = (λ − A)−1Ax, i. e. the resolventoperator commutes with the linear operator A on its domain D(A).

162

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Zusammenfassung in deutscherSpracheDiese Dissertation widmet sich der Untersuchung des Langzeitverhaltensder Lösungen von nicht-linearen abstrakten parabolischen Volterra Gleichun-gen. Im Mittelpunkt stehen dabei zum einen eine große Klasse semilinea-re parabolischer Gleichungen und zum anderen quasilineare fraktionelleEvolutionsgleichungen. Für diese zeitlich nicht-lokalen Gleichungen wirddas für klassische Evolutionsgleichungen bekannte Prinzip der linearisier-ten Stabilität verallgemeinert. Auf Grundlage der Stabilitätseigenschaftendes linearisierten Problems können Rückschlüsse über das Stabilitätsverhal-ten der nicht-linearen Gleichung gezogen werden. Die wesentliche Idee istfür beide Problemklassen identisch und besteht aus einem Störungsargu-ment, welches die Kontrolle von kleinen (verstanden in einem geeignetenSinn) nicht-linearen Termen ermöglicht. Die zur Umsetzung dieses Vorha-bens verwendeten analytischen Hilfsmittel und Methoden unterscheidensich wesentlich bei den betrachteten Klassen von Gleichungen. Für die qua-silineare Problemstellung wird insbesondere die Methode der maximalenLp-Regularität verwendet.

Abstrakte Semilineare Volterra Gleichungen. Wir formulieren das Prin-zip der linearisierten Stabilität für eine große Klasse von abstrakten semili-nearen Volterra Gleichungen, welche auch als abstrakte Integro-Differential-gleichungen formulierten werden können und insbesondere den Fall derfraktionellen Evolutionsgleichungen einschließt. Zunächst beschreiben wirdiese Klasse von Volterra Gleichungen. Es sei a ∈ L1,loc(R+) ein unbeschränk-ter, vollständig monotoner Kern und b ∈ L1,loc(R+) bezeichne den zugehörigenvollständig monotonen Kern mit der Eigenschaft a ∗ b ≡ 1 auf (0,∞); es ist be-kannt, dass es stets genau einen Kern b mit dieser Eigenschaft gibt. Ferner seiA ein linearer, abgeschlossener und dicht definierter Operator im BanachraumX, f ∈ C1(X;X) mit f (0) = 0 und f ′(0) = 0, sowie u0 ∈ X. Für die unbekannteFunktion u : J → X betrachten wir die semilineare Volterra Gleichung

u + a ∗Au = u0 + a ∗ f (u), t ∈ J ; (A.1)

wobei J entweder ein kompaktes Intervall [0,T ], T > 0, oder R+ ist. AufGrund der Kerneigenschaft a ∗ b ≡ 1 ist es möglich die Gleichung (A.1) zueiner Integro-Differentialgleichung umzuschreiben:

∂t [b ∗ (u −u0)] +Au = f (u), t ∈ J, u(0) = u0. (A.2)

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Zusammenfassung in deutscher Sprache

Ein Equilibrium der Volterra Gleichung (A.1) bzw. der Integro-Differential -gleichung (A.2) ist durch die stationäre Lösung u∗ = 0 gegeben.

Im Gegensatz zu klassischen Evolutionsgleichungen gibt es keine Halb-gruppen für Volterra Gleichungen. Daher wird ein verallgemeinertes Konzeptder sogenannten Resolventen und Integral-Resolventen benötigt. Diese beidenObjekte ermöglichen eine Lösungsdarstellung für die Volterra Gleichungenin ähnliche Weise zu der aus dem klassischen Fall bekannten Variation derKonstanten Formel.

Das hier formulierte Prinzip der linearisierten Stabilität für semilineareVolterra Gleichungen fordert für die Stabilität des Equilibriums u∗ = 0 dieExistenz einer beschränkten Resolvente und einer integrierbaren Integral-Resolvente. Darüber hinaus erhält man im Fall a < L1(R+) die asymptotischeStabilität mit exakt derselben Abklingrate wie für die skalare Resolvente.

Für parabolische Volterra Gleichungen kennt man einfache hinreichendeBedingungen für die Existenz einer beschränkten Resolvente, jedoch keine fürdie Existenz und Integrierbarkeit der Integral-Resolvente. Es werden dahereinfach überprüfbare Kriterien für die Existenz und Integrierbarkeit derIntegral-Resolvente im Fall von parabolischen Volterra Gleichungen bewiesen.

Unter Verwendung der hinreichenden Bedingungen für die Existenz einerbeschränkten Resolvente und integrierbaren Integral-Resolvente erkennt man,dass die wesentliche Bedingung für die Stabilität des Equilibriums u∗ = 0, ne-ben einem gewissen Abklingverhalten der Kerne, letztlich die Kompatibilitätdes Operators A mit dem Kern a, im Sinne der Parabolizität, ist.

Darüber hinaus können wir auch hinreichende Bedingungen für die In-stabilität des Equilibriums u∗ = 0 geben. Im Falle des Standardkerns erhältman die Instabilität sobald das Spektrum des Operators A einen kompaktenTeil im Komplement des Stabilitätssektors hat. Im allgemeinen Fall hängtdie Instabilitätsbedingung von den Abbildungseigenschaften der Laplace-Transformation des Kerns a ab.

Die zulässigen Paare von Kernen und Operatoren in dem hier formuliertenPrinzip der linearisierten Stabilität decken eine große Klasse von paraboli-schen Volterra Gleichungen ab.

Quasilineare Fraktionelle Evolutionsgleichungen. Der zweite Schwer-punkt dieser Arbeit liegt auf der Verallgemeinerung des Prinzips der linea-risierten Stabilität für quasilineare fraktionelle Evolutionsgleichungen. ZurBehandlung dieses Gleichungstypes wird die Methode der maximalen Lp-Regularität verwendet. Dabei seien X0 und X1 Banachräume der Klasse HTmit dichter Einbettung X1 → X0, p ∈ (1,∞) und α ∈ (1/p,1). Wir betrachtendie quasilineare fraktionelle Evolutionsgleichung

∂αt (u −u0) +A(u)u = F(u), u(0) = u0 ∈ V , (A.3)

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wobei V eine offene Teilmenge des reellen Interpolationsraumes (X0,X1)1− 1αp ,p

ist, und (A,F) : V →B(X1;X0)×X0. Wir untersuchen Lösungen mit maximalerLp-Regularität, d. h. u ∈Hα

p ([0,T ];X0)∩Lp([0,T ];X1).Zunächst nutzen wir Resultate zur maximal Lp-Regularität von linearen

Gleichungen, um die Existenz und Eindeutigkeit von Lösungen der quasili-nearen Gleichung zu beweisen.

Im Hauptresultat über die Stabilität von Equilibria für quasilineare fraktio-nelle Evolutionsgleichungen betrachten wir (A,F) ∈ C1(V ;B(X1;X0)×X0) unddas Equilibrium u∗ ∈ V ∩X1, d. h. A(u∗)u∗ = F(u∗). Unter der Voraussetzung,dass die Linearisierung von A(u)u −F(u) um u∗ maximale Lp-Regularität aufR+ hat, können wir die asymptotische Stabilität des Equilibriums u∗ zeigen.Unter der zusätzlichen Voraussetzung (A,F) ∈ C2(V ;B(X1,X0)×X0) könnenwir auch die optimale algebraische Abklingrate mit Exponent α beweisen.

Das in dieser Arbeit formulierte Theorem ist die einzige bekannte Ver-allgemeinerung des Prinzips der linearisierten Stabilität auf quasilinearefraktionelle Evolutionsgleichungen. Bis auf die zusätzliche technische Regula-ritätsannahme für die optimale Abklingrate verallgemeinert es das klassischeResultat in zu erwartender Art und Weise.

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Ehrenwörtliche ErklärungIch versichere hiermit, dass ich die Arbeit selbstständig angefertigt habeund keine anderen als die angegebenen Quellen und Hilfsmittel benutztsowie die wörtlich oder inhaltlich übernommenen Stellen als solche kenntlichgemacht habe und die Satzung der Universität Ulm zur Sicherung guterwissenschaftlicher Praxis in der jeweils gültigen Fassung beachtet habe.

Ulm, den 13. März 2018. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Marie-Luise Hein

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