is.cuni.cz 3 This habilitation thesis is based on the following research articles: [B1] T. B arta, R. Chill, and E. Fa sangov a, Every ordinary di erential equa-tion with a strict

Asymptotic behavior of gradient-like

systems

(Habilitation Thesis)

Tomas BartaKMA MFF UK

Subject: mathematics – mathematical analysis

Faculty of Mathematics and PhysicsCharles University

– September 2016 –

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I would like to thank Ralph Chill and Eva Fasangova for raising myinterest in the topic of this thesis and also for our collaboration. Further,I would like to express my gratitude to Petr Kaplicky and Jirı Spurny fortechnical support during preparation of this manuscript. Finally, I would liketo thank my family for their lasting support.

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This habilitation thesis is based on the following research articles:

[B1] T. Barta, R. Chill, and E. Fasangova, Every ordinary differential equa-tion with a strict Lyapunov function is a gradient system, Monatsh.Math. 166 (2012), 57–72.

[B2] T. Barta, Convergence to equilibrium of relatively compact solutions toevolution equations, Electron. J. Differential Equations 2014 (2014),No. 81, 1–9.

[B3] T. Barta and E. Fasangova, Convergence to equilibrium for solutions ofan abstract wave equation with general damping function, J. DifferentialEquations 260 (2016), no. 3, 2259–2274.

[B4] T. Barta, Rate of convergence and Lojasiewicz type estimates, J. Dyn.Diff. Equat., online first, 16 pages.

[B5] T. Barta, Decay estimates for solutions of an abstract wave equationwith general damping function, submitted 2016.

All these articles study convergence to equilibrium of bounded solutionsto gradient or gradient-like systems based on various generalizations of the Lojasiewicz gradient inequality. In [B1] we prove that every finite-dimen-sional gradient-like system is in fact a gradient system with respect to anappropriate Riemannian metric. This was an initial impulse to obtain newsufficient conditions for convergence to equilibrium of solutions to gradient-like systems that do not satisfy so called angle condition. In [B1] an abstractconvergence result for ODEs on manifolds is proved and it is applied to secondorder equations with weak damping. Rate of convergence for abstract finite-dimensional problems and also second order ODEs with weak damping isestimated in [B4]. Papers [B2], [B3], [B5] are devoted mainly to infinite-dimensional problems (but they can also be applied to ODEs). Article [B2]contains several abstract convergence results. In [B3], resp. [B5] we showconvergence to equilibrium resp. decay estimates for abstract wave equationswith weak damping.

The collection of articles is supplemented by an introductory commentary.In Chapter 1 we present the studied problem with all the settings we consider— ordinary differential equations in Rn, ordinary differential equations onmanifolds and partial differential equations (evolution equations in Hilbertor Banach spaces). Chapter 2 is devoted to abstract convergence results (and

4

corresponding decay estimates) in finite-dimensional and infinite-dimensionalcases. In Chapter 3 we present the results on damped second order equations,both ordinary and partial.

Contents

1 Introduction 71.1 Euclidean space setting . . . . . . . . . . . . . . . . . . . . . . 101.2 Manifold setting . . . . . . . . . . . . . . . . . . . . . . . . . . 121.3 Infinite-dimensional setting . . . . . . . . . . . . . . . . . . . . 12

2 Abstract convergence results 152.1 Finite-dimensional case . . . . . . . . . . . . . . . . . . . . . . 162.2 Infinite-dimensional case . . . . . . . . . . . . . . . . . . . . . 192.3 Decay estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3 Second order problems 293.1 Finite-dimensional case . . . . . . . . . . . . . . . . . . . . . . 313.2 Infinite-dimensional case . . . . . . . . . . . . . . . . . . . . . 373.3 Appendix to second order problems . . . . . . . . . . . . . . . 48

3.3.1 Well-posedness and existence of global solutions . . . . 483.3.2 Precompactness of bounded solutions . . . . . . . . . . 493.3.3 Functions satisfying the Lojasiewicz inequality . . . . . 51

Bibliography 53

4 Presented works 614.1 Research paper [B1] . . . . . . . . . . . . . . . . . . . . . . . 634.2 Research paper [B2] . . . . . . . . . . . . . . . . . . . . . . . 814.3 Research paper [B3] . . . . . . . . . . . . . . . . . . . . . . . 924.4 Research paper [B4] . . . . . . . . . . . . . . . . . . . . . . . 1104.5 Research paper [B5] . . . . . . . . . . . . . . . . . . . . . . . 128

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6 CONTENTS

Chapter 1

Introduction

Before we start with exact definitions and settings in Sections 1.1 – 1.3,let us write a few words without being precise to introduce the topic ofthis thesis. Let us consider a dynamical system governed by a differentialequation (ordinary or partial)

u+ F(u) = 0.

Here u is a function of time t, it has values in a state space X and u meanstime derivative of u. We assume that the system is dissipative in the sensethat energy of any nonstationary solution is decreasing, i.e., there exists astrict Lyapunov function to the system. Such systems are called gradient-like.The main problem studied in this thesis is, whether (under what conditions)every bounded solution of such a system has a limit as time goes to infinity.

As a special case we consider so called gradient systems, where F is thegradient of a potential E and the equation is then in the form

u+∇E(u) = 0.

Since −∇E(u) is a vector pointing in the direction of the steepest decay ofthe function E in point u, we can say that solutions of a gradient system aremaximizing loss of energy in every time t or that the system moves in thedirection of maximal energy decay.

There are many mathematical models of real-life processes that can bewritten in the form of a gradient or gradient-like system. Let us mentionseveral examples: second order equation describing an oscilating spring withnonlinear damping

x+ g(x) + kx = 0, (1.1)

7

8 CHAPTER 1. INTRODUCTION

the heat equation

ut −∆u = 0,

some semilinear heat equations, e.g. (see [24])

ut −∆u+ u− up = 0,

the wave equation with damping (see [24])

utt + αut −∆u = 0,

some semilinear wave equations with nonlinear damping (see [13])

utt + g(ut)−∆u+ |u|p−1u = 0, (1.2)

all of them with appropriate boundary conditions and on appropriate do-mains.

In many examples, energy (the Lyapunov function) E of the system iscoercive in the sense that the level sets

u ∈ X : E(u) ≤ K

are bounded. Then any solution to the gradient-like system is bounded. Ifwe are in a finite-dimensional space (and it is also true for some infinite-dimensional problems) then every bounded solution is relatively compact,i.e. the closure of

u(t) : t ∈ [0,+∞)is compact. Then the omega-limit set of u is nonempty, i.e. there exists

ϕ ∈ ω(u) := a ∈ X : ∃ tn +∞, u(tn)→ a.

Such ϕ is typically an equilibrium of the system. We ask (and this is themain task of this thesis), whether it neccessarily holds that

limt→+∞

u(t) = ϕ.

In general, the answer is negative. A simple counterexample in R2 is (inpolar coordinates)

r′ = (1− r)3, ϕ′ = 1− r.

9

Solutions of this system are spirals converging to the unit circle consisting ofequilibria. A Lyapunov function to this system is E(r, ϕ) = (r−1)2. The an-swer is negative even for gradient systems. There is a famous example called‘mexican hat’ by Palis and de Melo [55, page 14] and another one (whichlooks more difficult but is easier to handle) by Absil, Mahony and Andrews[1]. Further, Polacik and Rybakowski gave counterexamples in R2 with anyRiemannian metric and also for solutions of semilinear heat equations (see[56]). Jendoubi and Polacik presented in [49] an example of a bounded so-lution u to a semilinear wave equation with ω(u) containing a continuum offunctions.

So, an additional condition must be considered to obtain convergence to ϕ(i.e. ω(u) = ϕ). It was observed by Lojasiewicz in 1962 (see [53]) thatfor gradient systems in Rn a gradient inequality can be such a condition. Inparticular, if the potential E of a gradient system satisfies

|E(u)− E(ϕ)|1−θ ≤ C‖∇E(u)‖ (1.3)

for some θ ∈ (0, 12] and for all u from a neighborhood of ϕ ∈ ω(u), then

limt→+∞ u(t) = ϕ.

In this thesis we present some known generalizations to the Lojasiewicz in-equality (conditions that imply limt→+∞ u(t) = ϕ for ϕ ∈ ω(u)) for gradient-like systems in Rn, in Banach spaces, and on finitedimesional manifolds andwe introduce some new generalizations. We also show how these conditionsinfluence the speed of convergence to ϕ. Further we show that the new con-ditions/inequalities apply to second order equations with weak damping, i.e.partial differential equations of the type (1.2) or ordinary differential equa-tions (1.1) with g′(0) = 0 (so the damping is weaker than linear near zero).We show convergence and decay estimates for such equations.

For these results we do not need global (nor local) existence for everyinitial data, we neither need uniqueness. We only assume that we haveone precompact solution u : [0,+∞) → X to a gradient-like system andϕ ∈ ω(u), then we show that limt→+∞ u(t) = ϕ. In some abstract results weeven do not need that there is a differential equation behind.


1.1 Euclidean space setting

Let M ⊂ Rn be open and connected and let F : M → Rn be a continuousvector field and consider the following ordinary differential equation

u+ F(u) = 0. (1.4)

LetCr(F) = u ∈ Ω : F(u) = 0 (1.5)

be the set of stationary points of (1.4). A function E : M → R is called astrict Lyapunov function for (1.4) if

〈E ′(u),F(u)〉 > 0 for all u ∈M \ Cr(F) (1.6)

where E ′ denotes the derivative of E and the brackets denote the dualitybetween (Rn)′ and Rn. System (1.4) is called a gradient-like system if thereexists a strict Lyapunov function E for (1.4). If u : I → M is a solution to(1.4), then

d

dtE(u(t)) = E ′(u(t))u(t) = −E ′(u(t))F(u(t)) < 0, (1.7)

whenever u(t) is not a stationary point of (1.4). So, E is decreasing alongany nonstationary solution.

We say that E : M → R is a Lyapunov function for (1.4) if E is nonin-creasing along solutions, i.e. t 7→ E(u(t)) is nonincreasing for every solution uto (1.4). System (1.4) is called weakly gradient-like if there exists a Lyapunovfunction E for (1.4) satisfying

if E(u(·)) is constant on [t0,+∞), then u(·) is constant on [t0,+∞). (1.8)

Important examples of gradient-like systems are gradient systems. LetE : M → R be a continuously differentiable function. The following ordinarydifferential equation is called a gradient system

u+∇E(u) = 0. (1.9)

Of course, every gradient system is gradient-like. In fact, if F = ∇E , then

〈E ′(u),F(u)〉 = 〈E ′(u),∇E(u)〉 = ‖∇E(u)‖2 = ‖F(u)‖2 > 0

1.1. EUCLIDEAN SPACE SETTING 11

on M \ Cr(F).An important notion for studying asymptotic behavior is the omega-limit

set of a function u : R+ → Rn

ω(u) := ϕ ∈ Rn : ∃ tn +∞ s.t. limn→∞

‖u(tn)− ϕ‖ = 0.

Obviously, if u : R+ → Rn is a bounded solution to (1.4), then ω(u) isnonempty, so there exists ϕ ∈ ω(u). Further, if F is locally Lipschitz contin-uous on M \ Cr(F), then ω(u) ⊂ Cr(F). In fact, if ϕ ∈ ω(u) \ Cr(F), thenfor the solution starting at ϕ we have (1.7) with u(t) replaced by ϕ, so E isdecreasing along this solution and this is a contradiction with the fact that Eis constant on ω(u) and ω(u) is invariant. So, for gradient systems we haveω(u) ⊂ ϕ ∈M : E ′(ϕ) = 0.

Let ϕ ∈ ω(u). We are going to show that under additional conditions(gradient inequality) limt→+∞ u(t) = ϕ. In particular, for (weakly) gradient-like systems with a coercive Lyapunov function this means that every so-lution converges to an equilibrium. However, we often only assume that uis a solution to (1.4) and ϕ ∈ ω(u) to obtain minimal assumptions for theimplication

ϕ ∈ ω(u) ⇒ limt→+∞

u(t) = ϕ.

The results on (weakly) gradient-like systems can also be applied to sec-ond order equations. As an example let us mention the following ordinarydifferential equation describing damped oscilations of a spring (assume α,k > 0)

x+ αx+ kx = 0,

which can be rewritten as(xy

)+

(−y

αy + kx

)= 0.

We can see that E(x, y) = kx2+y2 is a Lyapunov function satisfying condition(1.8) (so we have a weakly gradient-like system) since

d

dtE(x(t), y(t)) = 2kxx+ 2yy = −2αy2 ≤ 0.

On the other hand, E(x, y) = kx2 + y2 + εxy, ε > 0 small enough is a strictLyapunov function (by easy computations), so the system is even gradient-like.


1.2 Manifold setting

Let (M, g) be a differentiable finite-dimensional Riemannian manifold witha Riemannian metric g. We denote by 〈·, ·〉 the duality between tangentand cotangent vectors, 〈·, ·〉g(u) the scalar product on the tangent space TuMin point u ∈ M , and ‖ · ‖g(u) is the norm on TuM generated by the scalarproduct. Sometimes, we write shortly 〈·, ·〉g and ‖ · ‖g. Also, if X, Y aretangent vector fields on M , we write 〈X, Y 〉g meaning 〈X(u), Y (u)〉g(u) forevery u ∈M and similarly we write ‖X‖g.

Let F : M → TM be a continuous tangent vector field on M (where TMis the tangent bundle) and assume that the differential equation

u+ F(u) = 0 (1.10)

is a gradient-like system, i.e., that there exists a differentiable function E :M → R such that

〈E ′,F〉 > 0 on M \ Cr(F) = u ∈M : F(u) 6= 0. (1.11)

As above, such function E is called a strict Lyapunov function. Definition ofa Lyapunov function and a weakly gradient-like system is the same as in theEuklidean space.

For a scalar valued differentiable function E : M → R we define itsgradient in u ∈M as a vector v representing the operator E ′(u), i.e.,

〈v, x〉g(u) = 〈E ′(u), x〉 for all x ∈ TuM . (1.12)

Since v depends on the scalar product g(u), we write ∇g(u)E(u) and ∇gE forthe corresponding gradient field.

As in Euclidean space, gradient systems

u+∇g(u)E(u) = 0 (1.13)

are important examples of gradient-like systems (with F = ∇g(u)E(u) and Ebeing a strict Lyapunov function).

1.3 Infinite-dimensional setting

We would like to generalize the concept of gradient-like systems and gradientsystems to infinite-dimensional spaces to study convergence to equilibria forsome partial differential equations.

1.3. INFINITE-DIMENSIONAL SETTING 13

Before we introduce the settings let us start with some general notations.If X is a Banach space, we denote by ‖ · ‖X the norm in X, X ′ the dual ofX and 〈·, ·〉X′,X the duality between X ′ and X. By BX(ϕ, r) we denote theclosed ball in X with radius r centered in ϕ. For a function u : R+ → X wedenote its omega-limit set in X by ωX(u), i.e.,

ωX(u) := ϕ ∈ Rn : ∃ tn +∞ s.t. limn→∞

‖u(tn)− ϕ‖X = 0.

We say that u has X-precompact range if u(t) : t ≥ 0 is precompact in X.Obviously, X-precompact range implies that ωX(u) 6= ∅. If X is a Hilbertspace, then we denote by 〈·, ·〉X the scalar product in X.

Our settings will be as follows. Let V ⊂ H be two Banach spaces, Vcontinuously and densely embedded in H. Let M ⊂ V be nonempty, openand connected and let F : M → H be a continuous map. We consider thefollowing evolution equation

u+ F(u) = 0. (1.14)

We say that a function u is a solution to (1.14) if u ∈ C(R+,V)∩C1(R+,H)and (1.14) is satisfied (in H) for every t > 0.

To define a strict Lyapunov function and gradient-like system we need togive a good sense to the computation

d

dtE(u(t)) = E ′(u(t))u(t) = −E ′(u(t))F(u(t)) < 0.

A continuously differentiable function E : M → R is called a strict Lyapunovfunction for (1.14) if

〈E ′(u),F(u)〉 > 0 for all u ∈M , s.t. F(u) ∈ V \ 0. (1.15)

If there exists a strict Lyapunov function for (1.14) then (1.14) is calleda gradient-like system. Definition of a Lyapunov function and a weaklygradient-like system is the same as in the Euklidean space.

Let us mention that if u is a solution to a weakly gradient-like systemthen E is constant on ωV(u). Moreover, if we have continuous dependenceon initial values, then ωV(u) is positively invariant and as a consequence wehave ωV(u) ⊂ Cr(F).

We now define gradient systems. In the literature (see e.g. [24]), by agradient system is often understood the equation (1.14) with H = V ′ and


F = E ′ for some E ∈ C1(M). However, we follow [23] and define gradient tobe a vector representing the linear functional E ′(u) via scalar product. Let Hbe a Hilbert spaces and E ∈ C1(M,R). For a fixed u ∈M let us assume thatE ′(u) extends to a bounded linear functional on H. Then v ∈ H is calledgradient of E in u if 〈v, x〉H = 〈E ′(u), x〉H′,H for every x ∈ H. We denote thegradient by ∇E(u) = v.

If E ∈ C1(M,R) is such that E ′(u) extends to a bounded linear functionalon H for every u ∈M , then the following equation is called a gradient system

u+∇E(u) = 0. (1.16)

A simple example of an infinite-dimensional gradient system is the heat equa-tion

ut −∆u = 0

on a bounded domain Ω ⊂ RN with Dirichlet boundary conditions. Tak-ing V = H1

0 (Ω) ∩ H2(Ω), H = L2(Ω) and E(u) = 12

∫Ω‖∇u‖2 we have

E ′(u)w =∫

Ω∇u · ∇w and for u ∈ V this functional can be extended to

H and represented via scalar product as E ′(u)w = 〈−∆u,w〉. It means that∇E(u) = −∆u = F(u) for all u ∈ V .

Chapter 2

Abstract convergence results

This chapter is devoted to abstract convergence results. The task is to findconditions (typically formulated in terms of a Lyapunov function E) thatimply convergence of u to some ϕ ∈ ω(u). We are not so much interestedhere, which differential equations satisfy these conditions. More about thisquestion (applications of these abstract results) can be found in Chapter 3.

As we mentioned in the Introduction, the first convergence result based ona gradient inequality was formulated by Lojasiewicz [53] for gradient systemsin Rn. The convergence result reads as follows and the inequality (LI) is calledthe Lojasiewicz gradient inequality.

Theorem 2.0.1 ( Lojasiewicz 1962). Let M ⊂ Rn be a nonempty open setand E ∈ C1(M). Let u : [0,+∞) → M be a solution to the gradient system(1.9) and ϕ ∈ ω(u). Assume that there exist θ ∈ (0, 1

2] and η > 0 such that

|E(u)− E(ϕ)|1−θ ≤ C‖∇E(u)‖ for all u ∈ B(ϕ, η). (LI)

Then ‖u(t)− ϕ‖ → 0.

Although Lojasiewicz’s main result was that inequality (LI) holds for anyanalytic function E in Rn and any ϕ, if we refer to Lojasiewicz’s result in thiswork we always mean the convergence result, i.e. Theorem 2.0.1.

Since 1962, there are many works generalizing this result in many waysin finite-dimensional and also infinite-dimensional spaces. It was applied notonly to semilinear heat or wave equations but also to Cahn–Hilliard equation[25], degenerate parabolic equations [28], or integrodifferential equations [63].

15

16 CHAPTER 2. ABSTRACT CONVERGENCE RESULTS

2.1 Finite-dimensional case

First generalization of the Lojasiewicz result we would like to mention is theresult by Kurdyka [50], who observed that the function s 7→ s1−θ in (LI) canbe replaced by a more general function s 7→ Θ(s) and the convergence resultremains valid. The inequality in [50] reads

‖∇(Ψ E)(u)‖ ≥ C for all u ∈ B(ϕ, η) (2.1)

for a class of positive increasing functions Ψ : R+ → R. If Ψ is differentiable,then (2.1) can be rewritten as

Θ(|E(u)− E(ϕ)|) ≤ C‖∇E(u)‖ for all u ∈ B(ϕ, η), (KLI)

where 1Θ

= Ψ′ and E(·) is replaced by E(·)− E(ϕ). Here, we need to assumethat 1/Θ is integrable on (0, ε) if we want to get convergence to equilibrium.If we take Θ(s) = s1−θ, then (KLI) becomes (LI). In fact, Kurdyka’s mainresult was that (2.1) holds in Rn for a much larger class of functions thananalytic functions (see [50] or Section 3.3.3 for more details).

Other generalizations go from gradient systems to gradient-like systems.If we consider a gradient-like system with a strict Lyapunov function E sat-isfying (KLI) or (LI), then an additional condition is needed to obtain con-vergence to equilibrium. This additional condition can be so called anglecondition

〈E ′(u),F(u)〉 ≥ α‖E ′(u)‖ · ‖F(u)‖ (AC)

for some α > 0 and all u ∈ M . This was observed by Absil, Mahony andAndrews [1] and then generalized by Lageman [51] to gradient-like systems onRiemannian manifolds. In the following theorem, inequality (KLI) is hiddenin the notion analytic-geometric category.

Theorem 2.1.1 ([51], Theorem 1.2). Let X be a Lipschitz continuous vectorfield on an analytic Riemannian manifold (M, g) with an associated Lyapunovfunction E satisfying: for every compact K ⊂M there exists α > 0 such that

〈∇g(u)E(u),F(u)〉g(u) ≥ α‖E ′(u)‖ · ‖F(u)‖. (2.2)

Assume that E belongs to an analytic-geometric category. Then the ω-limitset of any integral curve of X contains at most one point.

2.1. FINITE-DIMENSIONAL CASE 17

Neccessity of an additional condition (e.g. (AC)) is one reason why gra-dient systems are easier to handle than gradient-like systems. In [B1] wehave shown that every gradient-like system is a gradient system if we changethe Riemannian metric appropriately. Especially, every gradient-like systemin Euklidean space becomes a gradient system if we deform the geometry ofthe space.

Theorem 2.1.2 ([B1], Theorem 1). Let M be a differentiable finite-dimen-sional Riemannian manifold, F a continuous tangent vector field on M , andlet E : M → R be a continuously differentiable, strict Lyapunov function for(1.10). Then there exists a Riemannian metric g on the open set

M := u ∈M : F(u) 6= 0 ⊆M

such that ∇gE = F . In particular, the differential equation (1.10) is a gra-dient system on the Riemannian manifold (M, g).

Let us call the Riemannian metric g from Theorem 2.1.2 a gradient Rie-mannian metric. Let us mention that g is not uniquely determined.

It seems to be possible to obtain convergence to equilibrium without theangle condition (AC) if we change the Riemannian metric and transform thegradient-like system to a gradient system. But then we need the Kurdyka- Lojasiewicz inequality (KLI) to be satisfied with respect to the new norm(g-norm) and we would obtain convergence in g-norm, which is not alwaysequivalent to the original norm (g-norm on the tangent bundle TM or theEuclidean norm in Rn) on a neigborhood of stationary points. In fact, thefollowing theorem shows that equivalence of the new and the old norm isconnected to the angle condition.

Theorem 2.1.3 ([B1], Theorem 2). The metrics g and g are equivalent onM if and only if E ′ and F satisfy the conditions (AC) and

c‖E ′‖g ≤ ‖F‖g ≤ C‖E ′‖g (2.3)

holds with some c, C > 0.

As a consequence, if (AC) is not valid on a neighborhood of an equilib-rium, then the new norm is not equivalent to the original norm. However, wecan still obtain convergence to equilibrium even in the case when the angle


condition is not satisfied. Since F = ∇gE and by definition of a gradient〈∇gE , X〉g = 〈E ′, X〉 for any continuous vector field X, we obtain

‖∇gE‖g =1

‖F‖g‖∇gE‖2

g =1

‖F‖g〈E ′,∇gE〉 =

1

‖F‖g〈E ′,F〉 .

This computation leads us to a new condition (GenLI) under which we obtainconvergence in the original norm ‖ · ‖g.

Theorem 2.1.4 ([B1], Theorem 3). Let (1.10) be a gradient-like systemon a Riemannian manifold (M, g) with a strict Lyapunov function E. Letu : R+ →M be a global solution of (1.10) and ϕ ∈ ω(u). Assume that thereexist a neighborhood U of ϕ and Θ : R+ → R+ such that 1/Θ ∈ L1

loc([0,+∞)),Θ(s) > 0 for s > 0, and

Θ(|E(v)− E(ϕ)|) ≤⟨E ′(v),

F(v)

‖F(v)‖g

⟩for every v ∈ U ∩ M. (GenLI)

Then u has finite length in (M, g) and, in particular, limt→+∞

u(t) = ϕ in (M, g).

Remark 2.1.5. The theorem remains valid with the same proof if we assumeonly that (1.10) is a weakly gradient-like system and E a Lyapunov function.

A simple example in R2, where this result applies and the angle conditiondoes not hold, is ([B1], Example 2)

F(u) = F(u1, u2) = (‖u‖αu1 − u2, u1 + ‖u‖αu2), E(u) =1

2(u2

1 + u22).

For more details see Example 2.3.7 below where we also derive decay estimatefor this equation. A more interesting application is a second order equationwith weak nonlinear damping, which can be found in Chapter 3.

The gradient Riemannian metric g from Theorem 2.1.2 is defined on M =M \Cr(F). An interesting open question is, whether (under what conditions)it can be defined on the whole of M . It follows from Theorem 2.1.3 that gcan be continuously extended to a stationary point ϕ only if (AC) and (2.3)hold on a neighborhood of ϕ. So, not every g can be extended continuouslyto a stationary point. In Example 3 in [B1] we have found two gradientRiemannian metrics for a gradient-like system in R2 such that one of themcan be continuously extended to a stationary point ϕ and the other cannot.

2.2. INFINITE-DIMENSIONAL CASE 19

2.2 Infinite-dimensional case

The first generalization of the Lojasiewicz result to infinite-dimensional set-ting is due to L. Simon [60], who proved that the gradient inequality

|E(u)− E(ϕ)|1−θ ≤ C‖∇E(u)‖L2 for all u ∈ BC2,µ(ϕ, η) (2.4)

holds for E(u) =∫

ΩE(x, u,∇u), where E is analytic in the second and third

variable, and used this inequality to show convergence (in C2-norm) to equi-librium for solutions to the corresponding gradient system

u+∇E(u) = f

and also for solutions to the second order equation

u− u−∇E(u) = f,

which, in fact, becomes a gradient-like system. This fact was observed byJendoubi who unified the approach to the first and second order problemin [46] and simplified significantly Simon’s proof. Jendoubi and Haraux [37]finally came to the gradient inequality in most satisfactory setting

|E(u)− E(ϕ)|1−θ ≤ C‖E ′(u)‖V ′ for all u ∈ BV(ϕ, η), (LSI)

in their case V = H10 (Ω).

In [24], Chill, Haraux and Jendoubi proved the following abstract conver-gence result.

Theorem 2.2.1 ([24], Theorem 1). Let u ∈ C(R+,V) ∩ C1(R+,H) with V-precompact range and ϕ ∈ ωV(u). Let ρ > 0, c > 0 and E ∈ C2(V ,R) be suchthat t 7→ E(u(t)) is differentiable almost everywhere and

− d

dtE(u(t)) ≥ c‖E ′(u(t))‖V ′‖u(t)‖H (2.5)

for almost every t ∈ R+. Assume in addition that (1.8) holds and that E sat-isfies the Lojasiewicz–Simon gradient inequality (LSI). Then limt→+∞ ‖u(t)−ϕ‖V = 0.

We can see that u is not neccessarily connected to any differential equa-tion, but if it is a solution to the evolution equation (1.14), then (1.14) is


a weakly gradient-like system. Condition (2.5) is a kind of angle condition:if u ∈ C1(R+,V ′) is a solution of a gradient-like system (1.14) with a strictLyapunov function E satisfying the angle condition

〈E ′(u),F(u)〉V ′,V ≥ α‖E ′(u)‖V ′‖F(u)‖H (2.6)

for every u ∈ V s.t. F(u) ∈ V , then (2.5) is satisfied whenever u ∈ C1(R+,V)([24], Proposition 5). For gradient systems in the sense of [24] (i.e. H = V ′,F = E ′) and also for gradient systems in the sense of our definition the anglecondition (2.6) is satisfied automatically.

In [B2] we have generalized Theorem 2.2.1 in two ways. First, as wasmentioned above, there are important cases where the angle inequality is notsatisfied, e.g. second order equations with weak damping. In fact, what isreally needed to show convergence is

− d

dtE(u(t)) ≥ Θ(E(u(t)))‖u(t)‖H (2.7)

for some positive function Θ s.t. 1/Θ is integrable at zero or, equivalently,

− d

dtE(u(t)) ≥ ‖u(t)‖H. (2.8)

Clearly, these two conditions are equivalent. In fact, if E satisfy (2.7), then(2.8) is satisfied with E replaced by E := ΦΘ(E) where ΦΘ(t) =

∫ t0

1Θ(s)

ds.The second implication is trivial.

So, conditions (2.5) and (LSI) can be replaced by more general condition(2.8). Inequality (2.8) follows from (2.5) and (LSI) by taking E = E1−θ:

− d

dtE(u(t)) = − 1

E(u(t))θd

dtE(u(t)) ≥ ‖E

′(u)‖V ′E(u(t))θ

‖u‖H ≥ c‖u‖H.

Obviously, in Theorem 2.2.1 the Lojasiewicz–Simon inequality can be re-placed by Kurdyka– Lojasiewicz–Simon inequality

Θ(E(u)− E(ϕ)) ≤ ‖E ′(u)‖V ′ (KLSI)

for any function Θ > 0 with 1Θ∈ L1

loc([0,+∞)) (see [B2], Theorem 3.2).Second generalization is, that it is not neccessary to assume (2.8) on a

whole halfline t ≥ t0. It is enough to assume that E is nonincreasing alongsolutions (e.g. d

dtE(u(t)) < 0) for t ≥ t0 and the stronger estimate (2.8) holds

whenever u(t) is in a small neighborhood of ϕ. This assumption is easier toverify, e.g. for second order equations (ordinary or partial). We obtain thefollowing result.


Theorem 2.2.2 ([B2], Theorem 2.4). Let u ∈ C(R+,V) ∩ C1(R+,H) withV-precompact range and ϕ ∈ ωV(u). Let ρ > 0 and E ∈ C(V ,R) be such thatt 7→ E(u(t)) is nonincreasing on R+ and (2.8) holds for almost every t ∈ R+

such that u(t) ∈ B := BV(ϕ, ρ).Then limt→+∞ ‖u(t)− ϕ‖V = 0.

It was observed in [24] that the space H can be replaced by any largerspace with a weaker norm. In other words, it is sufficient to verify the decaycondition (2.8) or (2.5) for a very weak norm ‖ · ‖H and the convergenceis then obtained in the stronger norm ‖ · ‖V by a compactness argument(due to precompact range). On the other hand, estimates of the speed ofconvergence are lost while using the compactness argument, so one can obtainthese estimates in the norm of H only.

In applications to second order equations we work on a product spaceand we often need to control the first coordinate only. The following gen-eralization of Theorem 2.2.2 is appropriate for such situations. If we needconvergence of one coordinate, it is enough to assume (2.9) instead of (2.8).Unfortunately, the condition (2.9) does not imply any estimates of the speedof convergence.

Theorem 2.2.3 ([B2], Theorem 2.6). Let u = (u1, u2) be such that u1 ∈C(R+, V1) ∩ C1(R+, H1), u2 ∈ C(R+, V2) ∩ C1(R+, H2), H1 → V1 and let(u1(·), u2(·)) have a precompact range in V1×V2. Let ϕ ∈ ωV1(u1), ρ > 0 andE ∈ C(V1 × V2,R) be such that t 7→ E(u(t)) is nonincreasing on R+ and

− d

dtE(u(t)) ≥ ‖u1(t)‖H1 (2.9)

for almost every t ∈ R+ such that u1(t) ∈ B := BV1(ϕ, ρ).Then limt→+∞ ‖u1(t)− ϕ‖V1 = 0.

Remark 2.2.4. Theorems 2.2.2 and 2.2.3 imply convergence to equilibriumfor precompact solutions of weakly gradient-like systems with Lyapunov func-tions satisfying (2.8), resp. (2.9).

Gradient systems.

As in finite-dimensional case, the situation is easier for gradient systems (noangle condition is needed). Therefore, it may be of interest that similarly to


the finite-dimensional case, any gradient-like system can be transformed toa gradient system by taking an appropriate Riemannian metric.

It was mentioned above that gradient depends on the scalar product.Let H be a Hilbert space and let g be any scalar product on H, we definegradient of E in u with respect to g as a vector v ∈ H (if such v exists)satisfying 〈v, w〉g = E ′(u)w for all w ∈ H. Then we write v = ∇gE(u).

We define a Riemannian metric on H to be a continuous mapping r :V → InnerH where InnerH is the space of all bounded scalar products onH equipped with strong convergence topology, i.e., gn → g in InnerH if〈u, v〉gn → 〈u, v〉g for every u, v ∈ H. Then the equation

u+∇r(u)E(u) = 0 (2.10)

is called a gradient system with respect to the Riemannian metric r.Let us call a Riemannian metric g a gradient metric for a gradient-like

system (1.14) with a strict Lyapunov function E , if F(v) = ∇g(v)E(v) for allv ∈M .

Theorem 2.2.5. Let (1.14) be a gradient-like system with a strict Lyapunovfunction E such that ∇E is continuous on M and 〈∇E ,F〉H > 0 on M =M \ Cr(F). Then there exists a gradient metric g for (1.14) on M .

Since this result was not published we present a proof here.

Proof. For any w ∈ M we have 〈∇E(w),F(w)〉H = 〈E ′(w),F(w)〉 > 0 andtherefore 0 6= F(w) 6∈ ker E ′(w). As a consequence, for every w ∈ M we have

H = ker E ′(w)⊕ 〈F(w)〉. (2.11)

For every u ∈ H and w ∈ M let us define

uw0 := u− 〈E ′(w), u〉〈E ′(w),F(w)〉F(w) and uw1 :=

〈E ′(w), u〉〈E ′(w),F(w)〉F(w). (2.12)

Then uw0 ∈ ker E ′(w), uw1 ∈ 〈F(w)〉 and the mappings w 7→ uw0, w 7→ uw1

are continuous from V to H (the denominators are continuous since ∇E :M → H is continuous).

Now we choose an arbitrary Riemannian metric r on H. Starting fromthis metric, we define a new metric g on M by setting

〈u, v〉g(w) := 〈uw0, vw0〉r(w) +1

〈E ′(w),F(w)〉〈E′(w), u〉〈E ′(w), v〉

= 〈uw0, vw0〉r(w) +1

〈E ′(w),F(w)〉〈E′(w), uw1〉〈E ′(w), vw1〉.

(2.13)

2.3. DECAY ESTIMATES 23

Clearly, g(w) is a sesquilinear form on H and it is positive difinite due to〈E ′,F〉 > 0. Continuity of g follows from continuity of the mappings w 7→uw0, w 7→ uw1, continuity of r, ∇E and F and the fact (an easy 3ε argument)that 〈un, vn〉r(wn) → 〈u, v〉r(w) whenever wn → w in V , vn → v in H andun → u in H.

By definition of the metric g and by definition of the gradient ∇gE , wehave for every v ∈ H, w ∈ M

〈F(w), v〉g(w) = 0 + 〈E ′(w), v〉 = 〈∇g(w)E(w), v〉g(w),

so g is a gradient metric on M .

For more about infinite-dimensional gradient systems see Chill and Fa-sangova [23] and an existence result by Boussandel [17]. These works assumethe Riemannian metric r to be continuous in a stronger sense, in particularr : W → InnerH where W satisfying V → W → H is a natural domain ofthe Lyapunov function E (i.e. W is the domain of the closure of E ′). It is notclear, whether one can find a gradient Riemannian metric g continuous in thissense for any gradient-like system. The gradient metric found in Theorem2.2.5 is typically not continuous with respect to a weaker norm on V .

2.3 Decay estimates

This section is devoted to decay estimates, i.e. estimates of the speed ofconvergence to equilibrium for a given solution (or for a given function to itslimit). Such estimates usually follow from the proofs of convergence resultsbased on gradient inequalities. The original convergence result by Lojasiewiczis accompanied by the following decay estimates proved by Haraux and Jen-doubi in 2001.

Theorem 2.3.1 ([38], Theorem 2.2). Let u be a bounded solution to a gra-dient system with E satisfying the Lojasiewicz gradient inequality (LI) withsome θ ∈ (0, 1

2]. Then there exists ϕ ∈M such that for t→ +∞ we have

‖u(t)− ϕ‖ =

O(e−ct) if θ = 1

2,

O(t−θ/(1−2θ)) if θ < 12.


This result remains valid also for gradient systems in the infinite-dimen-sional setting. Chill and Fiorenza [22] proved decay estimates for an infinite-dimensional gradient system with E satisfying the Kurdyka– Lojasiewicz–Simon inequality (KLI). They formulated the result for a semilinear parabolicequation but in fact they proved the following abstract result.

Theorem 2.3.2 ([22], Theorem 2.1). Let E ∈ C2(V) satisfiy the Kurdyka– Lojasiewicz–Simon gradient inequality (KLI), u be a solution to the gradientsystem (1.16) with u(t) : t ≥ 1 being relatively compact in V. Then thereexists ϕ ∈ ωV(u) and t0 > 0 such that

|E(u(t))− E(ϕ)| = O(ψ−1(t− t0)), (2.14)

‖u(t)− ϕ‖H = O(Φ(ψ−1(t− t0))) (2.15)

as t → +∞, where ψ is a primitive function to −1/Θ2, ψ−1 the inversefunction to ψ and Φ a primitive function to 1/Θ.

An abstract result, which can be applied to gradient-like systems satisfy-ing an angle inequality can be found in [24].

Theorem 2.3.3 ([24], Theorem 2). If the assumptions of Theorem 2.2.1 holdand in addition

− d

dtE(u(t)) ≥ β‖E ′(u(t))‖2

V ′ for a.e. t ≥ 0. (2.16)

Then, for t→ +∞ we have

‖u(t)− ϕ‖H =

O(e−ct) if θ = 1

2,

O(t−θ/(1−2θ)) if θ < 12.

This theorem can be easily modified for E satisfying the Kurdyka– Lo-jasiewicz–Simon inequality to obtain the estimates (2.14), (2.15) with theconstants c, β from (2.5), (2.16) appearing somewhere. Estimates (2.14),(2.15) were also proved for finite-dimensional gradient and gradient-like sys-tems by Begout, Bolte and Jendoubi in [10] (see Theorems 3.5, 3.7 therein)with condition (2.16) replaced by

‖∇E(u)‖ ≤ β‖F(u)‖. (2.17)


This inequality and (2.16) are in fact comparability conditions that states therelation between ‖E ′‖ and ‖F‖ (compare to (2.3)). Of course, some kind ofcomparability condition is needed, since if we change the size of F and keepthe direction, then the solutions have the same trajectories but their speed isdifferent. The previous theorems show that gradient-like systems satisfying(AC), (2.3) have the same speed of convergence as the corresponding gradientsystem. We can see below (Example 2.3.7) that if the angle condition is notvalid, then the decay estimates become worse.

We generalize the finite-dimensional result to Riemannian manifolds andreplace the Kurdyka– Lojasiewicz inequality (KLI) by the inequality (GenLI)introduced in Theorem 2.1.4. We assume that the relation between ‖∇E‖and ‖F‖ is represented by a function α (see (2.18)), which may be arbitraryand appears also in the obtained decay estimates. Let us remark that thisresult can be applied to a second order equation with weak damping withα(s) = Θ(s)h(Θ(s)) as we can see in the next chapter.

Theorem 2.3.4 ([B4], Theorem 1). Let (1.10) be a gradient-like system on(M, g) with a strict Lyapunov function E, u : [0,+∞) → M be a solutionto (1.10) and ϕ ∈ ω(u). Let E and F satisfy (GenLI) with a function Θ :[0, 1) → R+ such that 1

Θ∈ L1

loc([0, 1)) and Θ(s) > 0 for s > 0. Then uhas finite length in (M, g) and, in particular, lim

t→+∞u(t) = ϕ. Moreover, if

α : (0, 1)→ (0,+∞) is nondecreasing and satisfies

α(E(u(t))− E(ϕ)) ≤ ‖F(u(t))‖ for all t large enough, (2.18)

then there exists t0 > 0 such that

|E(u(t))− E(ϕ)| ≤ ψ−1(t− t0) for all t > t0, (2.19)

‖u(t)− ϕ‖ ≤ Φ(ψ−1(t− t0)) for all t > t0, (2.20)

where

Φ(t) :=

∫ t

0

1

Θ(s)ds and ψ(t) :=

∫ 1/2

t

1

Θ(s)α(s)ds

and ‖a− b‖ is for a, b ∈M the g-distance of a and b.

Remark 2.3.5. The theorem remains valid if (1.10) is a weakly gradient-likesystem and E is a Lyapunov function satisfying (1.8).


We can see that (2.17) together with (KLI) imply that one can take α = Θand the definition of ψ becomes the same as in Theorem 2.3.2 (and the sameas in [10]).

All the decay estimates above are based on the inequality

‖u(t)− ϕ‖ ≤∫ +∞

t

‖u(s)‖ds, (2.21)

so they estimate the length of the remaining trajectory which can be muchlonger than the distance ‖u(t) − ϕ‖ (typically, for second order equationswith weak damping, it is much longer). Often, there are direct estimates of‖u− ϕ‖ in the form

‖u− ϕ‖ ≤ γ(E(u)− E(ϕ)). (2.22)

For example, E(u) =∑n

i=1 |ui|p, where u = (u1, u2, . . . , un), is a Lyapunovfunction for many ordinary differential equations and it satisfies (2.22) withγ(s) = cs1/p. A similar estimate holds e.g. for

ut −∆u+ |u|p−1u = 0

with

E(u) =1

2

∫

Ω

|∇u|2 +1

p+ 1

∫

Ω

|u|p+1

(see Example 3.2.8 or [13]). Inequality (2.22) gives in many cases betterdecay estimates than (2.21).

Corollary 2.3.6 ([B4], Corollary 3). Let the assumptions of Theorem 2.3.4hold and let γ : (0, 1)→ (0,+∞) be a nondecreasing function such that (2.22)holds for all u in a neigborhood of ϕ. Then there exist t0 > 0 such that

‖u(t)− ϕ‖ ≤ γ(ψ−1(t− t0)) for all t > t0.

Application to some second order equations with weak damping can befound in the next chapter. Now we present a simple example where Corol-lary 2.3.6 yields optimal decay estimates and Theorem 2.3.4 does not. Thismeans that estimating the distance from the equilibrium by the length ofthe remaining trajectory may be the only estimate which is not sharp in thewhole process.


Example 2.3.7 ([B4], Example 4). Let M ⊆ R2 be the open unit discequipped with the Euclidean metric. Let α ≥ 0, and let F (u) = F (u1, u2) =(‖u‖αu1−u2, u1 +‖u‖αu2) and E(u) = 1

2(u2

1 +u22). Then one can show that E

satisfies the Lojasiewicz inequality (LI) near the origin for θ = 12

but the anglecondition (AC) does not hold (unless α = 0). On the other hand, (GenLI)holds with Θ(s) = 1√

2s1−θ, θ = 1−α

2and (2.18) holds with α(s) = 2

√s. Then

Theorem 2.3.4 yields‖u(t)‖ ≤ C(t− t0)

1α−1.

If the angle condition (AC) were satisfied, the decay of u would be exponentialdue to the Lojasiewicz exponent equal to 1

2. Since the (AC) condition is not

satisfied, the decay is only polynomial. Further, we can apply Corollary 2.3.6with γ(s) =

√2s and obtain

‖u(t)‖ ≤ C(t− t0)−1α .

This is a better result since − 1α< 1

α− 1. Moreover, transformation to polar

coordinates show that this result is optimal. More details can be found in[B4].


Chapter 3

Second order problems

The main application of the abstract results from Chapter 2 are second orderequations with damping. In this chapter, we assume that V → H → V ′ areHilbert spaces with embeddings being dense and continuous (we identifyH = H ′). We consider problems in the form

u+ g(u) + E ′(u) = 0, (3.1)

where E : M ⊂ V → R and g : H → V ′ are two given functions, E ∈ C2(M).In the following, we write ‖ · ‖ instead of ‖ · ‖H , ‖ · ‖∗ instead of ‖ · ‖V ′ and〈·, ·〉, 〈·, ·〉∗ instead of 〈·, ·〉H , 〈·, ·〉V ′ respectively. In a special case V = H =V ′ = RN we have an ordinary differential equation of second order.

A typical example of such equation (and probably the most studied case)is a nonlinear wave equation with damping

utt + g(ut)−∆u+ f(x, u) = 0, t ≥ 0, x ∈ Ω ⊂ Rn, (3.2)

where g : R → R, f : R × R → R are continuous, H = L2(Ω), V = H10 (Ω).

This equation can be rewritten as (3.1) with

E(u) =1

2

∫

Ω

|∇u(x)|2dx+

∫

Ω

F (x, u(x))dx, F (x, s) =

∫ s

0

f(x, r)dr.

(3.3)Equation (3.1) can be written as a first order problem

U + F(U) = 0, (3.4)

where

U =

(u

v

), F(U) =

( −vg(v) + E ′(u)

),

29

30 CHAPTER 3. SECOND ORDER PROBLEMS

F : M ⊂ V → H, H = H × V ′, V = V × H, M = M × H. Our keyassumption is

〈g(v), v〉V ′,V > 0 for all v ∈ V , v 6= 0

which means that g has a damping effect. We denote by S := u ∈ M :E ′(u) = 0 the set of stationary points of (3.1). Then CrF = S × 0. Wedefine

E1(u, v) =1

2‖v‖2 + E(u),

then for any solution to (3.4) such that u = v ∈ L1loc(R+, V ) we have

d

dtE1(U(t)) = 〈E ′(u), v〉V ′,V + 〈v, v〉V,V ′

= 〈E ′(u), v〉V ′,V − 〈v, g(v)〉V,V ′ − 〈v, E ′(u)〉V,V ′= −〈v, g(v)〉V,V ′≤ 0.

We can see that E1 is a Lyapunov function (not neccessarily strict). More-over, condition (1.8) holds: if E1(u(·)) is constant, then 〈v, g(v)〉V,V ′ = 0,hence v = 0 and u(·), v(·) are constant. So, (3.1) is a weakly gradient-likesystem, but E1 typically does not satisfy (GenLI), (2.9), so Theorems 2.1.4,2.2.3 cannot be applied. The proofs of the convergence results in this chapterrest in finding another Lyapunov function E which satisfies these additionalconditions. This E is usually in the form Φ(E1), where E1 is a small pertur-bation of E1 and Φ : R+ → R+ is an increasing function.

In Section 3.1 we consider the finite-dimensional case, i.e., V = H =V ′ = RN and we obtain an ordinary differential equation with E ′ identifiedwith ∇E. In Section 3.2 we consider the general infinite-dimensional case.The results of both sections (convergence results and also results on decayestimates) usually assume that we have a solution with a precompact rangeand that E satisfies the Kurdyka– Lojasiewicz–Simon inequality. Thereforewe discuss in Section 3.3 the problem of precompact range of solutions and listsome known sufficient conditions on E to satisfy the Kurdyka– Lojasiewicz–Simon inequality.

Finally, let us mention that we mostly focus on the damping term. Ourintuition tells us that the smaller is the damping the slower is convergenceto an equilibrium, and if the damping is too small for v small, then it mayhappen that it is not strong enough to stabilize the system and the system(e.g. an oscilating spring) may keep oscilating. The results of this chapter


confirm this intuition, they show that convergence occurs if g(v) is largeenough for v near zero. We try to find as small lower bound for g as possible,in particular we focus on functions g with

limu→0

‖g(u)‖‖u‖ = 0.

The most typical example is g(u) = g(|u|)u (e.g. g(u) = |u|αu, α ∈ (0, 1)),which means that the damping force acts in the opposite direction to ve-locity and its size depends on the size of velocity only. But we also allowmore general cases, e.g. damping depending on the direction of velocity(which corresponds to motions in an anisotropic environment) or dampingdepending not only on u but also on u (which corresponds to inhomogeneousenvironment).

We do not present any nonconvergence results in the next sections. Thefollowing nonconvergence example is due to Haraux. He proved in [34, Propo-sition 5.1.2] that the equation

u+ (u)2 + f(u) = 0

with f = 0 on [a, b], f < 0 on (−∞, a) and f > 0 on (b,+∞) has boundedsolutions with [a, b] ⊂ ω(u). However, sharpness of the convergence resultsand optimality of decay estimates for weakly damped equations remain anopen question.

3.1 Finite-dimensional case

In this section we consider the finite-dimensional case, i.e., second orderordinary differential equation with damping

u+ g(u) +∇E(u) = 0. (3.5)

If g ∈ C1(Rn,Rn), E ∈ C2(Ω), Ω ⊂ Rn, then there exists a unique maximalsolution to (3.5) for any initial data (u(0), v(0)) ∈ Ω × Rn and the solutiondepends continuously on the initial data. As was mention above, this impliesω(U) ⊂ Cr(F), hence ψ = 0 and ∇E(ϕ) = 0 for every (ϕ, ψ) ∈ ω(U). Thena standard argument yields limt→+∞ u(t) = 0 whenever ω(U) is nonempty(in particular for any bounded solution U).

Probably the first convergence result based on the Lojasiewicz inequality(so without assuming a special structure of E) in finite-dimensional settingis due to Haraux and Jendoubi [36].


Theorem 3.1.1 ([36], Theorem 1.1). Assume that E : Rn → R is analyticand g : Rn → Rn is locally Lipschitz continuous and satisfies for all v ∈ RN

〈g(v), v〉 ≥ c‖v‖2, (3.6)

‖g(v)‖ ≤ C‖v‖, (3.7)

with 0 < c ≤ C < +∞ independent of v. Let u ∈ W 1,∞(R+,Rn) be a solutionto (3.5). Then there exists ϕ ∈ S such that

limt→+∞

‖u(t)‖+ ‖u(t)− ϕ‖ = 0.

In this theorem, analyticity of E can be replaced by E ∈ C2(RN) and Esatisfies the Lojasiewicz inequality (LI). The damping function g is nonlinear,but it is larger than a linear function (in fact, it satisfies c‖v‖ ≤ g(v) ≤ C‖v‖,so it is not a weakly damped equation. A similar result can be found inAlvarez et al. 2002 (see [3, Theorem 4.1]) for the equation

u+ (γ + β∇2E(u))u+∇E(u) = 0

with E : Rn → R analytic (hence satisfying (LI)).In 2015, Begout, Bolte and Jendoubi [10] consider linear damping g(u) =

γu and more general potential F satisfying the Kurdyka– Lojasiewicz inequal-ity with a function Θ s.t. 0 < Θ(s) ≤ c

√s for s ∈ (0, τ) and 1

Θ∈ L1

loc([0, 1)).Under these assumptions, they obtain convergence to an equilibrium and alsodecay estimates

‖u(t)− ϕ‖ = O(Φ(ψ−1(t− t0))),

i.e. the same decay estimates as Chill and Fiorenza in [22] for first orderproblems. In fact, second order problems with linear damping are gradient-like systems satisfying conditions (AC), (2.3), so they have the same decayas corresponding gradient systems (first order problems).

Concerning weak damping, in 2008 Chergui [19] proved convergence anddecay estimates for analytic E and g(u) = ‖u‖αu.

Theorem 3.1.2 ([19], Theorems 1.2, 1.3). Assume that E ∈ C2(Rn) andthat there exists θ ∈ [0, 1

2) such that for every ψ ∈ S there exists η > 0 such

that (LI) holds. Let α ∈ [0, θ1−θ ) and let u ∈ W 2,∞(R+,Rn) be a solution to

(3.5) with g(u) = ‖u‖αu. Then there exists ϕ ∈ S such that

limt→+∞

‖u(t)‖+ ‖u(t)− ϕ‖ = 0.


Moreover, there exists C > 0 such that for all t ∈ R+

‖u(t)‖+ ‖u(t)− ϕ‖ ≤ Ct−θ−(1−θ)α

1−2θ+(1−θ)α . (3.8)

We have generalized this convergence result in [B1, Theorem 4] (with-out decay estimates). We assumed E ∈ C2(RN) satisfying the Kurdyka– Lojasiewicz inequality (KLI) instead of the Lojasiewicz inequality and thedamping function being dependent on u (not only u) and having non-power-like growth, in particular we assumed

u+ g(u, u) +∇E(u) = 0, (3.9)

with g ∈ C2(Rn × Rn,Rn) satisfying for all u, v ∈ Rn

〈g(u, v), v〉 ≥ h(‖v‖) ‖v‖2,

‖g(u, v)‖ ≤ Ch(‖v‖) ‖v‖,‖∇g(u, v)‖ ≤ C h(‖v‖),

(3.10)

where C ≥ 0 is a constant and h : R+ → R+ is a nonnegative, concave,nondecreasing function, g(s) > 0 for s > 0. In fact, g(u, v) = ‖v‖αv sat-isfies (3.10) with h(s) = sα and condition α ∈ [0, θ

1−θ ) in Theorem 3.1.2corresponds to the condition (3.11) below.

Theorem 3.1.3 ([B1], Theorem 4). Let u ∈ W 2,∞(R+;Rn) be a global so-lution of (3.9) with g satisfying (3.10). Assume that there exist ϕ ∈ ω(u),η > 0 and a nonnegative, concave, nondecreasing function Θ : R+ → R+

such that (KLI) holds. Assume that Θ(s) ≤ c√s for some c > 0 and all

s ≥ 0 small enough and that

s 7→ 1/Θ(s)h(Θ(s)) ∈ L1loc([0,+∞)). (3.11)

Then u has finite length and, in particular,

limt→+∞

‖u(t)‖+ ‖u(t)− ϕ‖ = 0.

The proof of this theorem follows the idea from [19] but we show that thisproblem fits in the abstract framework described in the previous chapter. Infact, we have rewritten (3.9) as a first order equation on the product spaceand we have shown that the function

E(u, v) =1

2‖v‖2 + E(u) + ε 〈g(u,∇E(u)), v〉


is a strict Lyapunov function for the equation and that the generalized Lojasiewicz inequality (GenLI) is satisfied with Θ replaced by the functionΘ(s) = Θ(s)h(Θ(s)).

In Theorem 3.1.3 we still assume that ‖g(u, v)‖ lies between two multiplesof a function h(‖v‖)‖v‖ for all v ∈ Rn (similarly to Haraux and Jendoubi[36]). In [B3] we further investigated which assumpions on the dampingfunction g are important and which can be relaxed (primarily for a dampedwave equation) and we ended up with different estimates from above andfrom below.

(e) Let E ∈ C2(Rn,R) satisfy (KLI) with a function Θ : [0, 1) → [0,+∞)which is nondecreasing, sublinear (Θ(s+t) ≤ Θ(s)+Θ(t)), and it holdsthat 1

Θ∈ L1

loc([0, 1)) and 0 < Θ(s) ≤ c√s for all s ∈ (0, 1) and some

c > 0.

(g) The function g : Rn × Rn → Rn is continuous and there exists τ > 0such that

(g1) there exists C2 > 0 such that ‖g(w, z)‖ ≤ C2‖z‖ for all ‖z‖ < τ ,w ∈ Rn,

(g2) there exists C3 > 0 such that C3‖z‖ ≤ ‖g(w, z)‖ for all ‖z‖ ≥ τ ,w ∈ Rn,

(g3) there exists C5 > 0 such that 〈g(w, z), z〉 ≥ C5‖g(w, z)‖‖z‖ for allw, z ∈ Rn.

(h) For τ from condition (g) there exists a function h : [0,+∞)→ [0,+∞),which is concave and nondecreasing on [0, τ ] and satisfies

(h1) ‖g(w, z)‖ ≥ h(‖z‖)‖z‖ for all ‖z‖ < τ , w ∈ Rn,

(h2) the function s 7→ 1Θ(s)h(Θ(s))

belongs to L1((0, τ)),

(h3) the function ψ : s 7→ sh(√s) is convex on [0, τ 2].

In fact, condition (g2) can be weakened to ‘g(w, z) 6= 0 for all z 6= 0’ whichtogether with (g3) yields 〈g(w, z), z〉 > 0 for all z 6= 0. This last conditionimplies that u→ 0 for any bounded solution, so we do not need any furtherassumptions on g(w, z) for ‖z‖ > τ .


Theorem 3.1.4 ([B3], Theorem 6.1). Let the functions E and g satisfy (e),(g), and (h). Let u ∈ W 1,∞((0,+∞),RN) ∩W 2,1

loc ([0,+∞),Rn) be a solutionto (3.9) and let ϕ ∈ ω(u). Then limt→+∞ (‖u(t)− ϕ‖+ ‖u(t)‖) = 0.

Under the same assumptions we obtained decay estimates in [B4, Theo-rem 6]

Theorem 3.1.5 ([B4], Theorem 6). Let the functions E and g satisfy (e),(g), and (h). Let u ∈ W 1,∞((0,+∞),Rn) ∩W 2,1

loc ([0,+∞),Rn) be a solutionto (3.9) and ϕ ∈ ω(u). Then there exists t0 > 0 such that

|u(t)|+ |u(t)− ϕ|+∫ +∞

t

|u(s)|ds ≤ Φ(ψ−1(t− t0)), (3.12)

holds for all t > t0, some C1, C2 > 0 and

Φ(t) = C1

∫ t

0

1

Θ(s)h(Θ(s))ds, ψ(t) = C2

∫ 12

t

1

Θ2(s)h(Θ(s))ds. (3.13)

The proofs of these theorems are again based on the abstract results fromthe previous chapter. This time, we work with the energy function

E(u, v) := Φ(H(u, v)), H(u, v) =1

2‖v‖2 + E(u) + εh(‖v‖) 〈∇E(u), v〉

and show that

− d

dtE(u(t), v(t)) ≥ ‖u(t)‖,

which is the condition (2.8) from Theorem 2.2.2. In fact, this inequality canbe rewritten as

Θ(H(u, v)−H(ϕ, 0)) ≤⟨H(u, v),

F(u, v)

‖F1(u, v)‖

⟩(3.14)

which is almost (GenLI), in the denominator we have instead of F(u, v) itsfirst coordinate F1(u, v) = −v (we denote Θ = Θh(Θ) as above). Further,we have shown that condition (2.18) from Theorem 2.3.4 is valid with α(s) =Θ(s). To obtain decay estimates, we cannot apply Theorem 2.3.4 directlysince ‖F1‖ in (3.14) can be much smaller than ‖F‖. However, it holds ([B4,Lemma 8]) that ∫ t

t0

‖F1(u(s), v(s))‖α(H(u(s), v(s)))

ds ≥ L(t− t0),


which is enough to do the remaining step in the decay estimates.Let us remark that if g(u, v) = ‖v‖αv (then h(s) = sα) and Θ(s) =

s1−θ then Theorem 3.1.5 yields the same decay estimate as Theorem 3.1.2.Further, we can obtain more delicate estimates in the logarithmic scale. By[B5, Example 5.3, Lemmas 6.5, 6.6], if Θ(s) = s1−θ and

h(s) = sα lnr1(1/s)(lnr2 ln(1/s)) . . . (lnrk ln . . . ln(1/s)), (3.15)

on a neighborhood of zero, then for a < θ1−θ we obtain

‖u(t)− ϕ‖ ≤ Ct−θ−a(1−θ)

1−2θ+a(1−θ) ln−q1(t) ln−q2(ln(t)) . . . ln−qk(ln . . . ln(t)), (3.16)

where qk = rk(1−θ)1−2θ+a(1−θ) and for a = θ

1−θ , r1 = · · · = rj−1 = 1, rj > 1, rj+1,. . . , rk ∈ R we obtain

‖u(t)− ϕ‖ ≤ C ln1−rj(ln . . . ln(t)) ln−rj+1(ln . . . ln(t)) . . . ln−rk(ln . . . ln(t)).

If we have a direct estimate of ‖u‖ by the potential E due to specialstructure of E (see (3.17) and Example 3.1.7 below), we get better decayestimates by the following theorem.

Theorem 3.1.6 ([B4], Theorem 7). Let the assumptions of Theorem 3.1.5hold and let

Θ(|E(u)− E(ϕ)|) ≥ c‖∇E‖ for all u ∈ B(ϕ, η).

Moreover, let γ be a nondecreasing function satisfying

γ(|E(u)− E(ϕ)|) ≥ ‖u− ϕ‖ for all u ∈ N(ϕ). (3.17)

Then

|u(t)| ≤ C√ψ−1(t− t0) and |u(t)− ϕ| ≤ Cγ(ψ−1(t− t0)), (3.18)

holds for all t > t0 and some C > 0, ψ defined as in (3.13).

Example 3.1.7. If we consider

u+ g(u, v) + p‖u‖p−2u = 0, p ≥ 2

which corresponds to E(u) = ‖u‖p, then we have

Θ(E(u)) ≤ c‖∇E(u)‖ ≤ CΘ(E(u))


with Θ(s) = s1− 1p , i.e., the Lojasiewicz inequality holds with θ = 1

p. Moreover,

(3.17) holds with γ(s) = s1p . Then, for ‖g(u, v)‖ ≥ h(‖v‖)‖v‖ = ‖v‖α+1 we

have‖u(t)‖ ≤ Ct−

θ1−2θ+α(1−θ) , (3.19)

which is better estimate than (3.8). For the function h given by (3.15) weobtain

‖u(t)− ϕ‖ ≤ Ct−θ

1−2θ+a(1−θ) ln−q1(t) ln−q2(ln(t)) . . . ln−qk(ln . . . ln(t))???,(3.20)

where qi are as in (3.16).

In [35], Haraux has found optimal decay estimates for the damping func-tion g(u) = |u|βu and E(u) = ‖u‖α. It follows that the estimate (3.19) isoptimal only for p = 2. Optimality of the estimates for general E is open.

3.2 Infinite-dimensional case

In this section we study convergence of solutions to the second order evo-lution equation (3.1) in infinite-dimensional spaces. In contrast to finite-dimensional case, well-posedness of the problem is not always easy to proof.This is not a crucial problem for the following results since they consideronly one trajectory. On the other hand, well-posedness implies that ωV(U) ⊂Cr(F) = S × 0, so it reduces the set of points where E should satisfy theKurdyka– Lojasiewicz–Simon inequality and also some methods of verifyingthe precompact range condition need well-posedness. However, in many ex-amples one can show ωV(U) ⊂ Cr(F) or u(t) → 0 ad hoc. The followingcriterion applies in a large class of problems (in fact, the ∗-norm in (3.21)can be replaced by any weaker norm).

Theorem 3.2.1 ([B2], Theorem 2.8). Let g ∈ C(V × H, V ′), E ∈ C2(V )and assume that there exists a nondecreasing function h : (0,+∞)→ (0,+∞)such that

〈g(u, v), v〉V ′,V ≥ h(‖v‖∗) (3.21)

for all u, v ∈ V , v 6= 0. Let u ∈ C1(R+, V ) ∩ C2(R+, H) be a classicalsolution to

u(t) + g(u(t), u(t)) + E ′(u(t)) = 0, u(0) = u0 ∈ V, u(0) = u1 ∈ H (3.22)

such that (u, u) is precompact in V ×H. Then limt→+∞ ‖u(t)‖ = 0.


In the article [60] where Simon used for the first time the Lojasiewiczinequality in infinite-dimensional setting, he also proved a convergence resultfor a class of second order evolution equations with linear damping. Jendoubi[46] (see also [47]) proved convergence of solutions to

u+Bu+ Au = f(x, u)

where A is a self-adjoint linear operator associated with a coercive bilinearform on V → L2(Ω) and B is a bounded linear operator. The functionf is analytic and no global growth assumptions on f . On the other hand,precompactness of range in W 2,p ∩W 1,p was needed. In 1999 Haraux andJendoubi [37] extended the convergence result to weak solutions precompactin V × L2. Then the nonlinearity f is assumed to be analytic and satisfyf ∈ C1(B, V ′) for some ball B ⊂ V , which is in fact a growth condition. IfA = ∆, then f satisfying

|∂sf(x, s)| ≤ C(1 + |s|γ) with γ ≥ 0, (N − 2)γ < 2 (3.23)

are admissible. Moreover, they allowed the damping operator B : V → V ′

to be nonlinear (but still satisfying 〈B(v), v〉V ′,V ≥ c‖v‖2, so the dampingis not weak). Moreover, they have shown that every bounded solution hasprecompact range in V × L2.

An abstract wave equation with linear damping was cosidered in Chill,Haraux and Jendoubi [24], where convergence to equilibrium was proved forprecompact solutions to (3.1) with g(u) = γu if E satisfies the Lojasiewicz–Simon inequality and E ′′(u) = M ′(u) satisfies condition (E2) below. Rate ofconvergence is also estimated by an exponential or a polynomial (dependingon the Lojasiewicz exponent θ of the energy E).

A wave equation with weak damping was studied in 2009 by Chergui [20]and convergence to equilibrium was shown for H1

0×L2(Ω)-bounded solutionsto (3.2) with Dirichlet boundary conditions, g(ut) = |ut|αut and f analyticsatisfying f(x, 0) ∈ L∞(Ω) and (3.23) if N ≥ 2. The exponent α is assumedto belong to [0, θ

1−θ ) (θ is again the Lojasiewicz exponent of the energy E)

and to satisfy α < 4N−2

if N ≥ 3. In fact, Chergui shows that under theseassumptions on f , the Lojasiewicz–Simon inequality holds with an appropri-ate θ and that bounded solutions are in fact precompact. Once he has thesefacts, he proves convergence to equilibrium.

In 2011, Ben Hassen and Haraux [13] proved convergence to equilibriumand decay estimates for an abstract wave equation (3.1) with E bounded


from below and weak nonlocal damping g : V → V ′ with power-like behavior

〈g(v), v〉V ′,V ≥ c1‖v‖α+2, v ∈ V,‖g(v)‖V ′ ≤ c2‖v‖α+1, v ∈ H. (3.24)

It is assumed that E satisfies the Lojasiewicz–Simon inequality, condition(E2) below and a kind of inverse Lojasiewicz inequality

‖E ′(u)‖V ′ ≤ cE(u)γ for some γ ∈ [1/2− α(1− θ), 1− θ] (3.25)

in a ball B ⊂ V containing the whole solution u (then no precompactness ofthe trajectory is needed).

In 2013, Aloui, Ben Hassen and Haraux [2] generalized Chergui’s resultfrom [20] to abstract wave equations (3.1) with a large class of dampingoperators g similar to [13], in fact they replaced the second condition in(3.24) with

‖g(v)‖V ′ ≤ 〈g(v), v〉α+1α+2

V ′,V , v ∈ V,

which is a similar condition to (G1′) below (here E is not bounded frombelow and precompactness of the range is asssumed).

In [B3] we have generalized the result by Chergui [20] to abstract waveequations and to more general damping functions. Our assumptions on E arethe same as in [24] but we allow more general Kurdyka– Lojasiewicz–Simoninequality instead of Lojasiewicz–Simon inequality. Our assumptions on thedamping function g are similar to those in finite-dimensional case. Basically,on a neighborhood of zero g is bigger than a concave function h which isrelated to the function Θ from the Kurdyka– Lojasiewicz–Simon inequalityby condition (h2) below. This relation becomes α < θ

1−θ in Chergui’s case.In contrast to the finite-dimensional case, a growth condition in infinity isneeded, see (g3). This condition implies g : V → V ′.

In [B3, Theorem 2.1] we stated a convergence result for a ‘scalar-valued’damping function g(u)(x) = g(|u(x)|)u(x) and in [B3, Theorem 5.1] fora ‘vector-valued’ (but still pointwise) function g(u, u)(x) = G(u(x), u(x)),which may depend on u (not only u). Here are the assumptions and theresult.

(E) Assume that E ∈ C2(V ) satisfies:


(E1) there exists a function Θ : [0, 1) → [0,+∞) which is nonde-creasing, sublinear (Θ(s + t) ≤ Θ(s) + Θ(t)), and it holds that1Θ∈ L1

loc([0, 1)) and 0 < Θ(s) ≤ c√s for all s ∈ (0, 1) and some

c > 0 and such that E satisfies the Kurdyka– Lojasiewicz–Simongradient inequality with the function Θ in a neighbourhood of thecritical points of E, i.e., for each ϕ ∈ N := ψ ∈ V : E ′(ψ) = 0there exist η, C > 0 such that

‖E ′(u)‖∗ ≥ CΘ(|E(u)− E(ϕ)|), u ∈ BV (ϕ, η);

(E2) for all u ∈ V , the operator KE ′′(u) ∈ L(V ) extends to a boundedlinear operator on H and sup ‖KE ′′(u)‖L(H) is finite whenever uranges over a compact subset of V .

(G) The function G : Rn × Rn → Rn is continuous and there exists τ > 0such that

(G1) there exists C2 > 0 such that |G(w, z)| ≤ C2|z| for all z ∈BRn(0, τ), w ∈ Rn,

(G2) there exists C3 > 0 such that C3|z| ≤ |G(w, z)| for all z ∈ Rn \BRn(0, τ), w ∈ Rn,

(G3) if N = 2 then there exist C4 > 0, α > 0 such that |G(w, z)| ≤C4|z|α|z| for all z ∈ Rn \ BRn(0, τ), w ∈ Rn; if N > 2 then theinequality holds with α = 4

N−2,

(G4) there exists C5 > 0 such that 〈G(w, z), z〉 ≥ C5|G(w, z)||z| for allw, z ∈ Rn.

(H) For τ from condition (G) there exists a function h : [0,+∞)→ [0,+∞),which is concave and nondecreasing on [0, τ ] and satisfies

(H1) |G(w, z)| ≥ h(|z|)|z| for all z ∈ BRn(0, τ), w ∈ Rn,

(H2) the function s 7→ 1Θ(s)h(Θ(s))

belongs to L1((0, τ)),

(H3) the function ψ : s 7→ sh(√s) is convex on [0, τ 2].

Theorem 3.2.2 ([B3], Theorem 5.1). Let E and G satisfy (E), (G) and(H). Let u be a strong solution to

u+G(u, u) + E ′(u) = 0


such that (u, u) has V ×H-precompact range and let ϕ ∈ ωV (u). Then

limt→+∞

(‖u(t)− ϕ‖V + ‖ut(t)‖) = 0.

In [B5] we have generalized the result from [13] to more general damp-ing functions and E satisfying the Kurdyka– Lojasiewicz–Simon inequalityinstead of Lojasiewicz–Simon inequality and we combined this result witha previous one to weaken the assumptions on E (assuming on the otherhand precompactness of the trajectory). Conditions (E1), (E2) are similarto (E1), (E2) (but on a larger set), condition (E3) together with (G4) gen-eralizes (3.25) (let us mention, that in applications condition (E3) is oftensatisfied with G(s) = C

√s and in this case (G4) holds). Conditions (G3),

(G5) are the same as (H2), (H3) and conditions (G1), (G2) correspond to(G), (H1).

Our hypothesis below use the notion of admissible functions, which weak-en the assumptions on functions Θ and h from (E1), (H). We say thatf : R+ → R+ is admissible if it is nondecreasing and there exists cA ≥ 1such that for all s > 0 we have f(s) > 0 and sf ′(s) ≤ cAf(s). It holds thatevery nonnegative differentiable concave function is admissible with cA = 1.On the other hand, if f is admissible then f is C-sublinear, i.e. f(t + s) ≤C(f(t) + f(s)) for some C ≥ 0 and all t, s ≥ 0.

(E) Let E ∈ C2(V ), M = E ′ ∈ C1(V, V ∗) and let B be a fixed ball in V .Assume that:

(E1) E is nonnegative on B and there exists an admissible function Θsuch that Θ(s) ≤ CΘ

√s for all s ≥ 0 and some CΘ > 0, 1

Θis

integrable in a neighbourhood of zero and

‖M(u)‖∗ ≥ Θ(E(u)), for all u ∈ B, (KLS)

i.e., E satisfies the Kurdyka– Lojasiewicz–Simon gradient inequal-ity with function Θ on B.

(E2) There exists CM ≥ 0 such that

|〈M ′(u)v, v〉∗| ≤ CM‖v‖2 for all u ∈ B, v ∈ V ,

(E3) There exists a nondecreasing function Γ : R+ → R+ such that

‖M(u)‖∗ ≤ Γ(E(u)), for all u ∈ B. (3.26)


(G) The function g : V → V ∗ is continuous and there exists an admissiblefunction h such that

(G1) there exists C2 > 0 such that ‖g(v)‖∗ ≤ C2‖v‖ on V ∩B(0, R) forany R > 0 with C2 depending on R,

(G2) 〈g(v), v〉V ∗,V ≥ h(‖v‖)‖v‖2 on V ,

(G3) the function s 7→ 1Θ(s)h(Θ(s))

belongs to L1((0, 1)),

(G4) there exists CΓ > 0 such that Γ(s) ≤ CΓ

√s

h(Θ(s))on (0, K] for any

K > 0 with CG depending on K,

(G5) the function ψ : s 7→ sh(√s) is convex for all s > 0.

Theorem 3.2.3 ([B5], Theorem 2.1). Let E and g satisfy (E) and (G). Letu be a strong solution to (3.1) and there exists t1 > 0 such that u(t) ∈ B forall t ≥ t1. Then there exist ϕ ∈ B and t0 ≥ 0 such that

E(u(t)) ≤ 2Ψ−1(t− t0), (3.27)

‖u(t)− ϕ‖ ≤ Φ(Ψ−1(t− t0)), (3.28)

‖u(t)‖ ≤√

Ψ−1(t− t0)) (3.29)

hold for all t > t0, some C1, C1 > 0 and Φ, Ψ defined by (3.13)

Let us mention that if Θ(s) = cs1−θ, h(s) = sα, then we are in thesituation from [13] and the convergence rate we obtain is the same as in[13]. However, we can consider more general damping functions or we canget better decay estimates in the logarithmic scale as in finite-dimensionalcase. See Example 3.1.7 above or Example 3.2.8 below.

The next result combines the method from [20] (resp. [B3]) and [13] toobtain decay estimates for relatively compact solutions with (KLS) satisfiedonly on a small neighborhood of some ϕ ∈ ωV (u).

Theorem 3.2.4 ([B5], Theorem 2.2). Let u be a strong solution to (3.1)with (u, u) having V × H-precompact range and ϕ ∈ ωV (u) with E(ϕ) = 0.Let (E) and (G) hold with the following changes.

• (E1), (E3) hold with B replaced by BV (ϕ, δ) for some δ > 0,

• (E2) holds with B replaced by ‘any compact subset of V with CM de-pending on the subset’,


• h is admissible with cA = 1,

Then limt→+∞ ‖u(t) − ϕ‖V = 0 and there exists t0 ≥ 0 such that the decayestimates (3.27), (3.28) and (3.29) hold for all t > t0, some CΦ, CΨ > 0 andΦ, Ψ defined in (3.13).

The assumption (G) on a nonlocal damping function are not met by alllocal damping functions satisfying (G). However, if we replace (G1) by (G1′)

(G1′) for every R > 0 there exists a convex function γ : R+ → R+ withproperty (K) and such that γ(0) = 0, lims→+∞ γ(s) = +∞, γ(s) ≥ cs2

for some c > 0 and all s small enough, and γ(‖g(v)‖∗) ≤ 〈g(v), v〉V ∗,Von V ∩B(0, R),

then Theorems 3.2.3, 3.2.4 remain valid and the new assumptions follow from(G) as states the following Proposition. Similar condition appears in Aloui,Ben Hassen and Haraux [2].

Proposition 3.2.5 ([B5], Proposition 3.1). Let G : RN → RN satisfy (G),(H1) and define (g(v))(x) := G(v(x)) for v ∈ V . Then g(V ) ⊂ V ∗ and gsatisfies (G) with (G1) replaced by (G1′).

In [B5] this Proposition is formulated and proved for G independent of u(depending on u only), but it remains valid with the same proof if G dependson u.

Theorem 3.2.6 ([B5], Theorem 2.3). Theorems 3.2.3 and 3.2.4 remain val-id if we replace (G1) by (G1′).

As in finite-dimensional case, if we have direct estimates for ‖u − ϕ‖ byE(u), then we can obtain better convergence rates.

Corollary 3.2.7 ([B5], Corollary 2.4). Suppose that the hypotheses of Theo-rems 3.2.3, 3.2.4 or 3.2.6 are satisfied. Let α : R+ → R+ be a nondecreasingfunction such that α(E(u)−E(ϕ)) ≥ ‖u−ϕ‖ on a neighborhood of ϕ. Then

‖u(t)− ϕ‖ ≤ α(2Ψ−1(t− t0))

holds for some t0 and all t > t0.


Example 3.2.8. It is shown in [13] that the following two problems fit intothe framework considered in the theorems of this section: the Dirichlet prob-lem

utt + g(ut)−∆u− λ1u+ |u|p−1u = 0 in R+ × Ω,

u(t, x) = 0 on R+ × ∂Ω,(3.30)

and the Neumann problemutt + g(ut)−∆u+ |u|p−1u = 0 in R+ × Ω,∂∂nu(t, x) = 0 on R+ × ∂Ω,

(3.31)

where Ω ⊂ RN is a bounded domain with smooth boundary, λ1 is the firsteigenvalue of −∆ and p > 1 with (N−2)p < N+2. The corresponding energyfunctions E satisfy the Lojasiewicz inequality with θ = 1

p+1(so Θ(s) = Cs1−θ)

on any bounded subset of V and any strong solution to (3.30) is bounded in V .Moreover, Γ(s) = C

√s satisfies (E3) and (G4) and we have E(u) ≥ c‖u‖p+1

V .In contrast to [13] we can obtain convergence for a larger class of damping

functions, e.g. for (g(v)) = G(v(x)) with G having different growth/decay ineach direction and also for |s| large and |s| small, e.g.

G(s) =

|s|b1s, s > 1,

|s|a1s, s ∈ [0, 1],

|s|a2s, s ∈ [−1, 0),

|s|b2s, s < −1,

(3.32)

with 0 ≤ a1 < a2 <1p, b1, b2 ≤ 4

N−2if N > 2. Denoting A = maxa1, a2 we

have‖u(t)− ϕ‖V ≤ Ct−

1(A+1)p−1 , t ≥ t0

by Corollary 3.2.7. We can see that the rate of decay depends on the growthof G near zero only.

We can also consider

G(s) =

|s|as lnr(1/|s|) |s| ≤ 1,

c|s|bs |s| > 1,(3.33)

with b < 4N−2

, 0 < a < 1p, r ∈ R or a = 1

p, r > 1. In this case we obtain

more delicate decay estimates in the logarithmic scale, namely

‖u(t)− ϕ‖V ≤ C(Ψ−1(t− t0)

) 1p+1 ≤ Ct−

1(a+1)p−1 ln−

r(a+1)p−1 (t), t ≥ t0.


Let us replace |u|p−1u in (3.31) with f(u) defined as

f(u) =

(d1 − s)p−1(s− d1), s < d1,

0, s ∈ [d1, d2],

(s− d2)p−1(s− d2), s > d2

for some d1 < d2. Then all the assumptions remain valid except that we donot have E(u) ≥ c‖u‖p+1

V . In this case, we apply Theorem 3.2.6 instead ofCorollary 3.2.7 and obtain

‖u(t)− ϕ‖ ≤ Ct−1−Ap

(A+1)p−1 , t ≥ t0

for the damping function G given by (3.32). For G given by (3.33) we get incase a < 1

p

‖u(t)− ϕ‖ ≤ Ct−1−ap

(a+1)p−1 ln−pr

(a+1)p−1 (t), t ≥ t0

and in case a = 1p

‖u(t)− ϕ‖ ≤ C ln1−r(t), t ≥ t0.

See [B5, Examples 5.2, 5.3] for details.

Nonautonomous case.Further generalizations of the above results consider non-autonomous

equations of the type

u+ a(t)g(u) + E ′(u) = f(t) (3.34)

in finite-dimensional or infinite-dimensional settings. In order to obtain con-vergence to an equilibrium, we need to assume that f is not too large and ais not too small.

First results of this kind are due to Chill and Jendoubi [25] and BenHassen [12] for g(s) = s, a(t) = 1 (in Hilbert spaces) and Cabot, Engler andGadat for the case a(t) ≥ a0, g(s) = s, f = 0 (in Rn assuming that Cr(∇E)is finite).

In 2013, Haraux and Jendoubi [40] proved convergence to equilibriumand decay estimates in Rn for f = 0, g identity, E satisfying the Lojasiewicz


inequality (LI) and a(t) ≥ (1 + t)−β. Also vector-valued functions a wereconsidered.

The case a(t) ≡ 1, f 6= 0, g nonlinear was studied in 2011 by Haraux [35]for g satisfying

‖g(v)‖ ≤ K‖v‖α+1, 〈g(v), v〉 ≥ c‖v‖α+2 (3.35)

and E ∈ W 2,+∞(H) satisfying

‖E(u)‖ ≤ K‖u‖β+1, 〈∇E(u), u〉 ≥ c‖v‖β+2 (3.36)

on bounded sets in a Hilbert space H. Haraux’s result gives convergenceand optimal decay estimates for exponentially decaying functions f and forarbitrarily large α (i.e. very weak damping).

Theorem 3.2.9 ([35], Theorem 6.1 and 6.2). Let B1, B2 be two closed ballsin a Hilbert space H, g ∈ W 1,∞(B1, H), E ∈ W 2,∞(B2, R), f ∈ C(R+, H)and u ∈ C2(R+, H) be a solution to (3.34) such that (u(t), u(t)) ∈ B1 × B2

for all t > 0. Assume that (3.35) and (3.36) hold for all u ∈ B1, v ∈ B2. Letus define

E(t) :=1

2‖u(t)‖2 + E(u(t)).

If α < ββ+2

and λ ≥ (α+1)(β+1)β−α then

‖E(t)‖ ≤ Ct−(α+1)(β+1)

β−α , t ≥ 1.

If α ≥ ββ+2

and λ ≥ 1 + 1α

then

‖E(t)‖ ≤ Ct−2α , t ≥ 1.

Ben Hassen and Chergui [14, Theorem 1.6] showed convergence to equi-librium and decay estimates in Rn without assuming a special structure ofE (assumption (3.36) replaced by the Lojasiewicz inequality (LI)) and forpolynomially decaying f . Then the damping cannot be too small (α < θ

1−θ ).

For E(u) = ‖u‖β+1, the decay estimates in [35] are better than those in [14].

Theorem 3.2.10 ([14], Theorem 6.1). Let g ∈ C(RN ,RN), E ∈ C2(RN ,R),f ∈ C(R+,RN) and u ∈ W 1,+∞(R+,RN) be any solution to (3.34) Assumethat g satisfies (3.35) for all v in a bounded sets (with c, K depending on the


set), E satisfies (LI) for any ϕ ∈ S with θ ∈ (0, 12], η > 0 independent of ϕ,

and f satisfies

‖f(t)‖ ≤ C

(1 + t)1+δ+α, t ≥ 0

for appropriate C ≥ 0 and δ > 0. If α < θ1−θ then there exists ϕ ∈ S such

that limt→+∞ ‖u(t)‖+ ‖u(t)− ϕ‖ = 0 and

‖u(t)− ϕ‖ ≤ C(1 + t)−µ, t ≥ 0

with µ = min

1−(α+1)(1−θ)(α+2)(1−θ)−1

, δα+1

.

In [39], Haraux and Jendoubi proved weak convergence of solutions to

Au+ aAu+∇E(u) = f(t)

with a selfadjoint bounded linear operator A on a Hilbert space and a convexpotential E.

The case of nonconstant a and f 6≡ 0 was considered by Jendoubi and May[48] for convex potentials E on a Hilbert space and g being identity. Weakconvergence was obtained for appropriate polynomial decays of f and a (see[48, Theorem 1.3]). The case of nonlinear g and (nonconvex) E satisfyingthe Lojasiewicz inequality (LI) in Rn was solved in 2015 by Balti [4].

Theorem 3.2.11 ([4], Theorem 1.2, Remark 1.7). Let E ∈ W 2,∞loc (RN), γ ∈

L∞ be a positive function, u ∈ W 2,1loc ∩ L∞(R+,RN) be a solution to (3.34).

Let S = arg minE and E satisfies (LI) for all ϕ ∈ S with a fixed θ ∈ (0, 12]

and C > 0, η > 0 depending on ϕ. Let g satisfies (3.35) on RN ,

‖f(t)‖ ≤ d

(1 + t)1+δ, for all t ≥ 0

for some d, δ > 0 and let

‖a(t)‖ ≥ c

(1 + t)β, for all t ≥ 0

for some c > 0 and β ≥ 0 such that α + β ∈ (0,min θ1−θ , δ). Then there

exist ϕ ∈ S and M ≥ 0 such that

‖u(t)− ϕ‖ ≤Mt−µ, for all t ≥ 0,

where

µ = min

θ − (α + β)(1− θ)(1− θ)(α + 2)− 1

,δ − (α + β)

α + 1

.


3.3 Appendix to second order problems

3.3.1 Well-posedness and existence of global solutions

Although the convergence results in Section 3.2 hold for ill-posed problemsas well, we list here some results on well-posedness and global existence for(3.1).

A well-posedness result which includes also nonmonotone damping func-tions can be found in Haraux ’87 [33, Theorem II.2.2.1]. It concerns a problem

utt + Lu+ g(ut) + f(u) = cut + h(t, x) (3.37)

with V → H = L2(Ω), Ω ⊂ RN bounded domain, L : V → V ′ being a linearoperator associated with a coercive bilinear form on V , the nonlinearity f ∈C1(R) is such that u(·) 7→ f(u)(·) maps V into H (i.e. so called subcriticalcase) and it is Lipschitz continuous on bounded subsets of V . Functiong : R → R is assumed to be continuous and nondecreasing with g(0) = 0,h ∈ L1([0, T ], H) and c ≥ 0 (this means that the damping function s 7→g(s)− sc is not neccessarily nondecreasing).

Global existence for (3.37) is shown in [33, Theorem II.2.2.2] under addi-tional assumptions

C := inf

∫ u

0

f(z)dz +Mu2 : u ∈ R> −∞

for some M ∈ R and if Ω has infinite measure, then C ≥ 0. This assumptionmeans that the corresponding energy E given by (3.3) is bounded from below.

If the energy is not bounded from below, global existence depends onthe interplay between the source term f and the damping term g. In 1994,Georgiev and Todorova [29] studied (3.37) with L = −∆, V = H1

0 (Ω), f(u) =−|u|p−1u, g(v) = |v|m−1v, h ≡ 0, c = 0 for p ≤ N

N−2(which corresponds to

subcritical case). They have shown global existence for all initial values inV ×H for p ≤ m, blow-up of solutions with negative initial energy for m < p.

Levin, Park and Serrin in 1998 [52] studied global existence in subcriticalcase for source and damping terms depending on x (resp. x and t):

utt + g(t, x, ut)−∆u = f(x, u) (3.38)

with Ω = RN , g satisfying g(t, x, v)v ≥ 0 and some estimates from below(no monotonicity needed). They have shown that local existence implies

3.3. APPENDIX TO SECOND ORDER PROBLEMS 49

global existence, whenever so called continuation proprerty holds, i.e. ifevery bounded local solution can be continued. However, this continuationproperty is known (according to [52]) only for g(t, x, v) = a|v|m−2v.

In supercritical case, global existence for any initial data (and existenceof a global attractor) was proved by Feireisl in 1995 [27] for any boundedregular domain Ω ⊂ R3 and strictly increasing g ∈ C1(R) depending only onut satisfying g(0) = 0 (no additional assumptions at zero) and for appropriategrowth conditions of g and f in infinity (f independent of x).

Serrin, Todorova and Vittilaro in 2003 (see [59]) proved local and globalexistence in a more general (supercritical) case (g depending on t, x, f de-pending on x, Ω ⊂ RN) for compactly supported initial data, under somegrowth and regularity conditions on f and g, g increasing in the third vari-able and having the same power-like growth near zero and infinity. Furtherresults are due to Radu (see [57], [58]).

Benaissa and Mokeddem in 2004 (see [11]) showed global existence forΩ = Rn, f(x, u) = |u|p−1u − λ2(x)u (allowing also supercritical growth), gnondecreasing depending on ut only and satisfying c1|v|1/m ≤ |g(v)| ≤ c2|v|mfor small v and c3|v| ≤ |g(v)| ≤ c4|v|r for large v (for appropriate positiveconstants ci, r, m, p) and sufficiently small initial energy. They also provedecay estimates for the energy.

Global existence for an abstract problem u+A(u) +B(t)u+G(u) = f(t)in Banach spaces was proved by Biazutti in 1995 ([15]). Here B : V → V ′ isan operator associated with a (uniformly in t) positive definite bilinear form,so this part of damping is stronger than linear. On the other hand, G is anonlinear operator of lower order which can be negative near zero, so in factthere can be negative damping for small values of u.

3.3.2 Precompactness of bounded solutions

In this subsection we discuss the assumption on precompactness of solutions.Of course, in finite-dimensional case, all bounded solutions have precompactrange. The same implication holds in some infinite-dimensional problems.

An abstract result by Webb [61] states that if (S(t))t≥0 is a dynamicalsystem on a metric space X which can be written as a sum of S1(t), S2(t)with limt→+∞ ‖S1(t)‖ = 0 and S2(t) compact for all t large enough, then allbounded orbits are precompact. Another result by Webb says that if T is a


dynamical system which is bounded on bounded sets and satisfies

T (t) = S(t) +

∫ t

0

S(t− s)BT (s) (3.39)

with S that can be splitted as above and B a bounded operator on X, thenbounded orbits of T are precompact. In fact, the second results applies toperturbed problems — if A generates a linear semigroup S, then A + Bgenerates a semigroup T satisfying (3.39). Unfortunately, this result can beapplied only if we have well-posedness.

In 1980 Webb applied this perturbation result to a damped wave equation

utt −∆ut −∆u = f(u)

on a smooth bounded domain with Dirichlet boundary conditions with f ∈C1(R), |f ′| ≤ M , lim sup|x|→+∞ f(x)/x ≥ 0 and f(0) = 0 (see [62]). Let usmention that conditions on f imply that the energy is bounded from below.Webb proved that for any initial data, there exists a global solution and ithas precompact range.

In 1999, Haraux and Jendoubi [37] showed precompactness of boundedsolutions of a linearly damped equation

utt + cut −∆u = f(x, u)

with ∂uf only locally bouded and globally satisfying |∂uf(x, u)| ≤ C(1+|u|α)for some C > 0, α < 2

N−2(which means that the energy is not neccessarily

bounded from below but the growth is subcritical).In 2009, Chergui [20] proved precompactness of bounded solutions under

the same assumptions on f but with a nonlinear (possibly weak) dampingg(ut) , g : R+ → R+ increasing, g−1 uniformly continuous. In fact, Cher-gui has shown that the right-hand side h(t, x) = f(x, u(t, x)) satisfies theassumptions of a criterion by Haraux [31, Theorem 4.1].

Ben Hassen and Chergui in 2011 [14] further generalized this result to anon-autonomous equation

utt + |ut|αut −∆u+ f(u) = h(t, x).

Aloui, Ben Hassen and Haraux 2013 [2] proved precompactness of bound-ed trajectories for a large class of abstract semilinear problems

u+ g(u) + Au+ f(u) = h(t),


where A : V → V ′ is the duality mapping, f = ∇F : V → V ′ is a gradientfield and g : V → V ′ a nonlinear monotone damping operator with f beingLipschitz continuous on bounded sets from W to H for some W , V ⊂⊂ W ⊂H.

The compactness results from [20] and [2] can be helpful for verifying theassumptions of Theorems 3.2.2 — 3.2.6, also for damping functions that arenot covered by convergence results in [20] and [2].

3.3.3 Functions satisfying the Lojasiewicz inequality

In the results of Sections 3.1, 3.2 we assume that E satisfies the Lojasiewiczgradient inequality (or Lojasiewicz–Simon, Kurdyka– Lojasiewicz–Simon in-equality). Now we present some sufficient conditions on E to satisfy thesegradient inequalities.

By Lojasiewicz [53], any analytic function E : Ω ⊂ Rn → R satisfies (LI).Let us first stay in finite-dimensional spaces. In 1992, Kurdyka has shownthat (KLI) is satisfied by any function E, whose graph is a set belonging toan o-minimal structure. An example of an o-minimal structure are semial-gebraic sets, i.e. level sets of polynomials of several variables or their finiteintersections or unions. In particular, the graph of E(x, y) =

√x4 + y4 can

be written as (x, y, z) ∈ R3 : z2 − x4 − y4 = 0, z ≥ 0, this set is semial-gebraic. The set of polynomials can be replaced by other sets of functionsto obtain other o-minimal structures, e.g. analytic functions. More on thistopic can be found in [26]. Bolte et al. [16] gave some characterizations offunctions satisfying (KLI) and also an example of a smooth convex functionwhich does not satisfy (KLI) (several additional conditions to convexity areknown that imply (KLI), see Section 4 in [16]).

In 2003, Chill [21] proved many sufficient conditions for one-dimensionalcase E : R→ R to satisfy (LI), also with estimates of the Lojasiewicz expo-nent. These results are not interesting for the convergence of one-dimensionalODE’s, but they are important since Chill showed that if (LI) is satisfied ona so called critical manifold (which is often finite-dimensional), then it holdson the whole neighborhood of a critical point (even in infinite-dimensionalcase). We come back to this result below. Concerning one-dimensional case,Chill proved that if f ′(x) = g(x)+o(|x−a|p) with |g(x)| = c|x−a|p, then (LI)holds with θ = 1

p+1and if f ∈ Ck(B(a, δ)), f (k)(a) 6= 0 and f (j)(a) = 0 for

j = 1, 2, . . . , k−1, then (LI) holds with θ = 1k. Chill also gave some estimates


of the Lojasiewicz exponent for products and compositions of functions (e.g.for x 7→ f(ϕ(x)) where ϕ is a dipheomorphism on Ω ⊂ Rn and f satisfy(LI)).

Concerning generalizations to infinite-dimensional spaces, in 1983 Simon[60] proved (LI) for a class of analytic functions in C2,µ(Ω), in particularE(u) =

∫ΩE(x, u(x),∇u(x))dx with E analytic in the second and third

variables and satisfying some further properties.

In 1998, Jendoubi [47] proved the Lojasiewicz–Simon inequality (LSI)in L2-norm for E(u) =

∫Ω

12〈Au, u〉 + F (x, u)dx with a linear operator A

associated with a bilinear form on a subspace V ⊂ L2 and an analytic non-linearity F . In 1999, Haraux and Jendoubi [37] proved (LSI) for the same Ein V ′-norm, i.e.

|E(u)− E(ϕ)|1−θ ≤ C‖E ′(u)‖V ′ ,

which allows to work with weak solutions of damped wave equations.

In 2001, Huang and Takac [45] proved (LSI) in the abstract setting. Inparticular they showed that if E : V → R is analytic and E ′′(ϕ) : V → V ′ isa Fredholm operator, then (LSI) holds on a neighborhood of ϕ. Haraux,Jendoubi and Kavian ’03 [41] proved (LSI) for some nonanalytic energyfunctions. In particular for Au + f(x, u) on V ⊂ L2(Ω), (A,D(A)) linearself-adjoint with compact resolvent, f(x, s) = ∂sF (x, s) with F ∈ C2 andv 7→

∫ΩF (x, v(x))dx ∈ C2(V ). If ‖u‖V ≤ C|E ′(ϕ + u)|, then (LSI) holds

with θ = 12. Under some additional growth assumptions on f (no analytic-

ity) (LSI) holds with θ = 1p+1

. They also considered an abstract setting: if

E ′′(ϕ) : V → V ′ is an isomorphism, then (LSI) holds with θ = 12.

In 2003, Chill ([21]) proved that it is sufficient to verify (LI) on so calledcritical manifold. Here is the result.

Theorem 3.3.1 ([21], Theorem 3.10). Let V be a Banach space, U ⊂ V anopen subset and E ∈ C2(U). Let ϕ ∈ V satisfy the following hypotheses:

(L1) E ′(ϕ) = 0 and V0 = KerE ′′(ϕ) is a complemented subspace, i.e., thereexists a bounded linear projection P on V , such that V0 = P (V ). De-note V1 = KerP .

(L2) There exists W → V ′ such that the adjoint P ′ of P leaves W invariant,E ′ ∈ C1(U,W ) and E ′′(ϕ)(V ) = V ′1 ∩W .


(L3) E satisfies (LSI) on the critical manifold Sϕ with a Lojasiewicz expo-nent θ ∈ (0, 1

2], i.e. there exist C, η > 0 such that

|E(u)− E(ϕ)|1−θ ≤ C‖E ′(u)‖W (3.40)

holds for all u ∈ Sϕ, where Sϕ = u ∈ BV (ϕ, η) : E ′(u) ∈ V ′0.

Then E satisfies (3.40) for all u ∈ BV (ϕ, η) and a fixed η > 0 (with a differentconstant C > 0 but the same θ).


Bibliography

[1] P.A. Absil, R. Mahony, B. Andrews, Convergence of the iterates of de-scent methods for analytic cost functions. SIAM J. Optim. 16 (2005),no. 2, 531–547.

[2] F. Aloui, I. Ben Hassen and A. Haraux, Compactness of trajectoriesto some nonlinear second order evolution equations and applications. J.Math. Pures Appl. (9) 100 (2013), no. 3, 295–326.

[3] F. Alvarez, H. Attouch, J. Bolte, P. Redont, A second-order gradient-likedissipative dynamical system with Hessian-driven damping. Applicationto optimization and mechanics. J. Math. Pures Appl. (9) 81 (2002), no.8, 747–779.

[4] M. Balti, Asymptotic behavior for second-order differential equationswith nonlinear slowly time-decaying damping and integrable source. Elec-tron. J. Differential Equations 2015 (2015), No. 302, 11 pp.

[5] T. Barta, Convergence to equilibrium of relatively compact solutions toevolution equations. Electron. J. Differential Equations 2014 (2014), No.81, 1–9.

[6] T. Barta, Rate of convergence and Lojasiewicz type estimates. preprint2015.

[7] T. Barta, Decay estimates for solutions of an abstract wave equationwith general damping function. preprint 2016.

[8] T. Barta, R. Chill, and E. Fasangova, Every ordinary differential equa-tion with a strict Lyapunov function is a gradient system. Monatsh.Math. 166 (2012), 57–72.

55

56 BIBLIOGRAPHY

[9] T. Barta and E. Fasangova, Convergence to equilibrium for solutions ofan abstract wave equation with general damping function. J. DifferentialEquations 260 (2016), no. 3, 2259–2274.

[10] P. Begout, J. Bolte, M.A. Jendoubi, On damped second-order gradientsystems. J. Differential Equations 259 (2015), no. 7, 3115–3143.

[11] A. Benaissa, S. Mokeddem, Global existence and energy decay of solu-tions to the Cauchy problem for a wave equation with a weakly nonlineardissipation. Abstr. Appl. Anal. 2004, no. 11, 935–955.

[12] I. Ben Hassen, Decay estimates to equilibrium for some asymptoticallyautonomous semilinear evolution equations. Asymptot. Anal. 69 (2010),no. 1–2, 31–44.

[13] I. Ben Hassen, A. Haraux, Convergence and decay estimates for a classof second order dissipative equations involving a non-negative potentialenergy. J. Funct. Anal. 260 (2011), no. 10, 2933–2963.

[14] I. Ben Hassen, L. Chergui, Convergence of global and bounded solutionsof some non-autonomous second order evolution equations with non-linear dissipation. J. Dynam. Differential Equations 23 (2011), no. 2,315–332.

[15] A.C. Biazutti, On a nonlinear evolution equation and its applications.Nonlinear Anal. 24 (1995), no. 8, 1221–1234.

[16] J. Bolte, A. Daniilidis, O. Ley, L. Mazet, Characterizations of ojasiewiczinequalities: subgradient flows, talweg, convexity. Trans. Amer. Math.Soc. 362 (2010), no. 6, 3319–3363.

[17] S. Boussandel, Global existence and maximal regularity of solutions ofgradient systems. J. Differential Equations 250 (2011), no. 2, 929–948.

[18] A. Cabot, H. Engler, S. Gadat, On the long time behavior of secondorder differential equations with asymptotically small dissipation. Trans.Amer. Math. Soc. 361 (2009), no. 11, 5983–6017.

[19] L. Chergui, Convergence of global and bounded solutions of a secondorder gradient like system with nonlinear dissipation and analytic non-linearity. J. Dynam. Differential Equations 20 (2008), no. 3, 643–652.

BIBLIOGRAPHY 57

[20] L. Chergui, Convergence of global and bounded solutions of the waveequation with nonlinear dissipation and analytic nonlinearity. J. Evol.Equ. 9 (2009), 405–418.

[21] R. Chill, On the Lojasiewicz-Simon gradient inequality. J. Funct. Anal.201 (2003), no. 2, 572–601.

[22] R. Chill, A. Fiorenza, Convergence and decay rate to equilibrium ofbounded solutions of quasilinear parabolic equations. J. Differential Equa-tions 228 (2006), no. 2, 611–632.

[23] R. Chill, E. Fasangova, Gradient systems. Matfyzpress, 2010.

[24] R. Chill, A. Haraux, and M. A. Jendoubi, Applications of the Loja-siewicz-Simon gradient inequality to gradient-like evolution equations.Anal. Appl. 7 (2009), 351–372.

[25] R. Chill, M. A. Jendoubi, Convergence to steady states in asymptoticallyautonomous semilinear evolution equations. Nonlinear Anal. 53 (2003),no. 7–8, 1017–1039.

[26] L. van den Dries, C. Miller, Geometric categories and o-minimal struc-tures. Duke Math. J. 84 (1996), no. 2, 497540.

[27] E. Feireisl, Global attractors for semilinear damped wave equations withsupercritical exponent. J. Differential Equations 116 (1995), no. 2, 431–447.

[28] E. Feireisl, F. Simondon, Convergence for semilinear degenerate parabol-ic equations in several space dimensions. J. Dynam. Differential Equa-tions 12 (2000), no. 3, 647–673.

[29] V. Georgiev, G. Todorova, Existence of a solution of the wave equationwith nonlinear damping and source terms. J. Differential Equations 109(1994), no. 2, 295–308.

[30] J.K. Hale and G. Raugel, Convergence in gradient-like systems withapplications to PDE. Z. Angew. Math. Phys. 43 (1992), no. 1, 63–124.

[31] A. Haraux, Almost-periodic forcing for a wave equation with a nonlinear,local damping term. Proc. Roy. Soc. Edinburgh Sect. A 94 (1983), no.3–4, 195–212.

58 BIBLIOGRAPHY

[32] A. Haraux, Asymptotics for some nonlinear hyperbolic equations with aone-dimensional set of rest points. Bol. Soc. Brasil. Mat. 17 (1986), no.2, 51–65.

[33] A. Haraux, Semi-linear hyperbolic problems in bounded domains. Math.Rep. 3 (1987), no. 1, i–xxiv and 1–281.

[34] A. Haraux, Systemes dynamiques dissipatifs et applications. Recherchesen Mathematiques Appliquees 17, Masson, Paris, 1991.

[35] A. Haraux, Sharp decay estimates of the solutions to a class of nonlinearsecond order ODE’s. Anal. Appl. (Singap.) 9 (2011), no. 1, 49–69.

[36] A. Haraux and M.A. Jendoubi, Convergence of solutions of second-ordergradient-like systems with analytic nonlinearities. J. Differential Equa-tions 144 (1998), no. 2, 313–320.

[37] A. Haraux and M.A. Jendoubi, Convergence of bounded weak solutionsof the wave equation with dissipation and analytic nonlinearity. Calc.Var. Partial Differential Equations 9 (1999), no. 2, 95–124.

[38] A. Haraux and M.A. Jendoubi, Decay estimates to equilibrium for someevolution equations with an analytic nonlinearity. Asymptot. Anal. 26(2001), no. 1, 21–36.

[39] A. Haraux and M.A. Jendoubi, On a second order dissipative ODE inHilbert space with an integrable source term. Acta Math. Sci. Ser. BEngl. Ed. 32 (2012), no. 1, 155–163.

[40] A. Haraux and M.A. Jendoubi, Asymptotics for a second order differ-ential equation with a linear, slowly time-decaying damping term. Evol.Equ. Control Theory 2 (2013), 461–470.

[41] A. Haraux, M.A. Jendoubi, O. Kavian, Rate of decay to equilibrium insome semilinear parabolic equations. Dedicated to Philippe Bnilan. J.Evol. Equ. 3 (2003), no. 3, 463–484.

[42] K.H. Hoffmann, P. Rybka, Convergence of solutions to Cahn-Hilliardequation. Comm. Partial Differential Equations 24 (1999), no. 5-6, 1055–1077.

BIBLIOGRAPHY 59

[43] A. Haraux, E. Zuazua, Decay estimates for some semilinear dampedhyperbolic problems. Arch. Rational Mech. Anal. 100 (1988), no. 2, 191–206.

[44] S.-Z. Huang, Gradient inequalities. With applications to asymptotic be-havior and stability of gradient-like systems. Mathematical Surveys andMonographs 126, American Mathematical Society, Providence, RI, 2006.

[45] S.-Z. Huang, P. Takac, Convergence in gradient-like systems which areasymptotically autonomous and analytic. Nonlinear Anal. 46 (2001), no.5, Ser. A: Theory Methods, 675–698.

[46] M.A. Jendoubi, Convergence of global and bounded solutions of the waveequation with linear dissipation and analytic nonlinearity. J. DifferentialEquations 144 (1998), no. 2, 302–312.

[47] M.A. Jendoubi, A simple unified approach to some convergence theoremsof L. Simon. J. Funct. Anal. 153 (1998), no. 1, 187–202.

[48] M.A. Jendoubi, R. May, Asymptotics for a second-order differentialequation with non-autonomous damping and an integrable source term.Applicable Analysis 94 (2015), 435–443.

[49] M. A. Jendoubi and P. Polacik, Non-stabilizing solutions of semilinearhyperbolic and elliptic equations with damping. Proc. Royal Soc. Edin-burgh Sect A 133 (2003), no. 5, 1137–1153.

[50] K. Kurdyka, On gradients of functions definable in o-minimal structures.Ann. Inst. Fourier (Grenoble) 48 (1998), no. 3, 769–783.

[51] C. Lageman, Pointwise convergence of gradient-like systems. Math.Nachr. 280 (2007), no. 13-14, 1543–1558.

[52] H.A. Levine, S.R. Park, J. Serrin, Global existence and global nonexis-tence of solutions of the Cauchy problem for a nonlinearly damped waveequation. J. Math. Anal. Appl. 228 (1998), no. 1, 181–205.

[53] S. Lojasiewicz, Une propriete topologique des sous-ensembles analytiquesreels. Colloques internationaux du C.N.R.S.: Les equations aux deriveespartielles, Paris (1962), Editions du C.N.R.S., Paris, 1963.

60 BIBLIOGRAPHY

[54] S. Lojasiewicz, Sur les trajectoires du gradient d’une fonction analytique.Seminari di Geometria, Bologna (1982/83), Universita degli Studi diBologna, Bologna, 1984, pp. 115–117.

[55] J. Palis, W. de Melo, Geometric theory of dynamical systems. An intro-duction. Springer-Verlag, New York-Berlin, 1982.

[56] P. Polacik, K.P. Rybakowski, Nonconvergent bounded trajectories insemilinear heat equations. J. Differential Equations 124 (1996), no. 2,472–494.

[57] P. Radu, Weak solutions to the Cauchy problem of a semilinear waveequation with damping and source terms. Adv. Differential Equations10 (2005), no. 11, 1261–1300.

[58] P. Radu, Weak solutions to the initial boundary value problem for asemilinear wave equation with damping and source terms. Appl. Math.(Warsaw) 35 (2008), no. 3, 355–378.

[59] J. Serrin, G. Todorova, E. Vitillaro, Existence for a nonlinear waveequation with damping and source terms. Differential Integral Equations16 (2003), no. 1, 13–50.

[60] L. Simon, Asymptotics for a class of nonlinear evolution equations, withapplications to geometric problems. Ann. of Math. (2) 118 (1983), no.3, 525–571.

[61] G.F. Webb, Compactness of bounded trajectories of dynamical systemsin infinite-dimensional spaces. Proc. Roy. Soc. Edinburgh Sect. A 84(1979), no. 1-2, 19–33.

[62] G.F. Webb, Existence and asymptotic behavior for a strongly dampednonlinear wave equation. Canad. J. Math. 32 (1980), no. 3, 631–643.

[63] R. Zacher, Convergence to equilibrium for second-order differential equa-tions with weak damping of memory type. Adv. Differential Equations14 (2009), no. 7–8, 749–770.

Chapter 4

Presented works

61

62 CHAPTER 4. PRESENTED WORKS

4.1. RESEARCH PAPER [B1] 63

4.1 T. Barta, R. Chill, and E. Fasangova, Ev-

ery ordinary differential equation with

a strict Lyapunov function is a gradi-

ent system, Monatsh. Math. 166 (2012),

57–72.


Monatsh Math (2012) 166:57–72DOI 10.1007/s00605-011-0322-4

Every ordinary differential equation with a strictLyapunov function is a gradient system

Tomáš Bárta · Ralph Chill · Eva Fašangová

Received: 12 August 2010 / Accepted: 9 June 2011 / Published online: 21 June 2011© Springer-Verlag 2011

Abstract We explain and prove the statement from the title. This allows us toformulate a new type of gradient inequality and to obtain a new stabilization result forgradient-like ordinary differential equations.

Keywords Strict Lyapunov function · Gradient system ·Kurdyka–Łojasiewicz gradient inequality · Convergence to equilibrium ·Damped second order ordinary differential equation

Mathematics Subject Classification (2000) 37B25 · 34D05 · 34C40

Communicated by Josef Hofbauer.

T. Bárta was supported by the grant MSM 0021620839 from the Czech Ministry of Education.

T. Bárta · E. FašangováDepartment of Mathematical Analysis, Charles University,Sokolovská 83, 186 75 Praha 8, Czech Republice-mail: [email protected]

E. Fašangováe-mail: [email protected]

R. Chill (B)Laboratoire de Mathématiques et Applications de Metz et CNRS,Université Paul Verlaine, Metz, UMR 7122, Bât. A,Ile du Saulcy, 57045 Metz Cedex 1, Francee-mail: [email protected]

E. FašangováInstitut für Angewandte Analysis, Universität Ulm,89069 Ulm, Germany

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58 T. Bárta et al.

1 Introduction

The theory of Lyapunov functions is a fundamental part of the stability theory ofordinary differential equations [3,14], because many ordinary differential equationsnaturally admit an energy function which is nonincreasing along solutions. Moreover,La Salle’s invariance principle [13], [3, Chapter VIII], [8] and gradient inequalitiesfor the underlying Lyapunov function [1,5,9,11,12,15] provide easy and powerfulstabilization results.

Let F be a continuous tangent vector field on a manifold M . Following [8], we saythat a continuously differentiable function E : M → R is a strict Lyapunov functionfor the ordinary differential equation

u + F(u) = 0, (1)

if

〈E ′(u), F(u)〉 > 0 whenever u ∈ M and F(u) = 0. (2)

Every strict Lyapunov function is nonincreasing along solutions of (1), and ifit is constant on some solution then that solution must be stationary. Amongordinary differential equations, the gradient systems on a Riemannian manifold(M, g)

u + ∇gE(u) = 0 (3)

are the prototype examples of dissipative systems which admit a strict Lyapunovfunction: namely the function E itself. In Theorem 1 we observe the following sim-ple converse result, announced in the title: every ordinary differential equation (1)with a strict Lyapunov function is—on the open subset of nonequilibrium pointsof F—a gradient system. More precisely, it is a gradient system for the Lyapunovfunction itself and for an appropriate Riemannian metric. This observation, whichmay be deduced from [16,17], but which was formulated differently there, impliesthat the gradient systems are basically the only examples of dissipative systems withstrict Lyapunov function. As an application, we obtain that the damped second orderequation

u + G(u)+ ∇E(u) = 0

is a gradient system when it is rewritten as a first order equation.We apply this result in order to obtain a stabilization result for global solutions of

the general system (1) (Theorem 3). Strictly speaking, this result can be and actuallyis formulated independently of the gradient system structure in the appropriate ambi-ent metric constructed in Theorem 1. However, it can be better motivated with thisknowledge behind.

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Every ordinary differential equation 59

The classical Łojasiewicz gradient inequality (see [15])

|E(v)− E(ϕ)|1−θ ≤ C ‖E ′(v)‖g (4)

and the more general Kurdyka–Łojasiewicz gradient inequality (see [11], and [4,5]for the name)

(|E(v)− E(ϕ)|) ≤ ‖E ′(v)‖g (5)

have turned out to be particularly useful for proving that global solutions of gradientsystems converge to a single equilibrium. The first convergence result for gradientsystems is due to Łojasiewicz himself [15]; he also proved that every real analyticfunction satisfies the Łojasiewicz gradient inequality (4). Later, Haraux and Jendoubi[9] and Alvarez et al. [2] proved convergence results for damped second order ordinarydifferential equations by using a gradient inequality for a natural Lyapunov function.More recently, it has been shown that many convergence results can be unified into ageneral theorem for the ordinary differential equation (1), provided that that equationadmits a strict Lyapunov function, the Lyapunov function satisfies a gradient inequal-ity and an angle condition holds between E ′ and F [1,7,12]. This angle condition isautomatically satisfied for gradient systems but there are interesting examples whereit is not satisfied (see [6] and Sect. 5 below).

Now, if the problem of convergence to equilibrium—with respect to some givenmetric—is approached by knowing that the problem (1) has a gradient structure on theopen subset of nonequilibrium points—with respect to a different metric—, then oneis naturally led to the problem of comparing the two involved metrics. This problem iseven crucial since, by La Salle’s invariance principle, the only possible limit points ofsolutions are equilibrium points, and since equilibrium points lie on the boundary ofthe set of nonequilibrium points. The comparison of metrics motivates us to formulatethe gradient inequality

(|E(v)− E(ϕ)|) ≤⟨E ′(v), F(v)

‖F(v)‖g

⟩

and a convergence result (Theorem 3) which generalizes the results from [1,7,12] forordinary differential equations. With this view we can also give a new interpretationof a recent convergence result by Chergui [6] for a second order ordinary differentialequation with degenerate damping.

Of course, if the original and new metric are equivalent, then the induced distancefunctions are equivalent because lengthes of curves are equivalent. Since length isan important notion in the above cited convergence results, we characterize equiva-lence of the two metrics. We find that the two metrics are equivalent if and only if anangle condition and a comparability condition between E ′ and F holds (Theorem 2).These are exactly the conditions which appear in [1,7,12]. Finally, we formulate someopen questions concerning the metric with respect to which equation (1) is a gradientsystem.

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60 T. Bárta et al.

2 Main result

Before stating the main theorem, let us fix some notation. When we write manifoldin this article, we mean a differentiable and finite-dimensional manifold. The dualitybetween tangent vectors and cotangent vectors is denoted by 〈·, ·〉. Whenever g is aRiemannian metric, we write 〈·, ·〉g for the inner product on the tangent space; to beprecise, we should write 〈·, ·〉g(u) for the inner product on the tangent space Tu M , butthe variable u is dropped, and equalities and inequalities are usually to be understoodfor functions on M . We write ‖ · ‖g for the induced norm both on the tangent spaceand on the cotangent space. Given a differentiable function E : M → R, we denote byE ′ its derivative. The derivative E ′ is a cotangent vector field; the associated, uniquelydetermined tangent vector field ∇gE , given by

〈E ′, X〉 = 〈∇gE, X〉g for every tangent vector field X,

is called the gradient of E with respect to the metric g.The following is the theorem announced in the title of this article. It should be com-

pared to [16, Theorem 1], [17, Proposition 2.8], where it was stated that if the ordinarydifferential equation (1) (in RN ) admits a strict Lyapunov function E , then F = L∇E ,where L is a continuous function taking its values in the positive definite matricesand ∇E is the Euclidean gradient. Note that the multiplication by L corresponds to achange of the underlying metric.

Theorem 1 Let M be a manifold, F a continuous tangent vector field on M, and letE : M → R be a continuously differentiable, strict Lyapunov function for (1). Thenthere exists a Riemannian metric g on the open set

M := u ∈ M : F(u) = 0 ⊆ M

such that ∇gE = F. In particular, the differential equation (1) is a gradient systemon the Riemannian manifold (M, g).

Proof By assumption (2), 0 = F(u) ∈ ker E ′(u) and ker E ′(u) = Tu M for everyu ∈ M . As a consequence the tangent bundle T M is the direct sum of the bundleker E ′ and the bundle generated by the vector field F :

T M = ker E ′ ⊕ 〈F〉. (6)

For every continuous tangent vector field X on M , the vector fields

X0 := X − 〈E ′, X〉〈E ′, F〉 F and X1 := 〈E ′, X〉

〈E ′, F〉 F

are well-defined and continuous, X0 ∈ ker E ′ and X1 ∈ 〈F〉 (the one-dimensionalspace—or vector bundle—generated by F). Hence, the above decomposition of thetangent space T M is continuous in the sense that the projection onto ker E ′ along 〈F〉is a continuous function.

123


Now we choose an arbitrary Riemannian metric g on M . Starting from this metric,we define a new metric on M by setting

〈X,Y 〉g := 〈X0,Y0〉g + 1

〈E ′, F〉〈E′, X〉〈E ′, Y 〉

= 〈X0,Y0〉g + 1

〈E ′, F〉〈E′, X1〉〈E ′,Y1〉. (7)

Precisely at this point we use the assumption that E is a strict Lyapunov function, thatis, 〈E ′, F〉 > 0 on M , because this assumption implies that g really is a metric (inparticular: positive definite). Since the decomposition (6) is continuous, and since g,E ′ and F are continuous, g is continuous, too.

By definition of the metric g and by definition of the gradient ∇gE , we have forevery tangent vector field X

〈F, X〉g = 0 + 〈E ′, X〉 = 〈∇gE, X〉g,

so that F = ∇gE and the claim is proved. Remark 1 Note carefully that the metric g, for which we have the equality F = ∇gE ,is not unique. The inner product 〈X,Y 〉g is uniquely determined by the functions F , Eif one of the two vectors X or Y is a multiple of F ; this fact comes from the requirementthat one wants to have F = ∇gE . However, on ker E ′ × ker E ′ one has a free choice,how to define the metric g.

Theorem 1 says that equation (1) is a gradient system on (M, g). The set M beinga subset of M and the metric g being a new metric, several questions arise.

Question 1 When is the equation (1) a gradient system on the whole of M?

We ask this question in a slightly different way and in weaker forms. Recall fromRemark 1 above that the metric g from Theorem 1 is not uniquely determined. If weask whether the ordinary differential equation (1) is a gradient system associated withthe fixed Lyapunov function E , then Question 1 is closely related to the following one.

Question 2 Can we choose the metric g such that g extends to a Riemannian metricon M?

Example 1 (Not every metric g extends to a Riemannian metric on M) Let M = R2,and let F : R2 → R2 and E : R2 → R be given by

F(u1, u2) := (u1, 2 u2) and E(u1, u2) := 1

2(u2

1 + u22).

The origin u = (0, 0) is the only equilibrium point of F and for u = (0, 0) we have

〈E ′(u), F(u)〉 = u21 + 2u2

2 > 0.

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62 T. Bárta et al.

Hence, E is a strict Lyapunov function.Let g be the Euclidean metric, and define the new metric g by using the formula

(7), that is, for every u, X , Y ∈ R2,

〈X,Y 〉g(u) :=⟨

X − 〈E ′(u), X〉〈E ′(u), F(u)〉 F(u), Y − 〈E ′(u), Y 〉

〈E ′(u), F(u)〉 F(u)

⟩g(u)

+ 1

〈E ′(u), F(u)〉〈E′(u), X〉〈E ′(u), Y 〉.

Then, for X = (1, 0) and Y = (0, 1) a straightforward calculation shows that

limh→0

〈X,Y 〉g(0,h) = 0 and limh→0

〈X,Y 〉g(h,h) = − 19 .

Hence, the metric g does not have a continuous extension at the origin.Now, let g be the metric given by

〈X,Y 〉g = X1Y1 + 1

2X2Y2,

and define g starting with this metric g. Then g does have a continuous extension nearthe origin. In fact, in this case g = g.

3 Equivalence of metrics

Let g be any Riemannian metric on M and let g be the Riemannian metric on Mconstructed from g as in the proof of Theorem 1. If the metric g admits a continuousextension to M , then g and g are locally equivalent on M (and even on M). Theconverse is not true in general (take the metrics from Example 1). In this section, wecharacterize equivalence of the two metrics g and g on M in terms of an angle con-dition which has recently appeared in some articles on stabilization of gradient-likesystems [1,7,12] and in terms of a comparability condition.

We say that two Riemannian metrics g1, g2 on a manifold M are equivalent, if thereexist c1, c2 > 0 such that for every tangent vector field X

c1 ‖X‖g1 ≤ ‖X‖g2 ≤ c2 ‖X‖g1 on M. (8)

Note that if two Riemannian metrics are equivalent then the induced distance functionsare equivalent, too. In particular, the completions of (M, g1) and (M, g2) are the sameand the boundaries in this completion carry the same topology.

We say that E ′ and F satisfy the angle condition (see [1, Condition (2.2)], [12,Condition (AC), Definition 1.1], [7, Conditions (4), (9)]) if there exists α > 0 suchthat

〈E ′, F〉 ≥ α ‖E ′‖g ‖F‖g on M, (AC)

123


and we say that they satisfy the comparability condition if there exist constants c1,c2 > 0 such that

c1 ‖E ′‖g ≤ ‖F‖g ≤ c2 ‖E ′‖g on M . (C)

It is straightforward to check that the angle condition and the comparability conditiontogether are equivalent to the existence of a constant β > 0 such that

〈E ′, F〉 ≥ β (‖E ′‖2g + ‖F‖2

g) on M; (AC+C)

this condition appears in various examples in [7, Section 2].

Theorem 2 The metrics g and g are equivalent on M if and only if E ′ and F satisfythe conditions (AC) and (C).

Proof Assume first that the two metrics g and g are equivalent. Then there existconstants c1, c2 > 0 such that for every tangent vector field X

c1 ‖X‖g ≤ ‖X‖g ≤ c2 ‖X‖g on M . (9)

Then

‖E ′‖g = sup‖X‖g≤1

〈E ′, X〉 ≥ supc2 ‖X‖g≤1

〈E ′, X〉 = 1

c2‖E ′‖g on M .

Since F = ∇gE , the preceding inequalities imply

〈E ′, F〉 = 〈∇gE, F〉g = ‖∇gE‖g ‖F‖g = ‖E ′‖g ‖F‖g ≥ c1

c2‖E ′‖g ‖F‖g,

that is, E ′ and F satisfy the angle condition (AC). Moreover,

‖F‖g = ‖∇gE‖g = ‖E ′‖g

and the equivalence of g and g implies also the comparability condition (C).Now assume that E ′ and F satisfy the angle and comparability conditions (AC) and

(C). Given a tangent vector field X , we use the decomposition (6), that is, X = X0+X1with X0 ∈ ker E ′ and X1 ∈ 〈F〉. We have,

‖X‖2g = ‖X0‖2

g + 〈E ′, X1〉2

〈E ′, F〉 (definition of g)

≥ ‖X0‖2g + α2‖E ′‖2

g‖X1‖2g

‖E ′‖g‖F‖g(by (AC) and Cauchy-Schwarz)

≥ ‖X0‖2g + α2

c2‖X1‖2

g (by (C))

≥ 1

2min

1,α2

c2

‖X‖2

g (triangle inequality),

123

64 T. Bárta et al.

and

‖X‖2g = ‖X0‖2

g + 〈E ′, X〉2

〈E ′, F〉 (definition of g)

≤ 2 ‖X‖2g + 2 ‖X1‖2

g + 〈E ′, X〉2

〈E ′, F〉 (triangle inequality)

= 2 ‖X‖2g + 2

〈E ′, X〉2

〈E ′, F〉2 ‖F‖2g + 〈E ′, X〉2

〈E ′, F〉 (definition of X1)

≤ 2 ‖X‖2g + 2

α2 ‖X‖2g + ‖E ′‖2

g‖X‖2g

α‖E ′‖g‖F‖g(by Cauchy-Schwarz and (AC))

≤ 2 ‖X‖2g + 2

α2 ‖X‖2g + 1

α c1‖X‖2

g (by (C))

=(

2 + 2

α2 + 1

α c1

)‖X‖2

g,

that is, g and g are equivalent.

4 Asymptotics

In this section, (M, g) is a Riemannian manifold. We assume that the ordinary differ-ential equation (1) with initial condition u(0) = u0 ∈ M is uniquely solvable. Givena function u : R+ → M , the set of all accumulation points (as t → ∞)

ω(u) = ϕ ∈ M : there exists (tn) ∞ such that u(tn) → ϕ

is called the ω-limit set of u.

Theorem 3 Let F be a continuous tangent vector field on the Riemannian manifold(M, g). Let u : R+ → M be a global solution of the ordinary differential equation(1) and let E : M → R be a continuously differentiable, strict Lyapunov functionfor (1). Assume that there exist : R+ → R+ such that 1/ ∈ L1

loc([0,+∞)) and(s) > 0 for s > 0, ϕ ∈ ω(u) and a neighbourhood U ⊆ M of ϕ such that for everyv ∈ U ∩ M

(|E(v)− E(ϕ)|) ≤⟨E ′(v), F(v)

‖F(v)‖g

⟩. (10)

Then u has finite length in (M, g) and, in particular, limt→+∞ u(t) = ϕ in (M, g).

Proof Since E is a Lyapunov function, necessarily the function E(u) is nonincreasing.Hence, limt→∞ E(u(t)) exists in R ∪ −∞. By continuity of the function E and bydefinition of the ω-limit set ω(u), this limit equals E(ψ) for every ψ ∈ ω(u) (so,since ω(u) is non-empty by assumption, the limit of E(u) is finite and equals E(ϕ)).By changing E by an additive constant, if necessary, we may without loss of generalityassume that E(ϕ) = 0, so that E(u(t)) ≥ 0 and limt→∞ E(u(t)) = 0.

123


If E(u(t0)) = 0 for some t0 ≥ 0, then E(u(t)) = 0 for every t ≥ t0, and therefore,since E is a strict Lyapunov function, the function u is constant for t ≥ t0. In this case,there remains nothing to prove.

Hence, we may assume that E(u(t)) > 0 for every t ≥ 0. By unique solvability of(1) this implies that u(t) ∈ M for every t ≥ 0. Let σ > 0 be such that the closed ballB(ϕ, σ ) (with respect to the distance d induced by g) is contained in U . Let

(t) :=t∫

0

1

(s)ds,

and let t0 ≥ 0 be so large that

d(u(t0), ϕ) ≤ σ

3and (E(u(t0))) <

σ

3.

Let

t1 := inft ≥ t0 : d(u(t), ϕ) = σ .

By continuity of the function u, we have t1 > t0. Then for every t ∈ [t0, t1),

− d

dt(E(u(t))) = 1

(E(u(t))

(− d

dtE(u(t))

)(chain rule and def. of )

= 1

(E(u(t)) 〈E ′(u(t)), F(u(t))〉 (chain rule and (1))

≥ ‖F(u(t))‖g (gradient inequality (10))

= ‖u(t)‖g (equation (1)).

Hence, for every t ∈ [t0, t1),

d(u(t), ϕ) ≤ d(u(t), u(t0))+ d(u(t0), ϕ) (triangle inequality)

≤t∫

t0

‖u(s)‖g ds + d(u(t0), ϕ) (def. of distance)

≤ (E(u(t0)))+ d(u(t0), ϕ) (preceding estimates)

≤ 2

3σ (choice of t0). (11)

This inequality implies that t1 = ∞. But then ‖u‖g ∈ L1(R+), by the estimates (11).This means that u has finite length in (M, g). The existence of limt→∞ u(t) followsfrom Cauchy’s criterion and the fact that ϕ ∈ ω(u).

By applying Theorem 3 with(s) = 1C s1−θ for θ ∈ (0, 1), we obtain the following

corollary.

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66 T. Bárta et al.

Corollary 1 Let M, F and E be as in Theorem 3. Let u : R+ → M be a global solutionof (1). Assume that there exists ϕ ∈ ω(u), θ ∈ (0, 1), C ≥ 0 and a neighbourhoodU ⊆ M of ϕ such that for every u ∈ U

|E(v)− E(ϕ)|1−θ ≤ C

⟨E ′(v), F(v)

‖F(v)‖g

⟩. (12)

Then u has finite length in (M, g) and, in particular, limt→+∞ u(t) = ϕ in (M, g).

Note that the Łojasiewicz gradient inequality (4) and the Kurdyka–Lojasiewiczinequality (5) from the Introduction involve only the function E while the modifiedKurdyka-Łojasiewicz inequality (10) and inequality (12) involve in addition the vectorfield F . In fact, the norm of the derivative E ′ (which appears on the right-hand side in(4) and (5)) is replaced by the directional derivative in the normalized direction of F .Let us repeat the two modified inequalities in the following way, that is, from a moregeometric point of view:

⟨( (E − E(ϕ)))′, F

‖F‖g

⟩≥ 1

or

⟨((E − E(ϕ))θ )′, F

‖F‖g

⟩≥ 1/C

in a neighbourhood of ϕ. If g denotes the metric constructed in Theorem 1, then wecan write the inequality (10) also in the form

(|E(v)− E(ϕ)|) ≤‖F(v)‖2

g

‖F(v)‖g= ‖E ′(v)‖g

‖F(v)‖g

‖F(v)‖g,

in which the norm of E ′ with respect to the new metric g and the ratio of the twoinvolved metrics appear. Hence, if the two metrics g and g are equivalent, then (10)reduces to the Kurdyka–Łojasiewicz inequality (5).

As indicated in the Introduction, there are convergence results similar to Theorem 3in which—instead of the modified Kurdyka–Łojasiewicz inequality—a gradientinequality for the Lyapunov function E (or an assumption of analyticity) and the anglecondition (AC) appear; see [1, Theorem 2.2], [12, Theorem 1.2], [7, Theorem 1] (thelatter article considers also the case of differential equations in infinite dimensions).These results (in the case of ordinary differential equations in finite dimensions) canbe seen as a consequence of Theorem 3 or its Corollary 1 because the Łojasiewiczinequality (4) and the angle condition (AC) imply the modified Łojasiewicz inequality(12). There are situations, however, in which the modified Łojasiewicz inequality (12)is satisfied but the angle condition (AC) is not. The following elementary example issuch a case. A more sophisticated example is described in the following section.

123


Example 2 Let M ⊆ R2 be the open unit disk, equipped with the Euclidean metricg. Let α ≥ 0, and let F(u) = F(u1, u2) = (‖u‖αu1 − u2, u1 + ‖u‖αu2) and E(u) =12 (u

21 + u2

2). Then

〈E ′(u), F(u)〉 = ‖u‖2+α, ‖F(u)‖ = ‖u‖ ·√

1 + ‖u‖2α and ‖E ′(u)‖ = ‖u‖.

Hence, unless α = 0, the angle condition (AC) does not hold on any neighbourhoodof the critical point (0, 0) (and, by Theorem 2, the Euclidean metric g and the met-ric g from Theorem 1 are not equivalent). The function E satisfies the Łojasiewiczinequality (4) near the origin for θ = 1

2 , but we even have

1

‖F(u)‖〈E ′(u), F(u)〉 = ‖u‖1+α√1 + ‖u‖2α

≥ 1√2‖u‖2(1−θ) ≥ 1

4E(u)1−θ

provided 0 < θ ≤ 1−α2 . Hence, if 0≤α<1, then E satisfies the modified Łojasiewicz

gradient inequality (12).

5 Application to a second order problem

Let us consider the following second order problem

u + G(u, u)+ ∇E(u) = 0, (13)

where E : RN → R and G : RN × RN → RN are two given C2 functions. Weassume that the second term in this equation represents a damping in the sense thatfor every u, v ∈ RN

〈G(u, v), v〉 ≥ g(‖v‖) ‖v‖2,

‖G(u, v)‖ ≤ cg(‖v‖) ‖v‖, and (14)

‖∇G(u, v)‖ ≤ c g(‖v‖),

where c ≥ 0 is a constant and g : R+ → R+ is a nonnegative, concave, nondecreasingfunction, g(s) > 0 for s > 0. Throughout this section, ‖ · ‖ denotes the Euclideannorm. The first line in (14) is a lower estimate of the damping in the direction of thevelocity.

The following theorem yields convergence of solutions to a singleton.

Theorem 4 Let u ∈ W 2,∞(R+; RN ) be a global solution of (13). Assume that thereexist ϕ ∈ ω(u), a neighbourhood U ⊆ RN of ϕ and a nonnegative, concave, nonde-creasing function : R+ → R+ such that for every v ∈ U

(|E(v)− E(ϕ)|) ≤ ‖∇E(v)‖.

123

68 T. Bárta et al.

Assume that (s) ≤ c√

s for some c > 0 and all s ≥ 0 small enough and that

s → 1/(s)g((s)) ∈ L1loc([0,+∞)).

Then u has finite length and, in particular, limt→+∞ u(t) = ϕ.

Remark 2 If we take (s) = c s1−θ , G(u, v) := ‖v‖αv and g(s) := sα for α ∈[0, θ

1−θ ), θ ∈ (0, 12 ], then the equation (13) becomes

u + ‖u‖α u + ∇E(u) = 0

and Theorem 4 applies, thus generalizing a recent convergence result by Chergui(see [6]). However, our theorem allows more general nonlinearities and also growthorders of the damping that are closer to the critical case α = θ

1−θ . If (s) = s1−θ ,

g(s) = sθ/(1−θ) we cannot expect convergence in general as an example by Harauxshows [8]. But if the growth of or g is a little better, we obtain convergence. Forexample,

(s) = s1−θ , g(s) = sθ

1−θ ln1+ε(1/s)

or

(s) = s1−θ ln1−θ+ε(1/s), g(s) = sθ

1−θ

with ε > 0 satisfy the assumptions of Theorem 4.

Proof We first show that limt→∞ ‖u(t)‖ = 0. Multiplying the equation (13) by u,integrating from 0 to t , and using the assumption on G, we obtain

1

2‖u(0)‖2 + E(u(0)) = 1

2‖u(t)‖2 + E(u(t))+

t∫0

〈G(u(s), u(s)), u(s)〉 ds

≥ 1

2‖u(t)‖2 + E(u(t))+

t∫0

g(‖u(s)‖)‖u(s)‖2 ds.

Since u and u are globally bounded, this implies

∞∫0

g(‖u(s)‖)‖u(s)‖2 ds < +∞.

Since, moreover, u is uniformly continuous, we obtain that limt→∞ g(‖u(s)‖)‖u(s)‖2

=0. Since g(s) > 0 whenever s > 0, this implies that limt→∞ ‖u(t)‖ = 0.

123


Note that u is a (global) solution of (13) if and only if (u, u) is a (global) solutionto the following problem

d

dt

(u(t)v(t)

)+ F

(u(t)v(t)

)= 0, (15)

where

F(u, v) =( −v

G(u, v)+ ∇E(u)

).

Let M ⊆ RN ×RN be a suffiently large (closed) ball which is a neighbourhood of therange of (u, u). Note that, by the above argument and by assumption, (ϕ, 0) belongs tothe ω-limit set of (u, u). In the following we will often use boundedness of continuousfunctions on M , in particular there exists a constant K such that

g(‖v‖), ‖G(u, v)‖, ‖∇G(u, v)‖, ‖∇E(u)‖ ≤ K

and

g(‖∇E(u)‖), ‖G(u,∇E(u))‖, ‖∇G(u,∇E(u))‖ ≤ K

for all (u, v) ∈ M .Following the idea of the proof by Chergui [6], we define, for ε > 0 small and to

be chosen below,

E(u, v) := 1

2‖v‖2 + E(u)+ ε 〈G(u,∇E(u)), v〉.

We show that E is a strict Lyapunov function for (15). We compute

〈E ′(u, v), F(u, v)〉 = 〈G(u, v), v〉 − ε〈∇G(u,∇E(u))(I d,∇2 E(u))v, v〉+ ε〈G(u,∇E(u)),G(u, v)〉 + ε〈G(u,∇E(u)),∇E(u)〉.

By the assumption on G (first line of (14)), we have

〈G(u, v), v〉 ≥ g(‖v‖)‖v‖2 and

ε〈G(u,∇E(u)),∇E(u)〉 ≥ ε g(‖∇E(u)‖)‖∇E(u)‖2.

By the second line of (14) and by Cauchy-Schwarz, for every (u, v) ∈ M we canestimate

|ε〈G(u,∇E(u)),G(u, v)〉| ≤ c2ε g(‖∇E(u)‖)‖∇E(u)‖ g(‖v‖)‖v‖≤ 1

4εg(‖∇E(u)‖)‖∇E(u)‖2 + C εg(‖v‖)‖v‖2.

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70 T. Bárta et al.

Here and in the following, C ≥ 0 denotes a constant which may change from line toline, which depends on K , but which is independent from ε > 0. Again by using theassumptions on G, by using that g is nondecreasing, and by Lemma 1 (b), we obtainthat for every (u, v) ∈ M

|ε〈∇G(u,∇E(u))(I d,∇2 E(u))v, v〉|≤ C εg(‖∇E(u)‖)‖v‖2

≤⎧⎨⎩

1

4εg(‖∇E(u)‖)‖∇E(u)‖2 if 2

√C‖v‖ ≤ ‖∇E(u)‖,

C εg(2√

C‖v‖)‖v‖2 if 2√

C‖v‖ ≥ ‖∇E(u)‖≤ C εg(‖v‖)‖v‖2 + 1

4εg(‖∇E(u)‖)‖∇E(u)‖2.

Taking the preceding estimates together, we obtain for every (u, v) ∈ M

〈E ′(u, v), F(u, v)〉 ≥ (1 − 2Cε)g(‖v‖)‖v‖2 + 1

2ε g(‖∇E(u)‖)‖∇E(u)‖2). (16)

In particular, for ε > 0 small enough, E is a strict Lyapunov function for the problem(15) on M . From the preceding estimate, from the estimate

‖F(u, v)‖ ≤ C (‖v‖ + ‖∇E(u)‖),

and from the estimate

(‖v‖ + ‖∇E(u)‖)(g(‖v‖) ‖v‖ + g(‖∇E(u)‖) ‖∇E(u)‖)≤ 3 (g(‖v‖)‖v‖2 + g(‖∇E(u)‖)‖∇E(u)‖2)

(Lemma 2), we obtain the lower estimate

1

‖F(u, v)‖〈E ′(u, v), F(u, v)〉 ≥ α (g(‖v‖) ‖v‖ + g(‖∇E(u)‖) ‖∇E(u)‖) (17)

on M , where α > 0 is a constant depending only on uniform bounds of the functionsE , G and g.

Next, by the definition of E and by the assumptions on G,

|E(u, v)− E(ϕ, 0)| ≤ 1

2‖v‖2 + |E(u)− E(ϕ)| + εc g(‖∇E(u)‖) ‖∇E(u)‖ ‖v‖

≤ ‖v‖2 + |E(u)− E(ϕ)| + Cε ‖∇E(u)‖2.

By Lemma 1 (applied with the function h = ),

(|E(u, v)− E(ϕ, 0)|) ≤ (‖v‖2)+(|E(u)− E(ϕ)|)+ C (‖∇E(u)‖2).

123


By the assumption on and by the gradient inequality for E , we obtain that thereexists a neighbourhood U ⊆ M of (ϕ, 0) such that for every (u, v) ∈ U

(|E(u, v)− E(ϕ, 0)|) ≤ C (‖v‖ + ‖∇E(u)‖).

Now let : R+ → R+ be given by (s) := (s) g((s)). By Lemmas 1 and 2, wethen have

(|E(u, v)− E(ϕ, 0)|) ≤ C (‖v‖ + ‖∇E(u)‖) g(‖v‖ + ‖∇E(u)‖)≤ C (g(‖v‖)‖v‖ + g(‖∇E(u)‖)‖∇E(u)‖)

Taking this estimate and (17) together, we have proved that there exists a neighbour-hood U ⊆ RN × RN of (ϕ, 0) such that for every (u, v) ∈ U

(|E(u, v)− E(ϕ, 0)|) ≤ C

⟨E ′(u, v) F(u, v)

‖F(u, v)‖⟩.

Moreover, by assumption, 1/ ∈ L1loc([0,∞)). Hence, Theorem 3 can be applied

and limt→∞(u(t), u(t)) = (ϕ, 0).

Remark 3 If G(u, v) = ‖v‖αv for some α ≥ 0 and if g(v) = ‖v‖α , then one easilychecks that the lower estimate (16) is optimal. Moreover, the estimate ‖E ′(u, v)‖ +‖F(u, v)‖ ≤ C(‖v‖ + ‖∇E(u)‖) is optimal for small v and ∇E(u). Hence, if α > 0,then E ′ and F do not satisfy the angle condition (AC).

Lemma 1 Let h : R+ → R+ be a nonnegative, concave, nondecreasing function.Then:

(a) For every u, v ≥ 0 one has h(u + v) ≤ h(u)+ h(v).(b) For every C > 0 and every u ≥ 0 one has h(Cu) ≤ maxC, 1 h(u).

Proof (a) By concavity we have the inequalities

u

u + vh(u + v)+ v

u + vh(0) ≤ h(u) and

v

u + vh(u + v)+ u

u + vh(0) ≤ h(v).

Summing up the two inequalities and using that h(0) ≥ 0 yields the desired result.

(b) If C ≤ 1, the inequality follows from the assumption that h is nondecreasing. IfC ≥ 1, the concavity and the assumption h(0) ≥ 0 imply

1

Ch(Cu) ≤ 1

Ch(Cu)+ C − 1

Ch(0) ≤ h(u).

123

72 T. Bárta et al.

Lemma 2 Let r, s ∈ C1(R+) be nonnegative and nondecreasing. Then, for everyu, v ≥ 0,

r(u)s(v) ≤ r(u)s(u)+ r(v)s(v).

Proof For fixed u ≥ 0, consider the function v → r(u)s(u) + r(v)s(v) − r(u)s(v).Observe first that this function is nonnegative for v ≤ u, since r and s are nonnegativeand s is nondecreasing. Then observe that the derivative of this function is nonnegativefor v ≥ u by the assumptions on r and s. Acknowledgements The work on this article started while the first author visited the Paul VerlaineUniversity of Metz. He would like to thank Eva Fašangová and Ralph Chill for their kind hospitality duringhis stay in Metz.

References

1. Absil, P.-A., Mahony, R., Andrews, B.: Convergence of the iterates of descent methods for analyticcost functions. SIAM J. Optim. 16, 531–547 (electronic) (2005)

2. Alvarez, F., Attouch, H., Bolte, J., Redont, P.: A second-order gradient-like dissipative dynamicalsystem with Hessian driven damping: application to optimization and mechanics. J. Math. PuresAppl. 81, 747–779 (2002)

3. Bhatia, N.P., Szegö, G.P.: Stability Theory of Dynamical Systems. Springer Verlag, Berlin (1970)4. Bolte, J., Daniilidis, A., Lewis, A., Shiota, M.: Clarke subgradients of stratifiable functions. SIAM

J. Optim. 18(2), 556–572 (electronic) (2007)5. Bolte, J., Daniilidis, A., Ley, O., Mazet, L.: Characterizations of Łojasiewicz inequalities: subgradient

flows, talweg, convexity. Trans. Am. Math. Soc. 362(6), 3319–3363 (2010)6. Chergui, L.: Convergence of global and bounded solutions of a second order gradient like system with

nonlinear dissipation and analytic nonlinearity. J. Dyn. Diff. Equ. 20(3), 643–652 (2008)7. Chill, R., Haraux, A., Jendoubi, M.A.: Applications of the Łojasiewicz-Simon gradient inequality to

gradient-like evolution equations. Anal. Appl. 7, 351–372 (2009)8. Haraux, A.: Systèmes dynamiques dissipatifs et applications. Masson, Paris (1990)9. Haraux, A., Jendoubi, M.A.: Convergence of solutions to second-order gradient-like systems with

analytic nonlinearities. J. Diff. Equ. 144, 313–320 (1998)10. Huang, S.-Z.: Gradient Inequalities: with Applications to Asymptotic Behaviour and Stability of

Gradient-like Systems. Mathematical Surveys and Monographs, vol. 126, American MathematicalSociety, Providence (2006)

11. Kurdyka, K.: On gradients of functions definable in o-minimal structures. Ann. Inst. Fourier(Grenoble) 48, 769–783 (1998)

12. Lageman, Ch.: Pointwise convergence of gradient-like systems. Math. Nachr. 280(13–14), 1543–1558 (2007)

13. Lasalle, J.P.: Asymptotic stability criteria. Proc. Symp. Appl. Math., vol. XIII, American MathematicalSociety, Providence, 1962, pp. 299–307

14. Liapunov, A.M.: Stability of motion. With a contribution by V.A. Pliss and an introduction byV.P. Basov. Translated from the Russian by Flavian Abramovici and Michael Shimshoni. Mathematicsin Science and Engineering, vol. 30, Academic Press, New York (1966)

15. Łojasiewicz, S.: Une propriété topologique des sous-ensembles analytiques réels. Colloques interna-tionaux du C.N.R.S.: Les équations aux dérivées partielles, Paris (1962), Editions du C.N.R.S., Paris,1963, pp. 87–89

16. McLachlan, R.I., Quispel, G.R.W., Robidoux, N.: Unified approach to Hamiltonian systems, Pois-son systems, gradient systems, and systems with Lyapunov functions or first integrals. Phys. Rev.Lett. 81(12), 2399–2403 (1998)

17. McLachlan, R.I., Quispel, G.R.W., Robidoux, N.: Geometric integration using discrete gradients.R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 357(1754), 1021–1045 (1999)

123


4.2 T. Barta, Convergence to equilibrium of

relatively compact solutions to evolution

equations, Electron. J. Differential Equa-

tions 2014 (2014), No. 81, 1–9.


Electronic Journal of Differential Equations, Vol. 2014 (2014), No. 81, pp. 1–9.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

ftp ejde.math.txstate.edu

CONVERGENCE TO EQUILIBRIUM OF RELATIVELY

COMPACT SOLUTIONS TO EVOLUTION EQUATIONS

TOMAS BARTA

Abstract. We prove convergence to equilibrium for relatively compact solu-tions to an abstract evolution equation satisfying energy estimates near theomega-limit set. These energy estimates generalize Lojasiewicz and Kurdyka- Lojasiewicz-Simon gradient inequalities. We apply the abstract results to sev-eral ODEs and PDEs of first and second order.

1. Introduction

Convergence results of the type “if ϕ is in the omega-limit set of u : R+ → Xand a condition (C) holds, then limt→+∞ u(t) = ϕ” have been extensively studied(see, e.g., Haraux and Jendoubi [6], Albis et al. [1], Chill et al. [5], Lageman [7],Chergui [3, 4], Barta et al. [2]). Each of the proofs of these results can be split intotwo parts: the first part shows the key estimate

− d

dtE(u(t)) ≥ c‖u(t)‖ (1.1)

for some function E : X → R and the second part proves convergence with help ofthis estimate.

The second part of the proofs is always the same (see proof of Theorem 2.6below or corresponding parts of proofs in the articles mentioned above). The firstpart follows from condition (C). Examples of condition (C) are the Lojasiewiczinequality

|E(u)− E(ϕ)|1−θ ≤ c‖E′(u)‖ for all u near ϕ (1.2)

or the more general Kurdyka- Lojasiewicz-Simon inequality

Θ(|E(u)− E(ϕ)|) ≤ c‖E′(u)‖ for all u near ϕ. (1.3)

If u is a solution to the ordinary differential equation

u+ F (u) = 0, (1.4)

one can write

− d

dtE(u(t)) = −〈E′(u(t)), u(t)〉 = 〈E′(u(t)), F (u(t))〉. (1.5)

2000 Mathematics Subject Classification. 35R20, 35B40, 34D05, 34G20.Key words and phrases. Convergence to equilibrium; gradient system;Kurdyka- Lojasiewicz gradient inequality; gradient-like system.c©2014 Texas State University - San Marcos.

Submitted Oactober 29, 2013. Published March 21, 2014.

1

2 T. BARTA EJDE-2014/81

In many important examples (e.g. if (1.4) is a gradient system with F = ∇E) onecan continue with

〈E′(u(t)), F (u(t))〉 ≥ c‖E′(u(t))‖ · ‖F (u(t))‖. (1.6)

This inequality is known as angle condition and it plays an important role in proving(1.1).

For partial differential equations, the situation is more complicated since weusually have E′ : V → V ′ and u has values in V ′. So, already the first equality in(1.5) is often unclear, since the expression on the right-hand side has no meaning.

Therefore, it seems to be a good idea to formulate a general convergence resultassuming that (1.1) holds and then study, under which conditions (1.1) holds.Another reason for this splitting is that (1.1) is equivalent to the fact that u hasfinite length (and all the mentioned convergence results are based on proving thatu has finite length).

Let us mention that another approach to convergence of (weak) solutions of firstand second order evolution equations with maximal monotone operators can befound in the works by Djafari Rouhani and his co-workers, see [8] and referencestherein.

In Section 2 we formulate and prove general convergence results assuming that(1.1) holds. In Sections 3 and 4 we give several applications to first and secondorder equations, respectively. Although the results in Sections 3 and 4 are known,we present some proofs to illustrate the applicability of the results in Section 2.

2. General convergence results

Before we formulate and prove the main results, we introduce some notations.Let V , H, be Hilbert spaces such that V → H → V ′. Then ‖ · ‖, ‖ · ‖V , ‖ · ‖∗ willbe the norms in H, V , and V ′, respectively. Corresponding scalar products will bedenoted by the same subscripts. The open ball in V of radius r centered at φ ∈ Vis denoted by BV (φ, r).

If u : R+ → V then the omega-limit set of u in V is

ωV (u) := φ ∈ V : ∃tn +∞ such that ‖u(tn)− φ‖V → 0.

We say that u ∈ C1(R+, H) has finite length in H if∫ +∞

0‖u(s)‖ ds < +∞.

We say that a function E satisfies Lojasiewicz (or Simon- Lojasiewicz) inequalityon a neighborhood of ϕ, if there exists θ ∈ (0, 1/2] and c > 0 such that (1.2) holds(‘u near φ’ means u ∈ BV (φ, ε) for some ε > 0). We say that E satisfies Kurdyka- Lojasiewicz-Simon inequality on a neighborhood of ϕ, if there exists c > 0 and afunction Θ ∈ C([0,+∞)) satisfying Θ(s) > 0 for all s > 0, 1/Θ ∈ L1

loc([0,+∞))and condition (1.3). We will call functions Θ with the above properties Kurdykafunctions. Taking Θ(s) = s1−θ yields that Lojasiewicz inequality is a special caseof Kurdyka- Lojasiewicz-Simon inequality. If Θ is a Kurdyka function, we define

ΦΘ(t) :=∫ t

01/Θ(s) ds.

The following are well known results.

Lemma 2.1. If u has finite length in H, then it has a limit in H.

Lemma 2.2. Let u : R+ → V . If limt→+∞ u(t) = ψ in H and u has precompactrange in V , then limt→+∞ u(t) = ψ in V .

EJDE-2014/81 CONVERGENCE TO EQUILIBRIUM 3

Lemma 2.3. Let u : R+ → V . If u has finite length in H and precompact rangein V , then it converges in V (as t→ +∞).

We formulate the general convergence result proposed in the introduction. Itsproof follows immediately from Theorem 2.6. Let us emphasize that H can be anarbitrarily large space. So, in the applications, it is sufficient to verify (1.1) with avery weak norm on the right-hand side.

Theorem 2.4. Let u ∈ C(R+, V ) ∩ C1(R+, H) with V -precompact range and ϕ ∈ωV (u). Let ρ > 0 and E ∈ C(V,R) be such that t 7→ E(u(t)) is nonincreasing onR+ and (1.1) holds for almost every t ∈ s ∈ R+ : u(s) ∈ B := BV (ϕ, ρ). Thenlimt→+∞ ‖u(t)− ϕ‖V = 0.

Remark 2.5. By the previous Lemmas, it is sufficient to show that u has finitelength in H. One can see from the proof of Theorem 2.6 below, that the theoremremains valid if E is only defined on the closure of the range of u and continuousin V -norm on this set. Moreover, if u is injective, then this weaker condition isnot only sufficient but also necessary for u to have finite length in H. In fact, one

can define E(u(t)) :=∫ +∞t‖u(s)‖ ds, then (1.1) holds on R+, so t 7→ E(u(t)) is

nonincreasing on R+ and continuity of E also holds.

Theorem 2.4 does not speak about differential equations but it can be appliedimmediately to a solution of a first order equation

u(t) + F (u) = 0

if E is nonicreasing along the solution (e.g. a Lyapunov function) and (1.1) holds.Here F may be an unbounded nonlinear operator. Second order equations

u(t) + F (u(t), u(t)) +M(u(t)) = 0

can be reformulated as a first order equation on a product space denoting v := u.But then the energy or Lyapunov function typically depends on u and v but weare interested in convergence of the first coordinate u only (the second coordinateconverges to zero “automatically” — see Theorem 2.8). So, we will formulateTheorem 2.6 suitable for this situation. It is easy to see that Theorem 2.4 followsimmediately from Theorem 2.6 (take V2 = 0 = H2 and V := V1 × V2, H :=H1 ×H2), so we will not prove it.

Theorem 2.6. Let u = (u1, u2) satisfy u1 ∈ C(R+, V1) ∩ C1(R+, H1) and u2 ∈C(R+, V2) ∩ C1(R+, H2) with V1 → H1, and let (u1(·), u2(·)) have a precompactrange in V1 × V2. Let ϕ ∈ ωV1

(u1), ρ > 0 and E ∈ C(V1 × V2,R) be such thatt 7→ E(u(t)) is nonincreasing on R+ and

− d

dtE(u(t)) ≥ ‖u1(t)‖H1

(2.1)

for almost every t ∈ s ∈ R+ : u1(s) ∈ B := BV1(ϕ, ρ). Then limt→+∞ ‖u1(t) −

ϕ‖V1= 0.

Remark 2.7. (i) It will be clear from the proof that Theorem 2.6 remains valid if(2.1) holds only for almost every t ∈ s ∈ [T,+∞) : u1(s) ∈ B := BV1

(ϕ, ρ) forsome T > 0.

Proof of Theorem 2.6. Let tn +∞ be an increasing sequence such that ‖u1(tn)−ϕ‖V1 → 0. By precompactness of the range we may assume that ‖u2(tn)−ψ‖V2 → 0for some ψ ∈ V2 (passing to a subsequence of tn if necessary).


Since t 7→ E(u(t)) is nonincreasing it has a limit for t → +∞. Since it iscontinuous, we have limt→+∞ E(u(t)) = E(ϕ,ψ) and we can assume without lossof generality E(ϕ,ψ) = 0 and E(u(t)) ≥ 0 for all t ∈ R+ (redefining E(u) :=E(u)− E(ϕ,ψ)).

Since ‖u1(tn) − ϕ‖V1→ 0, we have u1(tn) ∈ B for all n ≥ n0. Let us denote

sn := infs≥tnu1(s) 6∈ B and assume for contradiction that sn < +∞ for all n.From continuity of u we have sn > tn and ‖u1(sn)− ϕ‖V1 = ρ.

For t ∈ (tn, sn) inequality (2.1) holds, so

E(u(tn))− E(u(t)) ≥∫ t

tn

‖u1(s)‖H1 ds.

So, we can estimate

‖u1(t)− ϕ‖H1 ≤ ‖u1(t)− u1(tn)‖H1 + ‖u1(tn)− ϕ‖H1

≤∫ t

tn

‖u1(s)‖H1ds+ ‖u1(tn)− ϕ‖H1

≤ E(u(tn))− E(u(t)) + ‖u1(tn)− ϕ‖H1

≤ E(u(tn)) + ‖u1(tn)− ϕ‖H1

and by continuity of u this inequality holds for t = sn. Hence, ‖u1(sn) − ϕ‖H1 ≤E(u(tn)) + ‖u1(tn)− ϕ‖H1 → 0 as n→∞ (since V1 → H1).

On the other hand, by continuity of u we have ‖u1(sn)−ϕ‖V1= ρ for all n ∈ N.

So, there is a subsequence of u1(sn) converging to some ϕ ∈ V1 (by precompactnessof the range), ϕ 6= ϕ, which is a contradiction with ‖u1(sn)− ϕ‖H1

→ 0.Hence, sn = +∞ for some n. Hence, u1 ∈ L1(R+, H1), it has finite length in H1

and converges to φ in the norm of V1 by Lemma 2.2. In case of second order equations, if a solution has a limit then its derivative

usually tends to zero. However, convergence of the derivative often needs muchweaker assumptions (or different assumptions) and it is helpful to know the conver-gence of the derivative a-priori, before one shows convergence of the function itself.Therefore, we formulate the following theorem.

Theorem 2.8. Let V → H → V ′ be Hilbert spaces, F ∈ C(V × H,V ′), E ∈C1(V,R) and M = E′ : V → V ′. Assume that there exists a nondecreasing functiong : (0,+∞)→ (0,+∞) such that

〈F (u, v), v〉V ′,V ≥ g(‖v‖∗)for all u, v ∈ V . If u ∈ C1(R+, V ) ∩ C2(R+, H) is a classical solution of

u(t) + F (u(t), u(t)) +M(u(t)) = 0,

u(0) = u0 ∈ V, u(0) = u1 ∈ H(2.2)

such that (u, u) is precompact in V ×H, then limt→+∞ ‖u(t)‖ = 0.

Proof. Since range of (u, u) is precompact in V × H, range of F (u, u) + M(u) isbounded in V ′. Hence, range of u is bounded in V ′ and u is Lipschitz continuousin V ′. Moreover, we have

− d

dt

1

2‖u(t)‖2 = −〈u(t), u〉V ′,V

= 〈F (u(t), u(t)), u〉V ′,V +d

dtE(u(t))


≥ g(‖u(t)‖∗) +d

dtE(u(t)).

Since |E(u(s))| ≤ K for some K > 0 and all s ≥ 0, integrating on [t0, t],

∫ t

t0

g(‖u(s)‖∗) ds ≤ 1

2(−‖u(t)‖+ ‖u(t0)‖)− E(u(t)) + E(u(t0))

≤ 1

2‖u(t0)‖+ 2K.

(2.3)

Hence, s 7→ g(‖u(s)‖∗) ∈ L1((0,+∞)) and due to Lipschitz continuity we havelimt→+∞ ‖u(t)‖∗ = 0. Since range of u is precompact in H, limt→+∞ ‖u(t)‖ =0.

Corollary 2.9. Let the assumptions of Theorem 2.8 be satisfied and let there existρ > 0 and E ∈ C(V ×H,R) such that t 7→ E(u(t), u(t)) is nonincreasing on (0,+∞)and

− d

dtE(u(t), u(t)) ≥ c‖u(t)‖∗ (2.4)

for almost every t ∈ s ∈ R+ : u(s) ∈ BV (ϕ, ρ) × BH(0, ε) where ε > 0 isarbitrary. Then limt→+∞ ‖u(t)− ϕ‖V + ‖u(t)‖ = 0.

Proof. The derivative converges to 0 by Theorem 2.8. Then u(t) ∈ BH(0, ε) forall t ≥ T . Then (2.1) is satisfied for t ∈ [T,+∞) and applying Theorem 2.6 withH1 = V ′ (see Remark 2.7) we obtain convergence of u(t).

Remark 2.10. We can see that the ∗-norm on the right-hand side of (2.4) can bereplaced by any other norm weaker than H-norm.

3. Applications to first order equations

In this section, we show several known results that are covered by Theorem 2.4.

3.1. Lojasiewicz convergence result. We start with the classical convergenceresult by Lojasiewicz. Let us remark that the following Proposition speaks aboutordinary differential equations (then u has values in a finite-dimensional space H =V and E ∈ C1(H)) and also about partial differential equations (then V → H areHilbert spaces, u ∈ C(R+, V ) ∩ C1(R+, H) and E ∈ C1(V )).

Proposition 3.1. Let u be a solution to the gradient system u+∇E(u) = 0, ϕ ∈ωV (u) and let E satisfy the Lojasiewicz or Kurdyka- Lojasiewicz-Simon inequalityon a neighborhood of ϕ. Then there exists a function E such that t 7→ E(u(t)) isnonincreasing and (1.1) holds on a neighborhood of ϕ.

Proof. It is sufficient to define E(u) := E(u)θ in case of Lojasiewicz inequality andE(u) := ΦΘ(E(u)) in case of Kurdyka- Lojasiewicz-Simon inequality.

3.2. Convergence result by Chill, Haraux, Jendoubi and its corollaries.Theorem 1 in [5] is another corollary of Theorem 2.4. If we replace Lojasiewiczinequality by the more general Kurdyka- Lojasiewicz-Simon inequality, then thetheorem in [5] reads as follows.


Theorem 3.2 ([5, Theorem 1]). Let u ∈ C(R+, V )∩C1(R+, H) with V -precompactrange and ϕ ∈ ωV (u). Let ρ > 0, c > 0 and E ∈ C2(V,R) be such that t 7→ E(u(t))is differentiable almost everywhere and

− d

dtE(u(t)) ≥ c‖E′(u(t))‖∗‖u(t)‖∗

for almost every t ∈ R+ with u(t) ∈ BV (ϕ, ρ). Assume in addition that

if E(u(·)) is constant for t ≥ t0, then u is constant for t ≥ t0and that E satisfies the Kurdyka- Lojasiewicz-Simon gradient inequality with a Kur-dyka function Θ. Then limt→+∞ ‖u(t)− ϕ‖V = 0.

Proof. We can assume that E(ϕ) = 0. If E(u(t)) = 0 for some t0, then u is constantfor all t > t0 and the assertion holds. Otherwise, E(u(t)) > 0 for all t ∈ R+. Inthis case, let us define E(u) := ΦΘ(E(u)). Then

− d

dtE(u(t)) ≥ 1

Θ(E(u(t)))· c‖E′(u(t))‖∗‖u(t)‖∗ ≥ c‖u(t)‖∗.

So, assumptions of Theorem 2.4 hold and ‖u(t)− ϕ‖V → 0.

For many applications and corollaries of Theorem 3.2 see [5].

3.3. Convergence result by Barta, Chill, Fasangova. In [2], Barta, Chill andFasangova proved a convergence theorem formulated on manifolds. If we reformu-late it for RN , it becomes a corollary of Theorem 2.4.

Theorem 3.3 ([2, Theorem 3]). Let F ∈ C(RN ,RN ), u : R+ → RN be a globalsolution of the ordinary differential equation

u(t) + F (u(t)) = 0 (3.1)

and let E : RN → R be a continuously differentiable, strict Lyapunov function for(3.1). Assume that there exist a Kurdyka function Θ, ϕ ∈ ω(u) and a neighbourhoodU of ϕ such that for every v ∈ U we have F (v) 6= 0 and

Θ(|E(v)− E(ϕ)|) ≤ 〈E′(v),F (v)

‖F (v)‖〉. (3.2)

Then u has finite length and, in particular, limt→+∞ u(t) = ϕ.

Proof. Let us recall that E is a strict Lyapunov function for (3.1), if 〈E′(u), F (u)〉 >0, whenever u ∈ RN , F (u) 6= 0. Since E(u(·)) is nonincreasing and continuous,it has a limit which is equal to E(ϕ). We can assume that E(ϕ) = 0, so thatE(u(t)) ≥ 0 for all t ∈ R+. If E(u(t0)) = 0 for some t0 ≥ 0, then E(u(t)) = 0 forevery t ≥ t0, and therefore, since E is a strict Lyapunov function, the function u isconstant for t ≥ t0. In this case, there remains nothing to prove.

Hence, we may assume that E(u(t)) > 0 for every t ≥ 0 and define E(u) :=ΦΘ(E(u)). Then

− d

dtE(u(t)) =

1

Θ(E(u(t))

(− d

dtE(u(t))

)

=1

Θ(E(u(t))〈E′(u(t)), F (u(t))〉

≥ ‖F (u(t))‖ = ‖u(t))‖


in a neighborhood of ϕ. Hence the assumptions of Theorem 2.4 are satisfied andlimt→∞ u(t) = ϕ.

4. Applications to second order equations

4.1. Second order ODE with weak nonlinear damping. The equation

u(t) + |u(t)|αu(t) +∇E((u(t))) = 0

with α > 0 was studied by Chergui in [3] and the convergence result was thenextended to more general dampings

u(t) +G(u(t), u(t)) +∇E((u(t))) = 0 (4.1)

by Barta, Chill and Fasangova [2], where G ∈ C2(RN×RN ) and for every u, v ∈ RNit holds that

〈G(u, v), v〉 ≥ g(‖v‖) ‖v‖2,‖G(u, v)‖ ≤ cg(‖v‖) ‖v‖,‖∇G(u, v)‖ ≤ c g(‖v‖),

(4.2)

where c ≥ 0 is a constant and g : R+ → R+ is a nonnegative, concave, nondecreasingfunction, g(s) > 0 for s > 0.

Under these assumptions we have

〈G(u, v), v〉 ≥ g(‖v‖) ‖v‖2 = g(‖v‖∗) ‖v‖2∗ =: g(‖v‖∗),so assumptions of Theorem 2.8 hold with g. By Corollary 2.9, it is sufficient toprove that

E((u, v)) := ΦΘ

(1

2‖v‖2 + E(u) + ε〈G(u,∇E(u)), v〉

)

satisfies the key estimate (2.4), which needs some work (see [2] for details).

4.2. A semilinear wave equation with nonlinear damping. The followingproblem was studied by Chergui in [4]. Consider the equation

utt + |ut|αut = ∆u+ f(x, u) (4.3)

in R+ × Ω with Dirichlet boundary conditions and initial values

u(0, ·) = u0 ∈ H10 (Ω), ut(0, ·) = u1 ∈ L2(Ω).

Function f : Ω× R→ R satisfies

• If N = 1: f , ∂2f are bounded in Ω× [−r, r] for all r > 0,• If N ≥ 2: f(·, 0) ∈ L∞(Ω) and |∂2f(x, s)| ≤ c(1 + |s|γ) on Ω× R,

where c ≥ 0, γ ≥ 0 and (N − 2)γ < 2.Then the main part of the proof of [4, Theorem 1.4] can be interpreted as proving

that (for appropriate α and θ and small ε > 0)

E((u(t), u(t)))

:=(1

2‖u(t)‖22 + E(u(t))− ε‖u(t)‖α∗ 〈∆u(t) + f(x, u(t)), u(t)〉∗

)θ−(1−θ)α

satisfies estimate (2.4), where

E(u) :=1

2‖∇u‖22 −

∫

Ω

F (x, u) dx, F (x, u) :=

∫ u

0

f(x, s) ds. (4.4)


Let us mention that Corollary 2.9 can be applied in this case, if we consider classicalsolutions (the result in [4] refers to weak solutions).

4.3. Abstract wave equation with linear damping. The following abstractsecond-order equation is studied in [5]. We have V → H → V ′, γ 6= 0, E ∈ C2(V ),M = E′ and consider the equation

utt + γut +M(u) = 0. (4.5)

Let us introduce the duality mapping K : V ′ → V given by 〈u, v〉∗ = 〈u,Kv〉V ′,V =〈u,Kv〉 for u ∈ H, v ∈ V ′.Theorem 4.1 ([5, Corollary 16]). Assume that γ > 0 and

(1) for every v ∈ V , the operator KM ′(v) extends to a bounded operator on Hand supv ‖KM ′(v)‖L(H) is finite when v ranges over a compact subset ofV , and

(2) u ∈ C1(R+, V )∩C2(R+, H) is a global solution to (4.5), (u, u) has precom-pact range in V ×H and there exist ϕ ∈ ω(u), C > 0, ρ > 0 and a sublinearKurdyka function Θ, such that E satisfies Kurdyka- Lojasiewicz-Simon gra-dient inequality in BV (ϕ, ρ).

Then limt→+∞ ‖u(t)− ϕ‖V = 0.

Since

〈γu, u〉 ≥ γc‖u‖2∗ =: g(‖u‖∗),the assumptions of Theorem 2.8 are satisfied and ‖u‖ → 0. It is not difficult toshow that function E(u, u) := ΦΘ(Ψ(u, u)) satisfies the key estimate (2.4), where

Ψ(u, u) :=1

2‖u‖2 + E(u) + ε〈M(u), u〉∗

and ε > 0 is small enough. Then Corollary 2.9 proves the assertion.

Acknowledgements. This work is supported by GACR 201/09/0917. The au-thor is a researcher in the University Centre for Mathematical Modeling, AppliedAnalysis and Computational Mathematics (Math MAC) and a member of the NecasCenter for Mathematical Modeling.

References

[1] P. A. Absil, R. Mahony, B. Andrews; Convergence of the iterates of descent methods foranalytic cost functions, SIAM J. Optim. 16 (2005), no. 2, 531–547.

[2] T. Barta, R. Chill, E. Fasangova; Every ordinary differential equation with a strict Lyapunovfunction is a gradient system, Monatsh. Math. 166 (2012), 57–72.

[3] L. Chergui; Convergence of global and bounded solutions of a second order gradient like systemwith nonlinear dissipation and analytic nonlinearity, J. Dynam. Differential Equations 20(2008), no. 3, 643–652.

[4] L. Chergui; Convergence of global and bounded solutions of the wave equation with nonlineardissipation and analytic nonlinearity, J. Evol. Equ. 9 (2009), 405–418.

[5] R. Chill, A. Haraux, M. A. Jendoubi; Applications of the Lojasiewicz-Simon gradient inequal-ity to gradient-like evolution equations, Anal. Appl. 7 (2009), 351–372.

[6] A. Haraux, M. A. Jendoubi; Convergence of solutions of second-order gradient-like systemswith analytic nonlinearities, J. Diff. Eqs 144 (1998), no. 2, 313–320.

[7] C. Lageman; Pointwise convergence of gradient-like systems, Math. Nachr. 280 (2007), no.13-14, 1543–1558.

[8] B. Djafari Rouhani, H. Khatibzadeh; A strong convergence theorem for solutions to a nonho-mogeneous second order evolution equation, J. Math. Anal. Appl. 363 (2010), no. 2, 648–654.


Tomas BartaDepartment of Mathematical Analysis, Faculty of Mathematics and Physics, CharlesUniversity, Sokolovska 83, 186 75 Prague 8, Czech Republic

E-mail address: [email protected]


4.3 T. Barta and E. Fasangova, Convergence

to equilibrium for solutions of an ab-

stract wave equation with general damp-

ing function, J. Differential Equations 260

(2016), no. 3, 2259–2274.


Available online at www.sciencedirect.com

ScienceDirect

J. Differential Equations 260 (2016) 2259–2274

www.elsevier.com/locate/jde

Convergence to equilibrium for solutions of an abstract

wave equation with general damping function

Tomáš Bárta a,∗, Eva Fašangová b

a Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University in Prague, Sokolovská 83, 18675 Praha 8, Czech Republic

b Technische Universität Dresden, Institut für Analysis, Fachrichtung Mathematik, Zellescher Weg 12-14, 01062 Dresden, Germany

Received 11 November 2014; revised 9 September 2015

Available online 23 October 2015

Abstract

We prove convergence to a stationary solution as time goes to infinity of solutions to abstract nonlinear wave equation with general damping term and gradient nonlinearity, provided the trajectory is precompact. The energy is supposed to satisfy a Kurdyka–Łojasiewicz gradient inequality. Our aim is to formulate conditions on the function g as general as possible when the damping is a scalar multiple of the velocity, and this scalar depends on the norm of the velocity, g(|ut |)ut . These turn out to be estimates and a coupling condition with the energy but not global monotonicity. When the damping is more general, we need an angle condition.© 2015 Elsevier Inc. All rights reserved.

Keywords: Abstract wave equation with damping; Convergence to equilibrium; Łojasiewicz inequality

1. Introduction

This work has been inspired by a result of Chergui presented in [4], where the following semilinear damped wave equation

utt (t, x) + |ut (t, x)|αut (t, x) = u + f (x,u(t, x)), t ≥ 0, x ∈ ⊂ RN, (1)

* Corresponding author.E-mail address: [email protected] (T. Bárta).

http://dx.doi.org/10.1016/j.jde.2015.10.0030022-0396/© 2015 Elsevier Inc. All rights reserved.

2260 T. Bárta, E. Fašangová / J. Differential Equations 260 (2016) 2259–2274

with zero boundary conditions on ∂ is studied. It is proved that every bounded solution has relatively compact trajectory and that for certain values of α every solution with relatively com-pact trajectory converges to a stationary solution. The set of admissible α’s depends on the Łojasiewicz exponent of the operator + f (x, ·).

The main goal of this paper is to study the above equation with a more general damping term g(|ut |)ut resp. G(u, ut ) instead of |ut |αut and to obtain convergence to equilibrium for solutions with relatively compact trajectory. We prove our result in a more general setting assuming an abstract gradient operator E′(u) instead of u + f (x, u). Thus, we study the equation

utt (t) + g(|ut (t)|)ut (t) = E′(u(t)), t > 0, (2)

where the damping is g(|ut |)ut , ut is the velocity and g is a scalar function. Our analysis shows also the way how to generalise the result to a more general model, an anisotropic, inhomogeneous medium where the damping need not point into the direction of the velocity, that is the equation

utt (t) + G(u(t), ut (t)) = E′(u(t)), t > 0. (3)

In this formulation we obtain a generalisation of [1, Theorem 4] for the ordinary differential equation

u(t) + G(u(t), u(t)) = E′(u(t)), t > 0, (4)

for u : [0, +∞) → RN , for more general damping than in [1]; see also [3].We denote V := H 1

0 (), H := L2(), V ∗ := H−1(), where ⊂ RN is open and bounded. Let E ∈ C2(V ) and let g : [0, +∞) → [0, +∞) be given. Consider the problem (2) with given initial values u(0) = u0 ∈ V, ut (0) = u1 ∈ H . Let us assume that there exists a solution u ∈ C1([0, +∞), H) ∩C([0, +∞), V ) such that |ut |2g(|ut |) ∈ L1((0, +∞), L1()) and assume that the trajectory (u(t), ut (t)) : t ≥ 0 is relatively compact in V × H . Then there exists a se-quence tn → +∞ such that (u(tn), ut (tn)) converges to some (ϕ, ψ) ∈ V ×H and one can show that ψ = 0 (see [4]). The question we are interested in is whether

limt→+∞u(t) = ϕ ?

In [4, Theorem 1.4] Chergui gave a positive answer to this question for the equation (1) under suitable assumptions on f provided α satisfies the following two conditions:

1. 0 < α < θ1−θ

, where θ is a Łojasiewicz exponent depending on E, 0 < θ ≤ 12 ,

2. α < 4N−2 .

The first condition says that the damping term g(|ut |)ut is not too small near zero (which seems to be a reasonable condition). It also estimates the growth at infinity but it can be seen from the proof that one does not need this estimate. The second condition says that the growth of g at +∞is not too fast and it stems from a Sobolev embedding needed in the proof. It also means that the growth of g at zero is not too small, but we show that this estimate at zero is not necessary. From the physical interpretation we would say that the bigger is the damping term, the better is the convergence or the stabilisation effect.

T. Bárta, E. Fašangová / J. Differential Equations 260 (2016) 2259–2274 2261

In general, convergence can fail. For example, for linear damping in equation (1) (i.e. α = 0, resp. g(|ut |)ut = ut ) see [8], for the damping g(|ut |)ut = |ut |ut in equation (4), N = 1, mono-tone right-hand side E′ see [7].

In this paper we give a positive answer to the question for equations (2), (3) and (4) under suitable assumptions. First we formulate the assumptions on g, E and a coupling condition be-tween them, and we formulate our main result for equation (2). In Section 3 we formulate an equivalent set of assumptions. In Section 4 we prove the result for equation (2). In Section 5we formulate the assumptions on G, E and prove the main result for equation (3). Finally we mention a corollary for the equation (4).

2. Main result for the equation with a scalar damping function

First of all we fix the notation. Let ⊂ RN be a bounded domain. We work with the Hilbert spaces H := L2() resp. L2(, Rn) with norm and scalar product denoted by ‖ · ‖ and 〈·, ·〉, and V := H 1

0 () resp. H 10 (, Rn) with norm ‖ · ‖V . We have the continuous embedding V → H ,

we identify H with its dual H ∗, and we denote V ∗ the dual space to V . In this way we have V → H → V ∗ and after identification 〈v, u〉V ∗,V = 〈v, u〉 for u ∈ V ⊂ H, v ∈ H ⊂ V ∗. The norm and the scalar product on V ∗ are denoted by ‖ · ‖∗ and 〈·, ·〉∗.

We denote by K : V ∗ → V the duality mapping given by

〈v,u〉∗ = 〈v,Ku〉V ∗,V , u, v ∈ V ∗.

For N > 2 the space V is embedded into Lq() resp. Lq(, Rn) provided q ≤ 2NN−2 . We

define p := 2NN+2 for N > 2 and p := 1 for N = 1, and have 1 ≤ p < 2,

V → Lp′→ H → Lp → V ∗

with p′ = 2NN−2 for N > 2, p′ = ∞ for N = 1, and

〈v,u〉V ∗,V = 〈v,u〉Lp,Lp′ = 〈v,u〉, u ∈ V, v ∈ H.

For N = 2 the above embeddings hold for all p ∈ (1, 2) (and corresponding p′) but not for p = 1. Since there is no minimal value of p in this case, we fix the value of p later, see the text below condition (g3). The norm on Lp is denoted by ‖.‖p .

We usually denote real numbers by s, r , vectors in Rn by z, w, and x ∈ RN . By |.| we denote the Euclidean norm on Rn (resp. absolute value on R). Letters u, v are used for members of V ∗ (and its subspaces V , H ) or for functions of two variables, e.g. u : [0, +∞) → H . If u is a function of t ∈ R and x ∈ RN , we often write u(t) instead of u(t, ·), and ut = du

dt= u for the time

derivative. By BV (ϕ, ε) we denote the open ball in V with radius ε and centered at ϕ.Now we introduce the assumptions. We say that a function : [0, +∞) → [0, +∞) has

property (KL) if it is nondecreasing, sublinear (i.e. (s + r) ≤ (s) +(r) for all r, s ≥ 0) and satisfies (s) > 0 for all s > 0, and (s) ≤ C

√s for all s ∈ [0, τ] for some constants τ > 0

and C > 0. For example, the function (s) = s1−θ with θ ∈ [0, 12 ] has property (KL).

Remarks. Since the assumptions on below ((e1) and (h2)) involve only arguments near zero, we could define the property (KL) on a neighbourhood of zero only (any such function can be extended by a constant to [0, +∞) such that it has the above properties on the whole [0, +∞)).


The sublinearity assumption could be weakened to (s + r) ≤ C((s) + (r)) for some C > 0and all r, s ≥ 0, and our results would remain valid.

Our assumptions on the operator E are the following.

(E) Assume that E ∈ C2(V ) satisfies:(e1) there exists a function with property (KL) such that 1

is integrable in a neigh-

bourhood of zero and such that E satisfies the Kurdyka–Łojasiewicz gradient inequal-ity with the function in a neighbourhood of the critical points of E, i.e., for each ϕ ∈ N := ψ ∈ V : E′(ψ) = 0 there exist η, C > 0 such that

‖E′(u)‖∗ ≥ C(|E(u) − E(ϕ)|), u ∈ BV (ϕ,η); (5)

(e2) for all u ∈ V , the operator KE′′(u) ∈ L(V ) extends to a bounded linear operator on Hand sup‖KE′′(u)‖L(H) is finite whenever u ranges over a compact subset of V .

In [4] Chergui works with E′(u) = u + f (x, u) which corresponds to E(u) = ∫

12 |∇u(x)|2 +

F(x, u)dx, where F(x, u) := ∫ u

0 f (x, s) ds (n = 1). By [4, Corollary 1.2], this function E satis-fies the Łojasiewicz gradient inequality

‖E′(u)‖∗ ≥ C|E(u) − E(ϕ)|1−θ (6)

with some θ ∈ [0, 12 ) in a neighbourhood of N , provided f satisfies certain assumptions. The Łojasiewicz inequality (6) is a special case of the Kurdyka–Łojasiewicz inequality (5) with the function (s) = s1−θ , θ being the Łojasiewicz exponent. It is easy to see that Chergui’s operator satisfies (e2) as well. The conditions (e1) and (e2) (with (6) instead of (5)) appear also in [5], where linear damping is considered.

Now we formulate the assumptions on the damping function.

(G) The function g : [0, +∞) → [0, +∞) is continuous on (0, +∞) and there exists τ > 0 such that(g1) there exists C2 > 0 such that g(s) ≤ C2 on [0, τ),(g2) there exists C3 > 0 such that C3 ≤ g(s) on [τ, +∞),(g3) if N = 2 then there exist C4 > 0 and α > 0 such that g(s) ≤ C4s

α on [τ, +∞), and if N > 2 then there exists C4 > 0 such that g(s) ≤ C4s

α on [τ, +∞) with α = 4N−2 .

If N > 2, then p = α+2α+1 holds. We set p := α+2

α+1 for N = 2, too.

(H) For τ from condition (G) there exists a function h : [0, +∞) → [0, +∞), which is concave and nondecreasing on [0, τ ] and satisfies(h1) g ≥ h on [0, τ ],(h2) the function s → 1

(s)h((s))belongs to L1((0, τ)),

(h3) the function ψ : s → sh(√

s) is convex on [0, τ 2].

Note that no monotonicity of the damping function g is assumed, only some estimates from above and from below. Clearly, Chergui’s damping function g(s) = sα , α < 4

N−2 satisfies (G)


and it also satisfies (H) with h(s) = sα for (s) = s1−θ , θ ∈ (0, 12 ), where our condition (h2) corresponds to Chergui’s condition 0 < α < θ

1−θ. This is the condition coupling the damping

function g with the operator E.We want to achieve that the damping term g(|v|)v lies in the largest Lp-space satisfying

V → Lp′, i.e., Lp → V ∗, for v ∈ Lp′

. This is guaranteed by the growth condition (g3). If N = 2, then there is no largest Lp-space with this property. However, there is an optimal (smallest) Orlicz space L, (t) = et2

satisfying V → L (see [6]). It would be a subject of further research to work with this embedding and to extend the results of this paper for exponentially growing functions g in case N = 2.

Our main result is formulated for solutions in the following sense. We say that u ∈W

1,1loc ([0, +∞), V ) ∩ W

2,1loc ([0, +∞), H) is a strong solution to (2) if (2) holds in V ∗ for almost

every t > 0. The omega-limit set of u is

ωV (u) = ϕ ∈ V : there exists a sequence tn +∞ such that limn→∞‖u(tn) − ϕ‖V = 0.

Condition (g1) and the choice of p imply that g(|ut (t)|)ut (t) ∈ Lp → V ∗ for almost every t > 0 for a strong solution. We analogously define a strong solution of equation (3) under the assumptions on G in Section 5.

Theorem 2.1. Let E and g satisfy (E), (G) and (H). Let u be a strong solution to (2) such that (u(t), ut (t)) : t ≥ 0 is relatively compact in V × H and ϕ ∈ ωV (u). Thenlimt→+∞ (‖u(t) − ϕ‖V + ‖ut (t)‖) = 0.

Remarks. Condition (H), which estimates g from below on a neighbourhood of zero, is more complicated than the others, but this condition is trivial if lim infs→0+ g(s) > 0, since then a small constant function h works ((h2) holds since 1/ is integrable due to (e1)). If lim infs→0+ g(s) = 0, then necessarily h(0) = 0. Condition (h2) says that the growth of h at zero must be steep enough. In fact, together with the condition (s) ≤ c

√s from property (KL) we

have that h′+(0) = +∞ and if lims→0+ g(s) = 0, then also g′+(0) = +∞. Finally, let us mention that every function

h(s) := sα

(ln

1

s

)α1(

ln ln1

s

)α2

. . .

(ln . . . ln

1

s

)αn

with α ∈ (0, 1), n ∈ N, αi ∈ R satisfies condition (h3). This can be shown by computing the first derivative of h and the second derivative of s → sh(

√s).

3. An equivalent set of assumptions

In this section we introduce another set of assumptions ((G), (H), ()) and show that these assumptions are equivalent to assumptions (G), (H). These new assumptions are motivated by the proof of Theorem 1.4 in [4]. Reading that proof carefully and analysing the assumptions needed lead us to this set and we prove the assertion of Theorem 2.1 under these new assumptions in the next section. Here we show that the old assumptions imply the new ones. And we also show the opposite implication which says in some sense that these assumptions are the best possible if we want to use the method from [4]. Here are the new assumptions:


(G) The function g : [0, +∞) → [0, +∞) is continuous on (0, +∞) and there exists τ > 0 such that (g3) holds.

(H) There exists a function h : [0, +∞) → [0, +∞), which is positive and concave on (0, +∞)

and satisfies(h1) g ≥ h on [0, +∞),(h2) the function s → 1

(s)h((s))belongs to L1((0, 1)),

(h3) the function ψ : s → sh(√

s) is convex on [0, +∞).() There exists a Young function γ : [0, +∞) → [0, +∞) (convex with γ (0) = 0,

lims→+∞ γ (s) = +∞) such that(γ 1) there exists D1 > 0 such that γ (g(s)s) ≤ D1g(s)s2 for s ≥ 0,(γ 2) there exists D2 > 0 such that γ (s) ≥ D2s

2 on a neighbourhood of zero,

(γ 3) the function γ : s → γ (s1p ) is convex on [0, +∞).

(γ 4) for every K > 0 there exists C(K) such that γ (Ks) ≤ C(K)γ (s) holds for all s ≥ 0.

We say that a function f : [0, +∞) → [0, +∞) has property (K) if for every K > 0 there exists C(K) such that f (Ks) ≤ C(K)f (s) for all s ≥ 0. So, (γ 4) says that γ has property (K). Typically, nondecreasing functions with polynomial growth do have this property, while functions with exponential growth do not.

Lemma 3.1. Condition (H) implies that

(h5) h is nondecreasing on [0, +∞),(h6) sh′±(s) ≤ h(s) for s ∈ [0, +∞),(h7) h has property (K),(h8) ψ has property (K).

Proof. (h5), (h6) follow immediately from concavity and positivity of h. (h7) holds with C(K) = 1 for K ≤ 1 since h is nondecreasing and C(K) = K for K > 1 since h is con-cave and h(s) ≥ 0. We show that (h8) follows from (h7). In fact, ψ(Ks) = Ksh(

√Ks) ≤

KsC(√

K)h(√

s) = KC(√

K)ψ(s). Lemma 3.2. Denote by δ the convex conjugate function to γ . Then (γ 2) is equivalent to δ(s) ≤d3s

2 on a neighbourhood of zero for some d3 > 0.

Proof. By definition δ(r) = sups≥0(rs − γ (s)). From the shape of γ it follows that the max-imizer s0 of rs − γ (s) is small if r is small. Hence, for small r , maxs≥0(rs − γ (s)) ≤maxs≥0(rs − D2s

2) = r2

(2D2)2 . For the converse implication we compute γ (s) = maxr≥0(sr −

δ(r)) ≥ maxr≥0(sr − d3r2) = s2

(2d3)2 .

Proposition 3.3. Given a function g : [0, +∞) → [0, +∞) and a function : [0, +∞) →[0, +∞), the set of assumptions (G), (H) is equivalent to the set of assumptions (G), (H), ().

Proof. First we show that (G), (H), () imply (G) and (H). For the moment, τ > 0 is arbitrary, later it will be chosen small enough. The condition (g3), upper bound on [τ, +∞), follows from (G) on a neighbourhood of infinity [K, +∞) and from continuity of g on the compact interval


[τ, K]. The condition (g2), lower bound on [τ, +∞), follows from positivity and concavity of hand inequality (h1). Concerning condition (g1), upper bound on [0, τ), we distinguish two cases. The first case lims→0+ sg(s) = 0 leads to a contradiction. In fact, taking sk → 0, sk > 0 with skg(sk) ≥ c > 0 and dividing the inequality in (γ 1) by g(s)s we obtain

γ (skg(sk))

skg(sk)≤ D1sk.

Here the right-hand side tends to zero as k → ∞ and the left-hand side does not since γ (r) ≥ ar

for r ∈ [c, +∞) for some a > 0 (γ is increasing and convex on a neighbourhood of +∞). In the second case lims→0+ sg(s) = 0 we have γ (g(s)s) ≥ D2s

2g(s)2 and γ (g(s)s) ≤ D1g(s)s2, hence g(s) ≤ D1

D2for s ∈ [0, τ), provided τ > 0 is small enough. Condition (H) follows immediately by

taking h := h on [0, τ ] and constant h(τ ) on (τ, +∞).Now we prove that (G), (H) imply (G), (H) and ().(G) follows immediately from (g3). To show () let us define

γ (s) :=

c1s2 for s ∈ [0, τ )

sp − τp + c1τ2 for s ∈ [τ,+∞),

where the constant c1 > 0 will be chosen such that c1 < τp−2, c1 ≤ p2 τ

p−2p , and even smaller

if necessary, see later. This function is nonnegative, continuous, and convex since p ≥ 1 and γ ′−(τ ) = 2c1τ < pτp−1 = γ ′+(τ ). Therefore γ is a Young function, and the property (γ 2) holds trivially. The property in (γ 4) holds with C(K) = 1 provided K ≤ 1, since γ is increasing. For K > 1 we distinguish three cases.

– If s < τK

, then γ (Ks) = c1K2s2 = K2γ (s).

– If s > τ , then

γ (Ks) = Kpsp − τp + c1τ2 = Kpsp − τp + c1τ

2

sp − τp + c1τ 2γ (s)

≤ supr∈(τ,+∞)

Kprp − τp + c1τ2

rp − τp + c1τ 2γ (s).

– If s ∈ [ τK

, τ ], then γ (Ks) = Kpsp−τp+c1τ2

c1s2 γ (s) ≤ maxr∈[ τ

K,τ ] Kprp−τp+c1τ

2

c1r2 γ (s).

Now, (γ 4) is proven with C(K) being the maximum of the three factors on the right-hand sides above. Concerning (γ 3), the function

γ (s) :=

c1s2p for s ∈ [0, τp)

s − τp + c1τ2 for s ∈ [τp,+∞)

is convex, since p < 2 and γ ′−(τ ) = 2c1p

τ2p

−1 ≤ 1 = γ ′+(τ ).


Now we look for a suitable constant D1 > 0 in order to satisfy (γ 1). We distinguish the cases.

– If g(s)s < τ then for s < τ by (g1) we have γ (g(s)s) = c1g(s)2s2 ≤ c1C2g(s)s2, while for s ≥ τ we have γ (g(s)s) = c1g(s)2s2 = c1

g(s)ss

g(s)s2 ≤ c1g(s)s2.– If g(s)s ≥ τ then we have

γ (g(s)s) = g(s)psp − τp + c1τ2 ≤ g(s)psp = g(s) (g(s)s)p−2 g(s)s2 ≤ C2τ

p−2g(s)s2

for s < τ , using (g2) and p−2 < 0. For s ≥ τ and N ≥ 2 we use (g3) and the fact that p > 1, α(p − 1) + p − 2 = 0 and obtain

γ (g(s)s) ≤ g(s)p−1sp−2g(s)s2 ≤ (C4s

α)p−1

sp−2g(s)s2 = Cp−14 g(s)s2.

For s ≥ τ and N = 1 we have p = 1 and γ (g(s)s) ≤ g(s)psp = 1sg(s)s2 ≤ 1

τg(s)s2.

Thus we can take D1 = maxc1C2, c1, C2τp−2, Cp−1

4 , 1τ.

Finally we turn to (H). If h(0) > 0, then g is bounded from below on [0, +∞) by a positive constant and we define h to be this constant. This function satisfies (h1) and conditions (h2), (h3) are obvious.

If h(0) = 0, take δ ∈ (0, τ) such that h(δ) < C3 and h′(δ) > 0. Then h′(δ) < h(δ)δ

since h is concave and h(0) = 0. Let us define

h(s) :=

h(s)2 for s ∈ [0, δ)

h(δ)2 + ( 1

δ− 1

s)h′(δ)δ2

2 for s ∈ [δ,+∞).

In any case h is positive and concave on (0, +∞). Further, h(s) ≤ h(δ) for all s and we can verify (h1) as follows. For s ∈ (0, δ) we have h(s) ≤ h(s) ≤ g(s). For s ∈ [δ, τ ] we have h(s) ≤h(δ) ≤ h(s) ≤ g(s). For s ∈ (τ, +∞) we have h(s) ≤ h(δ) ≤ C3 ≤ g(s).

We turn to (h3), the convexity of the function ψ : s → sh(√

s). On [0, δ2) convexity follows from (h3). For s > δ2 we have

ψ ′′(s) =(

s

(h(δ)

2+ 1

δ

h′(δ)δ2

2

)− s

12h′(δ)δ2

2

)′′= 1

4

h′(δ)δ2

2s− 3

2 > 0.

For s = δ2 we have

ψ ′−(δ2) = h(δ)

2+ δ2h′−(δ)

1

4δ= h(δ)

2+

(1

δ− 1

2δ

)h′+(δ)δ2

2= ψ ′+(δ2).

Consequently, ψ is convex on [0, +∞). Condition (h2) follows immediately from (h2).


4. Proof of convergence for the equation with scalar damping

Let the assumptions (E), (G), (H) or equivalently (E), (G), (H), () hold. We denote by c∗, cp

and c∗p the constants of the embeddings H → V ∗, H → Lp and Lp → V ∗, respectively. We

start with the following lemma (compare to [4, Proposition 1.5]).

Lemma 4.1. The following assertions hold for a strong solution u to (2) which satisfies the assumptions of Theorem 2.1:

(i) the function t → g(|ut (t)|)|ut (t)|2 belongs to L1((0, +∞), L1()),(ii) limt→+∞ ‖ut (t)‖ = 0,

(iii) ϕ ∈ ωV (u) implies E′(ϕ) = 0.

Proof. Multiplying the equation (2) by ut (t) with respect to the duality 〈., .〉V ∗,V we have

〈utt , ut 〉V ∗,V + 〈g(|ut |)ut , ut 〉 = 〈E′(u),ut 〉V ∗,V (7)

and integrating over [s, T ] we obtain

(1

2‖ut (T )‖2 − E(u(T ))

)−

(1

2‖ut (s)‖2 − E(u(s))

)= −

T∫s

∫

g(|ut (t)|)|ut (t)|2dt ≤ 0.

This implies that 12‖ut (·)‖2 − E(u(·)) is nonincreasing. Relative compactness of the trajectory

of u then yields (i). Part (ii) follows from [2, Theorem 2.8].To prove (iii) let us fix ϕ ∈ ωV (u) and tn → +∞ such that u(tn) → ϕ in V . Then

u(tn + s) = u(tn) +tn+s∫tn

ut (r)dr.

Since the integral tends to zero in H by (ii), relative compactness of the trajectory implies that u(tn + s) → ϕ in V for every s ∈ [0, 1]. The following equalities hold in V ∗ (the second equality follows from Lebesgue dominated convergence theorem and the last one from (ii)):

E′(ϕ) =1∫

0

E′(ϕ)ds = limn→∞

1∫0

E′(u(tn + s))ds

= limn→∞

1∫0

(utt (tn + s) + g(|ut (tn + s)|)ut (tn + s))ds

= limn→∞

⎛⎝ut (tn + 1) − ut (tn) +

tn+1∫tn

g(|ut (s)|)ut (s)

⎞⎠ds

= limn→∞

tn+1∫tn

g(|ut (s)|)ut (s)ds .


In the following estimates we use the embedding Lp → V ∗, the notation s,τ := x ∈ :|ut (x, s)| < τ , the assumptions (g1) and (g3), the relation α(p − 1) + p = 2 (if N = 1, the estimates hold with Cp−1

4 replaced by 1/τ ), Hölder’s inequality, the embedding H → Lp and

Jensen’s inequality (the function s → s1p is concave):

∥∥∥∥tn+1∫tn

g(|ut (s)|)ut (s)ds

∥∥∥∥∗≤ c∗

p

tn+1∫tn

‖g(|ut (s)|)ut (s)‖pds =

= c∗p

tn+1∫tn

⎛⎜⎝ ∫

s,τ

g(|ut (s)|)p|ut (s)|p +∫

\s,τ

g(|ut (s)|)p−1g(|ut (s)|)|ut (s)|p⎞⎟⎠

1p

ds

≤ c∗p

tn+1∫tn

⎛⎜⎝ ∫

s,τ

Cp

2 |ut (s)|p +∫

\s,τ

Cp−14 g(|ut (s)|)|ut (s)|2

⎞⎟⎠

1p

ds

≤ c∗p

tn+1∫tn

⎛⎜⎝C2

⎛⎝∫

|ut (s)|p⎞⎠

1p

+ C

p−1p

4

⎛⎝∫

g(|ut (s)|)|ut (s)|2⎞⎠

1p

⎞⎟⎠ds

≤ c∗pC2cp

tn+1∫tn

‖ut (s)‖ds + c∗pC

p−1p

4

⎛⎝ tn+1∫

tn

⎛⎝∫

g(|ut (s)|)|ut (s)|2⎞⎠ds

⎞⎠

1p

.

The last terms tend to zero by (ii) and (i), and consequently E′(ϕ) = 0. Lemma 4.2. There exists a constant C

h> 0 such that

h(‖v‖∗)‖v‖2 ≤ Ch

∫

g(|v(x)|)|v(x)|2dx, v ∈ H.

Proof. Let ψ be from (h3). The following computation holds, since by Lemma 3.1 the function h is nondecreasing (first inequality) and has property (K) (second inequality):

h(‖v‖∗)‖v‖2 ≤ h(c∗‖v‖)‖v‖2 ≤ C(c∗)h(‖v‖)‖v‖2 = C(c∗)ψ(‖v‖2).

Since ψ has property (K) we have

ψ(‖v‖2) = ψ

(|| 1

|| ‖v‖2)

≤ C(||)ψ(

1

|| ‖v‖2)

.


By Jensen’s inequality (ψ is convex) and assumption (h1) we have

ψ

(1

|| ‖v‖2)

= ψ

⎛⎝ 1

||∫

|v|2⎞⎠ ≤ 1

||∫

ψ(|v|2) = 1

||∫

h(|v|)|v|2 ≤ 1

||∫

g(|v|)|v|2.

Altogether, the assertion follows with Ch

= 1||C(||)C(c∗).

Proof of Theorem 2.1. Let the functions h, be given as in the assumptions and ε > 0(small enough), which will be specified later. For a strong solution u of (2) let us denote v(t, x) := ut (t, x) and for convenience we abbreviate M = E′. Let us assume that h is every-where differentiable (the other case is discussed at the end of the proof).

We define

E(u(t), v(t)) := (H(u(t), v(t))), where (s) :=s∫

0

1

(ξ)h((ξ))dξ, s ≥ 0,

and

H(u,v) = 1

2‖v‖2 + E(ϕ) − E(u) − εh(‖v‖∗)〈M(u), v〉∗, u ∈ V, v ∈ H.

(It follows from the computations below that H(u(t), v(t)) ≥ 0.) We show that E is nonincreasing along solutions and that

− d

dtE(u(t), v(t)) ≥ c‖v(t)‖∗

holds for almost all t ∈ [0, +∞) such that u(t) ∈ BV (ϕ, η), where η is taken from condition (e1), because then the convergence u(t) → ϕ as t → +∞ follows from [2, Corollary 2.9].

Let us fix t > 0 and write (u, v) instead of (u(t), v(t)). We take the scalar product in V ∗ of the equation (2) with v and with M(u),

〈vt , v〉∗ + 〈g(|v|)v, v〉∗ = 〈M(u), v〉∗,〈vt ,M(u)〉∗ + 〈g(|v|)v,M(u)〉∗ = 〈M(u),M(u)〉∗.

Inserting this and (7) into the derivative below we compute (here we use that u is a strong solu-tion, and (G) which guarantees that g(|v|)v ∈ Lp() → V ∗, since v ∈ V → Lp′

()):

d

dtH(u(t), v(t)) = 〈 d

duH(u(t), v(t)), ut 〉V ∗,V + 〈 d

dvH(u(t), v(t)), vt 〉

= −〈M(u),ut 〉V ∗,V − εh(‖v‖∗)〈M ′(u)ut , v〉∗+ 〈vt , v〉 − εh′(‖v‖∗)

〈v, vt 〉∗‖v‖∗

〈M(u), v〉∗ − εh(‖v‖∗)〈M(u), vt 〉∗

= −〈g(|v|)v, v〉V ∗,V − εh′(‖v‖∗)1

‖v‖∗〈M(u), v〉2∗


+ εh′(‖v‖∗)1

‖v‖∗〈g(|v|)v, v〉∗〈M(u), v〉∗ − εh(‖v‖∗)〈M ′(u)v, v〉∗

− εh(‖v‖∗)〈M(u),M(u)〉∗ + εh(‖v‖∗)〈g(|v|)v,M(u)〉∗ (8)

(here and in what follows, if v(t) = 0 then any term containing 1‖v‖∗ has to be replaced by 0). In

the last equality we keep the first and fifth terms and estimate the other terms from above. The first term is

−〈g(|v|)v, v〉V ∗,V = −〈g(|v|)v, v〉Lp,Lp′ = −

∫

g(|v(t, x)|)|v(t, x)|2dx.

The second term is less or equal to zero, due to Lemma 3.1 (h5). The third term can be estimated with the help of Lemma 3.1 (h6), the Cauchy–Schwarz inequality and the embedding Lp() →V ∗ as follows

|εh′(‖v‖∗)1

‖v‖∗〈g(|v|)v, v〉∗〈M(u), v〉∗| ≤ εh(‖v‖∗)‖M(u)‖∗c∗

p‖g(|v|)v‖p.

The last, sixth term is estimated by (here we use again the Cauchy–Schwarz inequality and Lp() → V ∗)

|εh(‖v‖∗)〈g(|v|)v,M(u)〉∗| ≤ εh(‖v‖∗)‖M(u)‖∗c∗p‖g(|v|)v‖p.

The fourth term is rewritten and is estimated with the help of the Cauchy–Schwarz inequality, (e2) and relative compactness of the trajectory u, then by Lemma 4.2, and finally by choosing εsmall enough, as follows

| − εh(‖v‖∗)〈M ′(u)v, v〉∗| = εh(‖v‖∗)|〈KE′′(u)v, v〉V,V ∗ | = εh(‖v‖∗)|〈KE′′(u)v, v〉|

≤ εh(‖v‖∗)C‖v‖2 ≤ εCCh

∫

g(|v|)|v|2 ≤ 1

4

∫

g(|v|)|v|2.

Altogether, we have

d

dtH(u(t), v(t)) ≤ −

∫

g(|v(t, x)|)|v(t, x)|2dx − εh(‖v‖∗)‖M(u)‖2∗

+ 2εc∗ph(‖v‖∗)‖M(u)‖∗‖g(|v|)v‖p + 1

4

∫

g(|v(t, x)|)|v(t, x)|2dx. (9)

We show that the third term on the right-hand side of (9) can be dominated by the sum of the first and second terms.

By Young’s inequality we have (δ is the convex conjugate to γ )

‖M(u)‖∗‖g(|v|)v‖p ≤ δ

(1

K‖M(u)‖∗

)+ γ

(K‖g(|v|)v‖p

).


Since ‖M(u)‖∗ is bounded, ‖M(u)‖∗K

is uniformly small if K is large enough, and by Lemma 3.2and (γ 2) we have

δ

(1

K‖M(u)‖∗

)≤ d3

K2‖M(u)‖2∗.

Moreover, we have (here the first and the third steps are the definition of γ , the second step is Jensen’s inequality (γ is convex), the fourth step is property (K) for γ and the last step uses (γ 1)):

γ (K‖g(|v|)v‖p) = γ

⎛⎝∫

Kp|g(|v|)v|p⎞⎠

≤ 1

||∫

γ(||Kp|g(|v|)v|p) = 1

||∫

γ(|| 1

p K|g(|v|)v|)

≤ 1

||C(K|| 1

p

)∫

γ (|g(|v|)v|) ≤ 1

||C(K|| 1

p

)D1

∫

g(|v|)|v|2.

Hence,

‖M(u)‖∗‖g(|v|)v‖p ≤ d3

K2‖M(u)‖2∗ + 1

||C(K|| 1

p

)D1

∫

g(|v|)|v|2.

Taking K so large that 2c∗p

d3K2 ≤ 1

2 and ε so small that

2εc∗p

D1

||C(K|| 1

p

)sups≥0

h(‖v(s)‖∗) ≤ 1

2

we obtain

2εc∗ph(‖v‖∗)‖M(u)‖∗‖g(|v|)v‖p ≤ ε

2h(‖v‖∗)‖M(u)‖2∗ + 1

2

∫

g(|v|)|v|2.

Inserting this inequality into (9) we obtain

d

dtH(u(t), v(t)) ≤ −1

4

∫

g(|v(t)|)|v(t)|2 − ε

2h(‖v(t)‖∗)‖M(u(t))‖2∗ ≤ 0.

In particular, t → H(u(t), v(t)) is nonincreasing. Since H(u(tn), v(tn)) → H(ϕ, 0) = 0 by Lemma 4.1, we have H(u(t), v(t)) ≥ 0 and the definition of E(u(t), v(t)) is correct. We can further estimate using Lemma 4.2

d

dtH(u(t), v(t)) ≤ −ch(‖v‖∗)

(‖v‖2 + ‖M(u)‖2∗

)≤ − c

2h(‖v‖∗) (‖v‖ + ‖M(u)‖∗)2 (10)

with the constant c = min 14C

h, ε2 .


Now we compute the derivative

d

dtE(u(t), v(t)) = 1

(H(u(t), v(t)))h((H(u(t), v(t))))· d

dtH(u(t), v(t)) (11)

and see that E is nonincreasing along solutions. We estimate the right-hand side. Let us assume that t > 0 is such that u = u(t) ∈ BV (ϕ, η). We can write

(H(u,v)) ≤ (1

2‖v‖2) + (|E(ϕ) − E(u)|) + (εh(‖v‖∗)‖M(u)‖∗‖v‖∗)

≤ (‖v‖2) + 1

C‖M(u)‖∗ + (εh(‖v‖∗)‖M(u)‖2∗) + (εh(‖v‖∗)‖v‖2∗)

≤ c1(‖v‖ + ‖M(u)‖∗) ,

since is nondecreasing and sublinear (first inequality), by the Kurdyka–Łojasiewicz gradient inequality, the Cauchy–Schwarz inequality, monotonicity and sublinearity of (second inequal-ity), since (s) ≤ C

√s and by boundedness of h(‖v‖∗)) (last inequality). Using Lemma 3.1

(h is nondecreasing and has property (K)) we have

(H(u,v))h((H(u, v))) ≤ c1C(c1)(‖v‖ + ‖M(u)‖∗)h(‖v‖ + ‖M(u)‖∗). (12)

Since ‖v‖ + ‖M(u)‖∗ ≥ c∗‖v‖∗ and since the function s → s

h(s)is nondecreasing (this follows

from (h6) if h is differentiable), we have, using also property (K) of h,

‖v‖ + ‖M(u)‖∗h(‖v‖ + ‖M(u)‖∗)

≥ c∗‖v‖∗h(c∗‖v‖∗)

≥ c∗‖v‖∗C(c∗)h(‖v‖∗)

. (13)

Altogether, inserting the estimates (10), (12) and (13) into the equality (11) we obtain the estimate for the derivative of E along solutions:

− d

dtE(u(t), v(t)) ≥

c2 h(‖v‖∗)(‖v‖ + ‖M(u)‖∗)2

c1C(c1)(‖v‖ + ‖M(u)‖∗)h(‖v‖ + ‖M(u)‖∗)≥ c2‖v(t)‖∗

for t satisfying u(t) ∈ BV (ϕ, η).Thus, the proof is done for h everywhere differentiable.If h is not everywhere differentiable, we need to correct two places in the above proof.

First, it is not clear that ddt

E(u(t), v(t)) exists for almost all t . But in fact, if ddt

‖v(t)‖∗ = 0, then d

dth(‖v(t)‖∗) = 0, since h has bounded difference quotients on a neighbourhood of t .

If ddt

‖v(t)‖∗ = 0, we can compute the left and right derivatives. If ddt

‖v(t)‖∗ < 0, then ddt +h(‖v(t)‖∗) = h′−(‖v(t)‖∗) d

dt(‖v(t)‖∗) and d

dt −h(‖v(t)‖∗) = h′+(‖v(t)‖∗) ddt

(‖v(t)‖∗) and

these two derivatives are equal in all points with countably many exceptions since h is con-cave. We can proceed similarly provided d

dt‖v(t)‖∗ > 0. Therefore h(‖v(t)‖∗) (hence also

E(u(t), v(t))) has a time derivative everywhere except on a countable set.Second, the equalities (8) hold only in the points t where h′(‖v(t)‖∗) exists. However, in the

points where ddt

‖v(t)‖∗ = 0, the equalities hold if we replace the terms containing h′ by zeros


(since ddt

h(‖v(t)‖∗) = 0) and these zeros can be estimated as the original terms, so the rest of the proof remains unchanged also for these points. As we have shown in the previous paragraph, this already covers all t ’s except countably many and the proof is completed. 5. Equation with general damping

In this section we replace the assumptions (G), (H) by the following ones. The function E is the same as above.

(GG) The function G : Rn × Rn → Rn is continuous and there exists τ > 0 such that(gg1) there exists C2 > 0 such that |G(w, z)| ≤ C2|z| for all z ∈ BRn(0, τ), w ∈ Rn,(gg2) there exists C3 > 0 such that C3|z| ≤ |G(w, z)| for all z ∈ Rn \ BRn(0, τ), w ∈ Rn,(gg3) if N = 2 then there exist C4 > 0, α > 0 such that |G(w, z)| ≤ C4|z|α|z| for all

z ∈ Rn \ BRn(0, τ), w ∈ Rn; if N > 2 then the inequality holds with α = 4N−2 ,

(gg4) there exists C5 > 0 such that 〈G(w, z), z〉 ≥ C5|G(w, z)||z| for all w, z ∈ Rn.(HH) For τ from condition (GG) there exists a function h : [0, +∞) → [0, +∞), which is con-

cave and nondecreasing on [0, τ ] and satisfies(hh1) |G(w, z)| ≥ h(|z|)|z| for all z ∈ BRn(0, τ), w ∈ Rn,(hh2) the function s → 1

(s)h((s))belongs to L1((0, τ)),

(hh3) the function ψ : s → sh(√


Note that (h2), (h3) remained unchanged, (g1), (g2), (g3), (h1) were naturally reformulated for a function G(w, z) corresponding to g(|z|)z. The angle condition (gg4) was added. It means that the angle between the direction of the velocity and the direction of the damping stays away from π2 and generalises the condition that g > 0. As in Section 3 we can define a global lower bound function h.

Theorem 5.1. Let E and G satisfy (E), (GG) and (HH). Let u be a strong solution to (3) such that (u(t), ut (t)) : t ≥ 0 is relatively compact in V × H and let ϕ ∈ ωV (u). Then limt→+∞ (‖u(t) − ϕ‖V + ‖ut (t)‖) = 0.

Proof. The computations in Sections 3 and 4 remain valid with g(|z|)z replaced by G(w, z), g(|z|)|z|2 replaced by 〈G(w, z), z〉 and with three further changes. First, the inequality in (h1)

has to be replaced by |G(w, z)| ≥ h(|z|)|z|, z, w ∈ Rn. Second, in Proposition 3.3, (γ 1) can be proved as follows (with the help of the angle condition (g4)).

– If |G(w, z)| < τ then we have

γ (|G(w,z)|) = c1|G(w,z)|2 ≤ c1

C5〈G(w,z), z〉 |G(w,z)|

|z| ≤ c1

C5maxC2,1〈G(w,z), z〉 .

– If |G(w, z)| ≥ τ then we have

γ (|G(w,z)|) = |G(w,z)|p − τp + c1τ2 ≤ |z| |G(w,z)| |G(w,z)|p−1

|z|≤ 1

C5maxC2τ

p−2,Cp−14 ,

1

τ〈G(w,z), z〉 .


Third, in the proof of Lemma 4.2 we replace the estimate ∫

h(|v|)|v|2 ≤ ∫

g(|v|)|v|2 by (for u ∈ V , v ∈ H ) ∫

h(|v|)|v|2 ≤ 1

C5

∫

〈G(u,v), v〉 .

6. Ordinary differential equation

We may change the setting of Sections 2, 3 and 4 in the following way. Let V = H = V ∗ =RN and all the norms and scalar products are the norm and the scalar product in RN . We take p = 1 (the only purpose of p was to make the embedding V ∗ → Lp() continuous, now the Lp-norm is replaced by the norm in RN ). The growth condition (g3) is not needed here, since it was needed only to show condition (γ 1) in case p > 1. Condition (e2) holds trivially in this finite-dimensional setting. Of course, all integrals over and the variable x have to be erased in the above sections. In this way we can obtain the following result.

Theorem 6.1. Let the functions E : RN → R, G : RN × RN → RN satisfy (e1), (gg1), (gg2),(gg4) of (GG), and (HH). Let u ∈ W 1,∞((0, +∞), RN) ∩ W

2,1loc ([0, +∞), RN) be a solution to

(4) and let ϕ ∈ ω(u). Then limt→+∞ (‖u(t) − ϕ‖ + ‖u(t)‖) = 0.

This result generalises our result [1, Theorem 4]. There it is assumed that G is estimated by multiples of a radially symmetric concave function g from below and from above, i.e. that cg(|z|)|z|2 ≤ 〈G(w, z), z〉 ≤ Cg(|z|)|z|2, and we had a condition on ∇G. Moreover, we assumed to be concave but in fact it is sublinearity what was needed in [1, Theorem 4]. In [3] the damping |u(t)|αu(t) was considered.

References

[1] T. Bárta, R. Chill, E. Fašangová, Every ordinary differential equation with a strict Lyapunov function is a gradient system, Monatsh. Math. 166 (2012) 57–72.

[2] T. Bárta, Convergence to equilibrium of relatively compact solutions to evolution equations, Electron. J. Differential Equations 2014 (81) (2014) 1–9.

[3] L. Chergui, Convergence of global and bounded solutions of a second order gradient like system with nonlinear dissipation and analytic nonlinearity, J. Dynam. Differential Equations 20 (3) (2008) 643–652.

[4] L. Chergui, Convergence of global and bounded solutions of the wave equation with nonlinear dissipation and analytic nonlinearity, J. Evol. Equ. 9 (2009) 405–418.

[5] R. Chill, A. Haraux, M.A. Jendoubi, Applications of the Łojasiewicz–Simon gradient inequality to gradient-like evolution equations, Anal. Appl. 7 (2009) 351–372.

[6] A. Cianchi, Optimal Orlicz–Sobolev embeddings, Rev. Mat. Iberoam. 20 (2004) 427–474.[7] A. Haraux, Systèmes dynamiques dissipatifs et applications, Masson, 1991.[8] M.A. Jendoubi, P. Polácik, Non-stabilizing solutions of semilinear hyperbolic and elliptic equations with damping,

Proc. Roy. Soc. Edinburgh Sect. A 133 (5) (2003) 1137–1153.


4.4 T. Barta, Rate of convergence and Lo-

jasiewicz type estimates, J. Dyn. Diff.

Equat., online first, 16 pages.


J Dyn Diff EquatDOI 10.1007/s10884-016-9549-z

Rate of Convergence to Equilibriumand Łojasiewicz-Type Estimates

Tomáš Bárta1

Received: 24 November 2015 / Revised: 10 July 2016© Springer Science+Business Media New York 2016

Abstract A well known result states that the Łojasiewicz gradient inequality implies someestimates of the rate of convergence to equilibrium for solutions of gradient systems. Wegeneralize this result to gradient-like systems satisfying certain angle condition andKurdyka–Łojasiewicz inequality and to even more general situation. We apply the results to a broadclass of second order equations with damping.

Keywords Gradient-like system · Kurdyka–Łojasiewicz inequality · Rate of convergenceto equilibrium · Second order equation with damping

1 Introduction

In this paper,we study rate of convergence to equilibriumof solutions to gradient-like ordinarydifferential equations based on some generalizations of the Łojasiewicz gradient inequality.

General assumptions. Throughout this paper, we assume that (M, g) is a smooth Riemannianmanifold, ‖ · ‖ is the norm on the tangent bundle T M induced by the Riemannian metric gand d(·, ·) is the distance on M induced by g. We assume that F : M → T M is a continuousvector field. We consider an ordinary differential equation

u + F(u) = 0, (1)

its bounded solution u ∈ W 1,1loc ([0,+∞), M), and a point ϕ in the omega-limit set of u,

ω(u) = ϕ ∈ M : ∃ tn +∞, u(tn) → ϕ.

B Tomáš Bá[email protected]

1 Department of Mathematical Analysis, Faculty of Mathematics and Physics,Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic

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J Dyn Diff Equat

Moreover, we assume that a continuously differentiable function E : M → R is a strictLyapunov function to (1), i.e.,

〈∇E(u), F(u)〉 > 0 whenever u ∈ M, F(u) = 0. (2)

There are many results saying that under additional conditions on E we have u(t) → ϕ

as t → +∞. The main goal of this paper is to find the rate of convergence, i.e. a function Ras small as possible such that

d(u(t), ϕ) ≤ R(t) for all t ≥ 0.

Let us start with a special case when F = ∇E , then (1) becomes a gradient system

u + ∇E(u) = 0.

The classical result of Łojasiewicz (see [15]) states that u(t) converges to ϕ if the Łojasiewiczgradient inequality

|E(u) − E(ϕ)|1−θ ≤ C‖∇E(u)‖ for all u ∈ N (ϕ) (LI)

holds with some C > 0 and θ ∈ (0, 12 ] (by N (ϕ) we denote some neigborhood of ϕ). In

fact, Łojasiewicz proved that any analytic function E satisfies (LI) on a neighborhood of anystationary point of ∇E . Speed of convergence to ϕ was estimated by Haraux and Jendoubi in[11]: there exist a, K > 0 such that for all t ≥ 0 it holds that

d(u(t), ϕ) ≤ K e−at if θ = 1

2

and

d(u(t), ϕ) ≤ K (1 + t)−θ

1−2θ if θ <1

2.

For some examples of nonanalytic functions satisfying (LI) see Chill [7].The result by Łojasiewicz was later generalized in several ways. First, the inequality (LI)

was generalized to the so called Kurdyka–Łojasiewicz inequality (see [13])

(|E(u) − E(ϕ)|) ≤ ‖∇E(u)‖ for all u ∈ N (ϕ) (KLI)

with a function positive on (0,+∞) and satisfying (0) = 0 and 1

∈ L1loc([0, 1)) (if

we take (s) = s1−θ , then (KLI) becomes (LI)). In this case, we have again convergenceu(t) → ϕ as t → +∞ with convergence rate given by

d(u(t), ϕ) ≤ K(ψ−1(t − t0)) for some K , t0 > 0 and all t > t0, (3)

with

(t) =∫ t

0

1

(s)ds and ψ(t) = −

∫1

2(t)dt.

The rate of convergence was proved by Chill and Fiorenza in [8]. The convergence resultis due to Kurdyka [13] who also proved that (KLI) holds for any function E definable in ano-minimal structure. See also Bolte et al. [5].

Second, it was generalized to gradient-like systems, i.e. ordinary differential equationswith a strict Lyapunov function E (not neccessarily satisfying F = ∇E). In this case, it is notsufficient to assume that E satisfies the Łojasiewicz or Kurdyka–Łojasiewicz inequality toobtain convergence u(t) → ϕ. We need to add so called angle condition (see Absil, Mahony

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J Dyn Diff Equat

and Andrews [1, Theorem 2.2], Chill, Haraux and Jendoubi [9, Proposition 5(a), Theorem4], Lageman [14, Definition 1.1])

〈∇E(u), F(u)〉 ≥ α‖∇E(u)‖ ‖F(u)‖ for some α > 0 and all u ∈ N (ϕ). (AC)

If we moreover assume the comparability condition

c1‖F(u)‖ ≤ ‖∇E(u)‖ ≤ c2‖F(u)‖ for some c1, c2 > 0 and all u ∈ N (ϕ), (C)

then we can obtain the same rate of convergence as for gradient systems. This decay estimateis due toBégout, Bolte and Jendoubi [4], they also applied the result to a second order problemwith linear damping. See also Corollary 2 below for the result in the manifold setting. It isnot surprising that we need condition (C) to estimate the speed of convergence, since theorbits depend on the direction of F only, the size of F determines how quickly the solutionmoves along the orbit. Let us mention that (AC) and (C) together are equivalent to the angleand comparability condition (see [2])

〈∇E(u), F(u)〉 ≥ c(‖∇E(u)‖2 + ‖F(u)‖2) for all u ∈ N (ϕ). (AC+C)

Further, in [2] we introduced a generalized Łojasiewicz condition

(|E(u) − E(ϕ)|) ≤ 1

‖F(u)‖ 〈∇E(u), F(u)〉 for all u ∈ N (ϕ), (GLI)

that generalizes (AC + C) and (KLI), i.e., (AC + C) and (KLI) imply (GLI). We have shownthat (GLI) is sufficient to obtain convergence u(t) → ϕ and that (GLI) is satisfied by alarge class of second order problems with (weak) damping. In the present paper, we give anestimate of the convergence rate for this case (see Theorem 1 below). Then we apply thisresult to a second order equation with a general (weak) damping function (Theorems 5, 6)and generalize the result by Chergui [6, Theorem 1.3].

Finally, we present better estimates of the convergence rate in some cases. In fact, all theestimates mentioned above use the inequlality

d(u(t), ϕ) ≤∫ +∞

t‖u(s)‖ds,

i.e., estimate the distance to the equilibrium by the length of the remaining trajectory. Thisestimate is far from being optimal if the solution u has a shape of a spiral, which is exactlythe case if we consider a second order equation with a weak damping (smaller than linear).We show that it can be better to estimate d(u(t), ϕ) directly by a function of E(u(t)).

Section 2 is devoted to the abstract results, while in Sect. 3 we apply these results to adamped second order equation and Section 4 contains some technical Lemmas.

2 Main Results

We formulate the main result of this paper (keeping in mind the general assumptions intro-duced in the previous section).

Theorem 1 Let E and F satisfy (GLI) with a function : [0, 1) → R+ such that 1

∈L1

loc([0, 1)) and (s) > 0 for s > 0. Then u has finite length in (M, g) and, in particular,lim

t→+∞ u(t) = ϕ in (M, g). Moreover, if α : (0, 1) → (0,+∞) is nondecreasing and satisfies

α(E(u(t))) ≤ ‖F(u(t))‖ for all t large enough, (4)

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then there exist t0 > 0 such that

d(u(t), ϕ) ≤ (ψ−1(t − t0)) for all t > t0,

where

(t) :=∫ t

0

1

(s)ds and ψ(t) :=

∫ 1/2

t

1

(s)α(s)ds.

Let us remark that an example of such function α (and, in fact, the best one) is

α(s) := min‖F(u(t))‖ : E(u(t)) − E(ϕ) ≥ s.This function is well defined. Since u(t) : t ∈ R+ ∪ ϕ is compact and so the level setu(t) : E(u(t)) ≥ s + E(ϕ) is also compact. Therefore ‖F(u(t))‖ attains its minimum onthis set. Positivity and monotonicity of α and (4) follow immediately.

Proof We have proved convergence in [2], Theorem 5. It remains to show the moreover part.Without loss of generality wemay assume E(ϕ) = 0. Since α is nondecreasing, the Lebesgueintegral in the definition ofψ exists andψ is decreasing, therefore invertible. We show belowthat lims→0+ ψ(s) = +∞, which implies that ψ−1 is defined on a neighborhood of +∞.

Let ε > 0 be small enough. For all t large enough we have E(u(t)) ∈ (0, ε) and for almostall such t’s it holds that (by definition of ψ , (GLI) and (4))

d

dtψ(E(u(t))) = − 1

(E(u(t)))α(E(u(t)))〈∇E(u(t)), u(t)〉

= 1

(E(u(t)))α(E(u(t)))〈∇E(u(t)), F(u(t))〉

≥ 1

(E(u(t)))α(E(u(t)))(E(u(t)))‖F(u(t))‖

≥ 1

(E(u(t)))α(E(u(t)))(E(u(t)))α(E(u(t)))

= 1.

Fix t0 large enough (such that ψ(E(u(t0))) > 0) and integrate this inequality from t0 tot > t0

ψ(E(u(t))) ≥ (t − t0) + ψ(E(u(t0))) ≥ t − t0.

From this inequality it follows that lims→0+ ψ(s) = +∞. Since ψ is decreasing, we haveE(u(t)) ≤ ψ−1(t − t0). Further, by (GLI), u = −F(u) and by definition of , we have

d(u(t), ϕ) ≤∫ +∞

t‖u(s)‖

≤∫ +∞

t− 1

(E(u(s)))〈∇E(u(s)), u(s)〉

= − lims→+∞ (E(u(s))) + (E(u(t)))

= (E(u(t)))

≤ (ψ−1(t − t0)).

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Corollary 2 Let E and F satisfy (AC), (C) and (KLI) with a function : [0, 1) → R+ suchthat 1

∈ L1

loc([0, 1)) and (s) > 0 for s > 0. Then u has finite length in (M, g) and, inparticular, lim

t→+∞ u(t) = ϕ in (M, g). Moreover, there exists t0 > 0 such that

d(u(t), ϕ) ≤ 1(ψ−11 (t − t0)) for all t > t0,

where

1(t) := c1

∫ t

0

1

(s)ds and ψ1(t) := c2

∫ 1/2

t

1

2(s)ds

for appropriate positive constants c1, c2.

Proof Conditions (AC) and (KLI) imply

1

‖F(u)‖〈∇E(u), F(u)〉 ≥ α‖∇E(u)‖ ≥ α(|E(v) − E(ϕ)|),

so (GLI) holds with replaced by := α. Since (|E(u) − E(ϕ)|) ≤ ‖∇E(u)‖ ≤c2‖F(u)‖ by (KLI) and (C), we can take α(s) = 1

c2(s) and apply Theorem 1.

The above results estimate the distance from the equilibrium by the length of the remainingtrajectory, i.e.

‖u(t) − ϕ‖ =∫ ∞

tu(s)ds ≤

∫ ∞

t‖u(s)‖ds. (5)

This estimate seems to be quite bad if the trajectory looks like a spiral; then the remainingtrajectory can be much longer than the distance to the equilibrium.

Let us assume that (M, g) be an open subset of Rn with the Euclidean metric. We denotethe Euclidean norm by | · |. It is easy to show that if

〈F(u), u〉 ≥ α|F(u)| |u| (6)

for some fixed α > 0, then the estimate (5) is optimal. In fact,

− d

dt|u(t)| = −

⟨u(t)

|u(t)| , u(t)

⟩=

⟨u(t)

|u(t)| , F(u(t))

⟩≥ α|F(u(t))| = α|u(t)|

and after integration from T to +∞ we obtain

|u(T )| ≥ α

∫ +∞

T|u(t)|dt.

The estimate (6) means that E(u) = |u|2 is a Lyapunov function and ∇E and F satisfy (AC).So, the estimate (5) is optimal even for some spirals (logarithmic spiral). However, in manycases the estimate (5) is not optimal and the following corollary yields a better result.

Corollary 3 Let the assumptions of Theorem 1 hold and let γ : (0, 1) → (0,+∞) be anondecreasing function satisfying γ (E(u) − E(ϕ)) ≥ |u − ϕ| for all u in a neigborhood ofϕ. Then there exist t0 > 0 such that

|u(t) − ϕ| ≤ γ (ψ−1(t − t0)) for all t > t0.

Proof As in the proof of Theorem 1 (assuming E(ϕ) = 0) we obtain E(u(t)) ≤ ψ−1(t − t0).Further,

|u(t) − ϕ| ≤ γ (E(u(t))) ≤ γ (ψ−1(t − t0))

since γ is nondecreasing.

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Let us mention that an example (and, in fact, the best one) of a function γ from Corollary3 is

γ (s) := sup|u(t) − ϕ| : E(u(t)) − E(ϕ) ≤ s.Let us recall the Example 7 from [2]:

Example 4 Let M ⊆ R2 be the open unit disk, equipped with the Euclidean metric. Let

α ≥ 0, and let F(u) = F(u1, u2) = (|u|αu1 − u2, u1 + |u|αu2) and E(u) = 12 (u

21 + u2

2).Then

〈∇E(u), F(u)〉 = |u|2+α, |F(u)| = |u| ·√1 + |u|2α and |∇E(u)| = |u|.

The function E satisfies the Łojasiewicz inequality (LI) near the origin for θ = 12 . But the

angle condition (AC) does not hold on any neighbourhood of the critical point (0, 0), soCorollary 2 does not apply (unless α = 0). On the other side, we have

1

|F(u)| 〈E′(u), F(u)〉 = |u|1+α√

1 + |u|2α ≥ 1√2|u|2(1−θ) ≥ 1√

2E(u)1−θ

provided 0 < θ ≤ 1−α2 . Hence, if 0 ≤ α < 1, then E satisfies (GLI) with (s) = 1√

2s1−θ ,

θ = 1−α2 .

We can apply Theorem 1 with α(s) = 2√

s since for small s we have

inf|u(t)|√1 + |u(t)|2α : 1

2|u(t)|2 ≥ s = √

2s√1 + (2s)α ≤ 2

√s.

Hence, ψ ′(s) = csθ−3/2 and ψ(s) = csθ−1/2 and ψ−1(s) = cs1

θ−1/2 and = csθ (withvarious constants c). Then,

|u(t)| ≤ (ψ−1(t − t0)) ≤ C(t − t0)θ

θ−1/2 = C(t − t0)1− 1

α .

If (AC+C) condition were satisfied, the decay of u would be exponential due to theŁojasiewicz exponent equal to 1

2 . Since the (AC+C) condition is not satisfied, the decay isonly polynomial.

However, the above estimate is not optimal and we can get a better one from Corollary 2.In fact, taking

γ (s) = sup

√x2 + y2 : 1

2(x2 + y2) ≤ s

= √

2s

we obtain

|u(t)| ≤√2C(t − t0)

1θ−1/2 = C(t − t0)

− 1α .

This is a better result since− 1α

< 1− 1α. Moreover, transformation to polar coordinates show

that this result is optimal. In fact, we obtain r ′ = −rα+1, which yields r(t) = c(t − t0)−1/α .

3 Second Order Equation with Damping

In this section we apply the previous results to a damped second order equation

u + G(u, u) + ∇E(u) = 0. (7)

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Of course, Eq. (7) can be reduced to a first order system(uv

)+ F(u, v) = 0 with F(u, v) =

( −v

G(u, v) + ∇E(u)

).

If there exists c > 0 such that〈G(u, u), u〉 ≥ c|u|2 (8)

(in particular, when G(u, u) = γ u) then this system is gradient-like and satisfies the anglecondition (AC). This case was studied by Haraux and Jendoubi in [10] and by Bégout,Bolte and Jendoubi [4]. The case when (8) is not valid was studied by Chergui in [6] forG(u, u) = |u|α u, α ∈ (0, 1) and E satisfying (LI) and later by Ben Hassen and Haraux in[12] for more general G satisfying

C |u|α+2 ≥ 〈G(u, u), u〉 ≥ c|u|α+2. (9)

Both papers contain decay estimates

|u(t) − ϕ| + |u(t)| ≤ C(1 + t)−θ−α(1−θ)

1−2θ+α(1−θ) . (10)

In [2, Theorem 4] and [3, Theorem 6.1] we proved convergence to equilibrium under weakerassumptions onG and for E satisfying (KLI). Now,we complement these convergence resultswith decay estimates.

Let us formulate our assumptions on E and G. We start with a first set of assumptionsand apply the approach from [2]. Then we introduce more general assumptions and use theresult from [3].

(E) Let E ∈ C2(Rn,R) satisfy (KLI) with a function : [0, 1) → [0,+∞) which isnondecreasing, sublinear ((s + t) ≤ (s)+(t)), and it holds that 1

∈ L1

loc([0, 1))and 0 < (s) ≤ c

√s for all s ∈ (0, 1) and some c > 0.

(G) The function G : Rn × R

n → Rn is continuous and there exists a function h :

[0,+∞) → [0,+∞), which is concave and nondecreasing and it holds that

(g1) there exists C2 > 0 such that |G(w, z)| ≤ C2|z|h(|z|) for all z, w ∈ Rn ,

(g2) there exists C3 > 0 such that |G(w, z)| ≥ C3|z|h(|z|) for all z, w ∈ Rn ,

(g3) there exists C4 > 0 such that 〈G(w, z), z〉 ≥ C4|G(w, z)||z| for all w, z ∈ Rn .

(g4) there exists C5 > 0 such that |∇G(w, z)| ≤ C5h(|z|) for all w, z ∈ Rn .

(g5) the function s → 1(s)h((s)) belongs to L1((0, τ )),

Let us comment on these assumptions. First, function (s) = s1−θ , θ ∈ (0, 12 ] satisfies

the assumptions in (E), in this case (KLI) reduces to (LI). Concerning assumptions on G, letus first consider G(w, z) = g(|z|)z. Then conditions (g1),(g2) say that the damping functiong is between two multiples of a concave function h (g can even oscilate between them butnot much, due to (g4)). Condition (g5) is a connection between E and G. If (s) = s1−θ

and g(z) = |z|α , then (g5) reduces to α < θ1−θ

, which is the condition from [6]. In this case,the following theorem gives the same rate of convergence as [6, Theorem 1.3].

If G(w, z) = g(|z|)z, then the damping force acts in the direction opposite to velocity.For general G, condition (g3) is an angle condition which says that the angle between thedamping force and minus velocity is less than π

2 uniformly.

Theorem 5 Let E and G satisfy (E) and (G). Let u ∈ W 1,∞((0,+∞),Rn)∩W 2,1loc ([0,+∞),

Rn) be a solution to (7) and ϕ ∈ ω(u). Then there exists t0 > 0 such that

|u(t)| + |u(t) − ϕ| +∫ +∞

t|u(s)|ds ≤ (ψ−1(t − t0)), (11)

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holds for all t > t0, some C1, C2 > 0 and

(t) = C1

∫ t

0

1

(s)h((s))ds and ψ(t) = C2

∫ 12

t

1

2(s)h((s))ds. (12)

Proof By [2, Theorem 4], the left-hand side of (11) tends to zero as t → +∞. Let us assumewithout loss of generality that ϕ = 0 and E(ϕ) = 0 and denote v(t) := u(t). In the proof of[2, Theorem 4] we have shown that (for ε > 0 small enough)

E(u, v) = 1

2|v|2 + E(u) + ε〈G(u,∇E(u)), v〉

is a strict Lyapunov function and satisfies ([2, last inequality of the proof])

(E(u, v)) ≤ 1

|F(u, v)| 〈∇E(u, v), F(u, v)〉

with (s) := (s)h((s)). Further, we have shown ([2, p. 71, first inequality])

(E(u, v)) ≤ C(‖v‖ + ‖∇E(u)‖). (13)

From the definition of F we have immediately

|F(u, v)| = (|v|2 + |G(u, v) + ∇E(u)|2)1/2 ≥ 1

2(|v| + |G(u, v) + ∇E(u)|).

Now, we show that|F(u, v)| ≥ c(|v| + |∇E(u)|). (14)

By the assumptions on G we have |G(u, v)| ≤ C |v|h(|v|). Now, we distingiush two cases:

1. If (u, v) is such that C |v|h(|v|) ≤ (1 − α)|∇E(u)| for some α ∈ (0, 1), then

|F(u, v)| ≥ 1

2(|v| + |∇E(u)| − (1 − α)|∇E(u)|) ≥ α

2(|v| + |∇E(u)|).

2. If (u, v) is such that C |v|h(|v|) ≥ (1 − α)|∇E(u)|, then for |v| small enough (we areinterested for small |v| only) we have h(|v|) ≤ c. Then

|v| + |∇E(u)| ≤ |v| + cC

1 − α|v| = C · 1

2|v| ≤ C |F(u, v)|

We now have (14) and using (13) we obtain

|F(u, v)| ≥ c

C(E(u, v)).

It remains to apply Theorem 1 with α(s) = cC (s) and the proof is complete (with C1 = 1

and C2 = Cc ).

Now, let us further relax the assumptions on the damping function G. In particular, ifG(w, z) = g(|z|)z, then the following assumption say that g is still bigger than a concavefunction h on (0, τ ), but not necessarily less than a multiple of h. Since we do not have anycondition on ∇G, function g can oscilate arbitrarily between h and a constant function on(0, τ ).

(GG) The function G : Rn × Rn → R

n is continuous and there exists τ > 0 such that

(gg1) there exists C2 > 0 such that |G(w, z)| ≤ C2|z| for all |z| < τ , w ∈ Rn ,

(gg2) there exists C3 > 0 such that C3|z| ≤ |G(w, z)| for all z ≥ τ , w ∈ Rn ,

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(gg3) there exists C5 > 0 such that 〈G(w, z), z〉 ≥ C5|G(w, z)||z| for all w, z ∈R

n .

(HH) For τ from condition (G) there exists a function h : [0,+∞) → [0,+∞), which isconcave and nondecreasing on [0, τ ] and satisfies

(hh1) |G(w, z)| ≥ h(|z|)|z| for all |z| < τ , w ∈ Rn ,

(hh2) the function s → 1(s)h((s)) belongs to L1((0, τ )),

(hh3) the function ψ : s → sh(√


Theorem 6 Theorem 5 remains valid if E and G satisfy weaker assumptions (E), (GG),(HH).

Proof By [3, Theorem 6.1], the left-hand side of (11) tends to zero as t → +∞. Let usassume without loss of generality that ϕ = 0 and E(ϕ) = 0. Denote v(t) := u(t). In theproof of [3, Theorem 6.1] we have defined for ε > 0 small enough

H(u, v) := 1

2|v|2 + E(u) + εh(|v|) 〈∇E(u), v〉

In fact, h used in [3] is equal to h on a neighborhood of zero and E in [3] has the oppositesign. In the following we write u, v instead of u(t), v(t). In [3] we have shown that (see [3,inequality (10)])

− d

dtH(u, v) ≥ ch(|v|)(|v| + |∇E(u)|)2

and (see [3, inequality (12)])

(H(u, v))h((H(u, v))) ≤ C(|v| + |∇E(u)|)h(|v| + |∇E(u)|),and (see [3, the inequality below (13)])

− 1

(H(u, v))h((H(u, v)))· d

dtH(u, v) ≥ c|v|.

Since

− d

dtH(u, v) = −〈∇ H(u, v), (u, v)〉 = 〈∇ H(u, v), F(u, v)〉 ,

we have 〈∇ H(u, v), F(u, v)〉c · (H(u, v)) · h((H(u, v)))

≥ |v|, (15)

which is almost (GLI) with E = H , (s) = c(s)h((s)); the only difference is that thereis |v| instead of |F(u, v)| on the right-hand side of (15). It holds that

(H(u, v)) ≤ d(|v| + |∇E(u)|) ≤ d|F(u, v)|, (16)

the first inequality is proven in [3, the inequality before (12)], the second inequality followsby the same arguments as in the proof of Theorem 5. Therefore, we can continue similarlyto the proof of Theorem 1

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d

dtψ(H(u, v)) = −ψ ′(H(u, v)) 〈∇ H(u, v), F(u, v)〉

≥ cC2

(H(u, v))|v|

≥ cC2

d· |v||v| + |∇E(u)| ,

where the first inequality follows from (15) and the definition ofψ and the second inequalityfollows from (16). In Lemma 8 (proven in the appendix) we show that for an appropriate L ,t0 > 0 we have ∫ t

t0

|v||v| + |∇E(u)| ≥ L(t − t0) for all t > t0.

Then we can complete the proof similarly to Theorem 1. We have (taking t0 large enoughand C2 := d(cL)−1)

ψ(H(u(t), v(t))) ≥ cC2Ld−1(t − t0) + ψ(H(u(t0), v(t0))) ≥ t − t0,

so H(u(t), v(t)) ≤ ψ−1(t − t0). Finally, we complete the proof by the estimates

|u(t)| + |v(t)| ≤∫ +∞

t|u(s)| + |v(s)|

≤ √2

∫ +∞

t|F(u(s), v(s))|

≤ √2K

∫ +∞

t|v(s)|

≤ √2K

∫ +∞

t

〈∇ H(u(s), v(s)), F(u(s), v(s))〉c(H(u(s), v(s)))h((H(u(s), v(s)))

= (H(u(t), v(t))) − lims→+∞ (H(u(s), v(s)))

≤ (ψ−1(t − t0)).

Here the first and second inequality are obvious, we applied Lemma 9 in the third inequality(Lemma 9 is proven in the appendix), in the forth inequality we used (15) and in the nextequality we set C1 := K

c

√2 in the definition of .

Now, we improve the above estimates by applying Corollary 3. In the simplest case

u + |u|αu + u = 0,

where u is a scalar function, the solutions are spirals. So, it is reasonable to assume that thelength of the remaining trajectory is much bigger than the distance from the equilibrium.So, we improve the estimates from Theorem 6 in case that the energy function E is nice. Inparticular, in addition to the Kurdyka–Łojasiewicz inequality (KLI) we assume that also theopposite is true. Moreover, we assume that |u| can be estimated by an appropriate functionof E .

(E1) Let E and be the functions from (E) and ϕ be the point from Theorem 5. Then thereexists c > 0 such that

(E(u) − E(ϕ)) ≥ c|∇E(u)| for all u ∈ N (ϕ).

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Theorem 7 Let the assumptions of Theorem 5 hold with (G) replaced by (GG), (HH) andlet (E1) hold. Moreover, let γ be a nondecreasing function satisfying

γ (E(u) − E(ϕ)) ≥ |u − ϕ| for all u ∈ N (ϕ). (17)

Then

|u(t)| ≤ C√

ψ−1(t − t0) and |u(t) − ϕ| ≤ Cγ (ψ−1(t − t0)), (18)

holds for all t > t0 and some C > 0, ψ defined as in (12).

Before we prove this theorem, let us mention that if p ≥ 2, then E(u) = ∑ni=1 |ui |p is a

prototype of an energy satisfying the Łojasiewicz estimate with θ = 1p (i.e., (s) = s

p−1p ).

Moreover, this function E satisfies (E1) and function γ (t) = t(t) = t

1p satisfies (17). If we

take a damping function G(u, v) = |v|αv for α < 1p−1 as in [6] or any larger admissible

function, we obtain

|u(t)| ≤ C(1 + t)−12

1−2θ+α(1−θ) and |u(t)| ≤ C(1 + t)−θ

1−2θ+α(1−θ) ,

which improves the convergence rate from [6, Theorem 1.3]. In particular, if α = 0 (lineardamping), then this result is equal to the one in [6, Theorem 1.3] and the weaker is thedamping, the bigger is the difference between the two results.

Proof As in the proof of Theorem 6 we show H(u(t), v(t)) ≤ ψ−1(t − t0). Now, we showthat (assuming WLOG ϕ = 0, E(ϕ) = 0)

H(u, v) ≥ c(|v|2 + E(u)). (19)

By definition of H (see proof of Theorem 6) we have

H(u, v) ≥ 1

2|v|2 + E(u) − εh(|v|)|∇E(u)||v|.

Further,

|∇E(u)||v| ≤ C(E(u))|v| ≤ C√

E(u)|v| ≤ C

2(|v|2 + E(u)).

Since h is bounded on a neighborhood of 0, by taking ε > 0 small enough we obtain (19).Now, we have

|v(t)| ≤ 1√c

√H(u(t), v(t)) ≤ 1√

c

√ψ−1(t − t0),

which is the first estimate in (18). Further, by monotonicity of γ we have

|u| ≤ γ (E(u)) ≤ γ (c−1H(u, v)) ≤ γ (c−1ψ−1(t − t0)),

which is the second estimate in (18), when we change the constant C2 in the definition of ψ .

Appendix

In this section we prove two technical Lemmas.

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Lemma 8 There exist L, t0 > 0 such that∫ t

t0

|v(s)||v(s)| + |∇E(u(s))|ds ≥ L(t − t0) for all t > t0.

Proof We will show that if p(t) := |v(t)||∇E(u(t))| is small on an interval I , then we can find a

comparably long interval immediately before I , where p(t) is large. In particular, assumethat for some tp > t0 it holds that |v(tp)| < κ|∇E(u(tp))|, where κ is a small constant,which will be specified later. Define

t1 := supt < tp : |v(t)| ≥ κ|∇E(u(t))|,t2 := sup

t < t1 : |v(t)| ≥ 3

2κ|∇E(u(t))| or |v(t)| ≤ 1

2κ|∇E(u(t))|

.

We show below that tp − t1 < 5(t1 − t2). If we start in t0 where |v(t0)| > 32κ|∇E(u(t0))|,

then for any tp is the corresponding t2 larger than t0. Therefore, for any t > t0 we have|v(s)| ≥ 1

2κ|∇E(u(s))| for s ∈ Mt ⊂ (t0, t) and measure of Mt is at least 16 (t − t0).

Therefore∫ t

t0

|v(s)||v(s)| + |∇E(u(s))|ds ≥

∫Mt

|v(s)||v(s)| + |∇E(u(s))|ds

≥∫

Mt

12κ|∇E(u(s))|

12κ|∇E(u(s))| + |∇E(u(s))|ds

≥ 1

6(t − t0)

κ

κ + 2,

whatwewanted to prove (in the second inequalitywe used the fact that x → xx+a is increasing

for x ≥ 0 if a > 0).So, it remains to show tp − t1 ≤ 5(t1 − t2). The idea is that in the points where v(t) is

almost zero and ∇E(u(t)) is large in comparison to v(t) it holds that

v(t) = −∇E(u(t)) + O(|v|) andd

dt∇E(u(t)) = ∇2E(u(t))v(t) = O(|v|)

(the first equality follows from the differential equation (7) and |G(u, v)| ≤ c|v|). It meansthat ∇E(u) changes very slowly and the change of v is relatively fast and (almost) constantconcerning size and also direction.

Let tp be such that |v(tp)| < κ|∇E(u(tp))| and consider the scalar product of the equation(7) with −∇E(u(tp))

⟨v(t),−∇E(u(tp))

⟩ = ⟨G(u(t), v(t)),−∇E(u(tp))

⟩ + ⟨−∇E(u(t)),−∇E(u(tp))⟩. (20)

First, let us denote K = |∇E(u(tp))| and estimate (for t ∈ (t1, tp))

∣∣∣∣ d

dt|∇E(u(t))|

∣∣∣∣ ≤ |∇2E(u(t))||v(t)| ≤ Cκ|∇E(u(t))|.

Solving this differential inequality we obtain for t ∈ (t1, tp)

e−Cκ(tp−t)K ≤ |∇E(u(t))| ≤ eCκ(tp−t)K .

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To estimate the right-hand side of (20) we employ

| ⟨G(u(t), v(t)),−∇E(u(tp))⟩ | ≤ |G(u(t), v(t))| · |∇E(u(tp))|

≤ c|v(t)||∇E(u(tp))|≤ cκK 2e2Cκ(tp−t)

and⟨∇E(u(t)),∇E(u(tp))

⟩ = |∇E(u(tp))|2 − ⟨∇E(u(tp)) − ∇E(u(t)),∇E(u(tp))⟩

≥ K 2 − K |∇2E(u)(ξ)||v(ξ)||tp − t |≥ K 2 − K Cκ(tp − t)K eCκ(tp−t)

= K 2(1 − κC(tp − t)eCκ(tp−t)).

If t ∈ (t1, tp), t > tp − 1 and κ is small enough, then right-hand side of (20) is larger than

K 2(1 − κ(C(tp − t)eCκ(tp−t) + ce2Cκ(tp−t))) ≥ 99

100K 2.

Integrating (20) from t to tp , we obtain

99

100K 2(tp − t) ≤ ⟨

v(tp) − v(t),−∇E(u(tp))⟩

≤ K |v(t) − v(tp)|≤ K (|v(t)| + |v(tp)|)≤ K (κK eCκ(tp−t) + κK )

≤ κK 2(1 + eCκ(tp−t))

≤ 201

100κK 2

if κ is small enough. So,

tp − t ≤ 201

99κ.

It means that taking κ small enough we have tp − t << 1 and the restriction t > tp − 1 isredundant.

Now, we need to do similar estimates on (t2, t1). Let us denote K1 := |∇E(u(t1))|.Similarly as above we obtain

e− 32 Cκ(t1−t)K1 ≤ |∇E(u(t))| ≤ e

32 Cκ(t1−t)K1

Further, we have

|v(t)| ≤ |G(u(t), v(t))| + |∇E(u(t))|≤ c|v(t)| + |∇E(u(t))|≤ c

3

2κ|∇E(u(t))| + |∇E(u(t))|

≤ (1 + 3

2cκ)e

32 Cκ(t1−t)K1

≤ 101

100K1,

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J Dyn Diff Equat

provided t > t1 − 1 and κ is small enough. Integrating this inequality we obtain

|v(t) − v(t1)| ≤∫ t1

t|v(s)| ≤ 101

100K1(t1 − t) (21)

Let us first assume that t2 > t1 − 1. Since |v(t2)| = 12κ|∇E(t2)| or |v(t2)| = 3

2κ|∇E(t2)|,we have (for κ small enough)

|v(t2) − v(t1)| ≥ |v(t1)| − |v(t2)|= κ|∇E(u(t1))| − 1

2κ|∇E(u(t2))|

≥ κK1 − 1

2κe

32 κc(t1−t2)K1

≥ 49

100κK1

or

|v(t2) − v(t1)| ≥ |v(t2)| − |v(t1)|= 3

2κ|∇E(u(t2))| − κ|∇E(u(t1))|

≥ 3

2κK1e− 3

2 κc(t1−t2) − κK1

≥ 49

100κK1

and together with (21) we have

t1 − t2 ≥ 49

101κ.

If t2 ≤ t1 −1, then t1 − t2 ≥ 1 ≥ 49101κ if κ is small enough. Together with the upper estimate

of tp − t1 we have

t1 − t2 ≥ 49

101κ ≥ 49

101· 99

201(tp − t1) >

1

5(tp − t1)

and the proof is complete. Lemma 9 There exists K , t0 > 0 such that for all t > t0 it holds that

∫ +∞

t|F(u(s), v(s))| ≤ K

∫ +∞

t|v(s)|.

Proof Since |F(u, v)| ≤ C(|v| + |∇E(u)|), it remains to estimate∫ +∞

t |∇E(u(s))|. Letκ > 0 be small enough. The interval (t,+∞) is covered by the union of

M1 := s > t : |v(s)| ≤ κ|∇E(u(s))|,M2 :=

s > t : |v(s)| ≥ 1

2κ|∇E(u(s))|

.

Then ∫M2

|∇E(u(s))|ds ≤ 2

κ

∫M2

|v(s)|ds.

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If tp ∈ M1, then we can find

t1 := inft > tp : |v(s)| ≥ κ|∇E(u(s))|t2 := inf

t > t1 : |v(s)| ≥ 3

2κ|∇E(u(s))| or |v(s)| ≤ 1

2κ|∇E(u(s))|

and we can proof that t2 − t1 ≥ 49101κ > 1

3κ and t1 − tp ≤ 20199 κ < 3κ similarly to the proof

of Lemma 8 (here we have t2 > t1 > tp unlike in Lemma 8, where we had t2 < t1 < tp , butthe situation is symmetric). Denote K1 := |∇E(u(t1))|. Similarly to the proof of Lemma 8we can prove on (tp, t1) the inequality

|∇E(u(t))| ≤ K1eCκ(t1−t)

and since the length of the interval (tp, t1) is less than 3κ we obtain

|∇E(u(t))| ≤ 2K1

if κ is small enough. On the other hand, on (t1, t1 + 13κ) ⊂ (t1, t2) it holds that

|∇E(u(t))| ≥ K1e− 32 Cκ(t−t1) ≥ 1

2K1

if κ is small enough. It follows that∫ t1

tp

|∇E(u(s))| ≤ (t1 − tp)2K1

≤ 12κ · 12

K1

≤ 24∫ t1+ 1

3 κ

t1

1

2K1

≤ 24∫ t1+ 1

3 κ

t1|∇E(u(s))|

The set M1 is (at most countable) union of intervals (tpn, t1n) which are followed bycorresponding intervals (t1n, t1n + 1

3κ) ⊂ M2. Therefore, we have∫M1

|∇E(u(s))|ds ≤ 24∫

M2

|∇E(u(s))|ds ≤ 48

κ

∫M2

|v(s)|ds

and the proof is complete.

References

1. Absil, P.A., Mahony, R., Andrews, B.: Convergence of the iterates of descent methods for analytic costfunctions. SIAM J. Optim. 16, 531–547 (2005)

2. Bárta, T., Chill, R., Fašangová, E.: Every ordinary differential equation with a strict Lyapunov functionis a gradient system. Monatsh. Math. 166, 57–72 (2012)

3. Bárta, T., Fašangová, E.: Convergence to equilibrium for solutions of an abstract wave equation withgeneral damping function. J. Differ. Equ. 260(3), 2259–2274 (2016)

4. Bégout, P., Bolte, J., Jendoubi, M.A.: On damped second-order gradient systems. J. Differ. Equ. 259(7),3115–3143 (2015)

5. Bolte, J., Daniilidis, A., Ley, O., Mazet, L.: Characterizations of Łojasiewicz inequalities: subgradientflows, talweg, convexity. Trans. Am. Math. Soc. 362(6), 3319–3363 (2010)

123

J Dyn Diff Equat

6. Chergui, L.: Convergence of global and bounded solutions of a second order gradient like system withnonlinear dissipation and analytic nonlinearity. J. Dynam. Differ. Equ. 20(3), 643–652 (2008)

7. Chill, R.: On the Łojasiewicz–Simon gradient inequality. J. Funct. Anal. 201(2), 572–601 (2003)8. Chill, R., Fiorenza, A.: Convergence and decay rate to equilibrium of bounded solutions of quasilinear

parabolic equations. J. Differ. Equ. 228(2), 611–632 (2006)9. Chill, R., Haraux, A., Jendoubi, M.A.: Applications of the Łojasiewicz–Simon gradient inequality to

gradient-like evolution equations. Anal. Appl. 7, 351–372 (2009)10. Haraux,A., Jendoubi,M.A.: Convergence of solutions of second-order gradient-like systemswith analytic

nonlinearities. J. Differ. Equ. 144(2), 313–320 (1998)11. Haraux, A., Jendoubi,M.A.: Decay estimates to equilibrium for some evolution equations with an analytic

nonlinearity. Asymptot. Anal. 26(1), 21–36 (2001)12. Hassen, I.B.,Haraux,A.:Convergence anddecay estimates for a class of secondorder dissipative equations

involving a non-negative potential energy. J. Funct. Anal. 260(10), 2933–2963 (2011)13. Kurdyka, K.: On gradients of functions definable in o-minimal structures. Ann. Inst. Fourier (Grenoble)

48(3), 769–783 (1998)14. Lageman, Ch.: Pointwise convergence of gradient-like systems. Math. Nachr. 280(13–14), 1543–1558

(2007)15. Łojasiewicz, S.: Une proprit topologique des sous-ensembles analytiques rels. Colloques internationaux

du C.N.R.S.: Les quations aux drives partielles, Paris (1962), Editions du C.N.R.S., Paris, p. 8789 (1963)

123


4.5 T. Barta, Decay estimates for solutions

of an abstract wave equation with gen-

eral damping function, submitted 2016.


Decay estimates for solutions of an abstract waveequation with general damping function

Tomas Barta∗,

March 31, 2016

Abstract

In this paper we prove convergence to equilibrium and decay estimates for a wideclass of damped abstract wave equations. We focus on the damping term to beas general as possible. We allow e.g. damping functions that oscilate between twopositive functions in a neighborhood of the origin and/or behave differently in eachdirection.keywords: abstract wave equation, convergence to equilibrium, decay estimates, Lojasiewicz inequality

1 Introduction

In this paper, we prove convergence to equilibrium and show decay estimates for solutionsof the second order equation

u+ g(u) +M(u) = 0 (1)

on a Hilbert space H for a broad class of damping functions g and (unbounded) nonlinearoperators M = E ′ satisfying Kurdyka– Lojasiewicz–Simon estimates.

There are many convergence results for second order equations with linear dampingand various operators M , see [10], [14], [11] for M in the form −∆u+ f(x, u) and [9] for amore general theory. Some decay estimates were shown in [12] for −∆u + f(x, u), and in[8] for a general nonlinear operator M = E ′ satisfying the Lojasiewicz gradient inequality.Convergence and decay estimates for nonlinear damping and a linear operator M = −∆uand the right-hand side h(x, t) was shown in [13]. An example, where bounded solutions donot converge to equilibrium, can be found in [15] (a nonlinear wave equation on a boundeddomain with Dirichlet boundary conditions and linear damping).

Concerning nonlinear damping and a nonlinear operator M , the equation

utt + |ut|αut −∆u = f(x, u) (2)

∗Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University inPrague, Sokolovska 83, 18675 Praha 8, Czech Republic

1

was studied by Chergui in [6], where convergence to equilibrium was proved. Later, BenHassen and Haraux in [5] proved convergence to equilibrium and decay estimates in theabstract setting (1) with M = E ′ ∈ C1(V, V ∗) where V → H → V ∗ are Hilbert spaces,and for damping functions g : V → V ∗ satisfying

c1‖v‖α+2 ≤ 〈g(v), v〉V ∗,V and ‖g(v)‖∗ ≤ c2‖v‖α+1,

which implies

c1‖v‖α+1 ‖v‖‖v‖V

≤ ‖g(v)‖∗ ≤ c2‖v‖α+1. (3)

In [3], Fasangova and the author of this paper showed that the upper and lower estimatesfor g can be independent, they proved convergence to equilibrium (without decay estimates)for pointwise damping operators g(v)(x) = G(v(x)) on V = H1

0 (Ω) with G estimated frombelow and above by two independent functions.

In the present paper we combine ideas from [5] and [3] and prove convergence and decayestimates for g : V → V ∗ where V is an arbitrary Hilbert space, g satisfying

h(‖v‖)‖v‖ ≤ 〈g(v), v〉V ∗,V and ‖g(v)‖∗ ≤ c2‖v‖,

where h is a positive function (not neccessarily a power sα+1). We also show that theupper estimate for g can be replaced by γ(‖g(v)‖∗) ≤ 〈g(v), v〉V ∗,V , which is satisfied bya wide class of poinwise damping operators. Moreover, we assume that M = E ′ satisfiesKurdyka– Lojasiewicz–Simon inequality (see Kurdyka [16])

Θ(E(u)) ≤ ‖M(u)‖∗,

which is a generalization of the Lojasiewicz gradient inequality (see Lojasiewicz [17]) con-sidered in [5] and [6].

The present conditions on g allow much more general damping functions than theprevious results. In particular, if we focus on the special case g(v)(x) = G(v(x)), then thefollowing cases are covered by the present paper and not by [5]:

• growth of G near zero and near infinity are different, e.g. G(s) = |s|as for small sand G(s) = |s|bs for large s,

• steeper growth of G in infinity than in [5, Example 3.1], e.g. G(s) = |s|bs for b ≤ 4N−2

,

• G with different behavior in every direction around zero, e.g. for a scalar valued vone allows G(s) = |s|as for s > 0 and G(s) = |s|bs for s < 0, a 6= b,

• G with non-power-like behavior, e.g. G(s) = |s|a lnb(1/|s|) lnc(ln(1/|s|))s for small s.

Moreover, the present results

• show that the decay estimates depend on the growth of G near zero only (this is notobvious since ‖v‖ < ε does not imply that |v(x)| is small for every x ∈ Ω),

2

• yield more delicate decay estimates, e.g. in the logarithmic scales ‖u(t) − ϕ‖ ≤C|t|a lnb(1/|t|) lnc(ln(1/|t|)).

In fact, similar decay estimates (based on Kurdyka– Lojasiewicz–Simon inequality) wasshown in [4] and [2] for ordinary differential equations of second order and in [7] for firstorder partial differential equations.

We present two kinds of results. The first kind (Theorems 2.1, 2.3) applies if weknow a-priori that the whole solution (for all t ≥ t0) lies in a ball where the Kurdyka– Lojasiewicz–Simon estimates are satisfied. In the second one (Theorems 2.2, 2.3) we haveKurdyka– Lojasiewicz–Simon estimates only in a small neighborhood U of an omega-limitpoint of the solution and we assume that the solution is relatively compact, but we do notknow a-priory that it is contained in U for all t ≥ t0.

The paper is organized as follows. In Section 2 we introduce our settings and assump-tions and formulate the main results. Sections 3 and 4 are devoted to proofs of the twomain Theorems. In Section 5, the results are applied to some semilinear wave equations.Section 6 is an appendix where we prove some technical lemmas.

2 Assumptions and the main result

Let V → H → V ∗ be Hilbert spaces with the embedding being dense, we identify〈v, u〉V ∗,V = 〈v, u〉H for u ∈ V ⊂ H, v ∈ H ⊂ V ∗. The norm and the scalar producton V ∗ (resp. on H, V ) are denoted by ‖ · ‖∗ and 〈·, ·〉∗ (resp. ‖ · ‖ and 〈·, ·〉, ‖ · ‖V and〈·, ·〉V ). By B(0, R) we denote the ball in H of radius R centered in 0, while BV (0, R) isthe corresponding ball in V . In the whole paper, C denotes a generic constant which maychange from line to line or from expression to expression.

Now, we define several properties of real functions. We say that a differentiable functionf : R+ → R+

• is admissible if f is nondecreasing and there exists cA ≥ 1 such that f(s) > 0 andsf ′(s) ≤ cAf(s) for all s > 0.

• has property (K) if for every K > 0 there exists C(K) > 0 such that f(Ks) ≤C(K)f(s) holds for all s > 0.

• is C-sublinear if there exists C > 0 such that f(t+ s) ≤ C(f(t) + f(s)) holds for allt, s > 0.

It is shown in the Appendix that the first property implies the other two. It is easy to seethat any nonnegative increasing concave function is admissible with cA = 1 provided it iseverywhere differentiable (otherwise sf ′±(s) ≤ f(s) holds, which would be also sufficientfor our purpose).

Let us introduce our assumptions on the operator E.

(E) Let E ∈ C2(V ), M = E ′ ∈ C1(V, V ∗) and let B be a fixed ball in V . Assume that:

3

(e1) E is nonnegative on B and there exists an admissible function Θ such thatΘ(s) ≤ CΘ

√s for all s ≥ 0 and some CΘ > 0, 1

Θis integrable in a neighbourhood

of zero and‖M(u)‖∗ ≥ Θ(E(u)), for all u ∈ B, (KLS)

i.e., E satisfies the Kurdyka– Lojasiewicz–Simon gradient inequality with func-tion Θ on B.

(e2) There exists CM ≥ 0 such that

|〈M ′(u)v, v〉∗| ≤ CM‖v‖2 for all u ∈ B, v ∈ V ,

(e3) There exists a nondecreasing function G : R+ → R+ such that

‖M(u)‖∗ ≤ G(E(u)), for all u ∈ B. (4)

Let us comment on these assumptions. In [6] Chergui works with H = L2(Ω), V =H1

0 (Ω), E ′(u) = ∆u+f(x, u) which corresponds to E(u) =∫

Ω12|∇u(x)|2 +F (x, u)dx, where

F (x, u) :=∫ u

0f(x, s) ds. By [6, Corollary 1.2], this function E satisfies the Lojasiewicz

gradient inequality‖E ′(u)‖∗ ≥ C|E(u)− E(ϕ)|1−θ (5)

with some θ ∈ [0, 12) in a neighbourhood of stationary points, provided f satisfies certain

assumptions. The Lojasiewicz inequality (5) is a special case of the Kurdyka– Lojasiewicz–Simon inequality (KLS) with the function Θ(s) = s1−θ, θ being the Lojasiewicz exponent.It is easy to see that Chergui’s operator satisfies (e2) as well. The conditions (e1) and (e2)(with (5) instead of (KLS)) appear also in [8], where linear damping is considered.

Concerning assumption (e3), there is one more condition (g4) below, which connectsfunctions G and Θ with a function h defined below. Let us mention than (e3) is oftensatisfied with G(s) = C

√s, in particular in all applications in [5] and in finite-dimensional

case for any E ∈ C1,1loc (Rn) satisfying that E(u) = 0 for all critical points u (see [4, Lemma

2.7]).

We now formulate the assumptions on the damping function.

(G) The function g : V → V ∗ is continuous and there exists an admissible function hsuch that

(g1) there exists C2 > 0 such that ‖g(v)‖∗ ≤ C2‖v‖ on V ∩ B(0, R) for any R > 0with C2 depending on R,

(g2) 〈g(v), v〉V ∗,V ≥ h(‖v‖)‖v‖2 on V ,

(g3) the function s 7→ 1Θ(s)h(Θ(s))

belongs to L1((0, 1)),

(g4) there exists CG > 0 such that G(s) ≤ CG√s

h(Θ(s))on (0, K] for any K > 0 with

CG depending on K,

(g5) the function ψ : s 7→ sh(√s) is convex for all s > 0.

4

Let us comment on these assumptions. If we take (g(v))(x) = |v(x)|α, we obtainequation (2) studied by Chergui in [6] and (g2) holds with h(s) = sα. Chergui’s conditionα < 4

N−2(and also condition (g3) in [3]) implies g(v) ∈ V ∗. Moreover, taking Θ(s) = s1−θ

(e.g. the Lojasiewicz inequality instead of (KLS)), then (g3) corresponds to condition0 < α < θ

1−θ in [6] and [5]. Condition (g3) is a condition coupling the damping functiong with the operator E. Another condition coupling g and E is (g4). But (as was saidabove) in many applications G(s) = C

√s, and in this case (g4) holds for any h and Θ

since h(Θ(s)) is bounded on (0, 1).In [5] the authors work with (g2) for h(s) = sα and (g1) replaced by ‖g(v)‖∗ ≤

C2‖v‖1+α. It is easy to modify the proof in [5] in such a way that the upper boundfor ‖g(v)‖∗ can be relaxed to (g1) (it is easy to show that ‖v‖ → 0, so ‖v‖1+α < ‖v‖).After doing this, one can apply the result in [5] e.g. to

g(v)(x) = |v(x)|α ln(1/|v(x)|)v(x)

with h(s) = s1+α. However, applying Theorem 2.1 below one can take h(s) = s1+α ln(1/s)in (g2) and get better convergence rates.

One can show (by differentiating), that functions

h(s) = sa lnr1(1/s) lnr2(ln(1/s)) . . . lnrk(ln . . . ln(1/s))

are positive increasing and concave on (0, ε) for a ∈ (0, 1), ri ∈ R. So, they becomeadmissible with cA = 1 after redefining them appropriately on (ε,+∞). In Section 5 wegive some examples of decay estimates in these scales of functions.

Our main results are formulated for solutions in the following sense. We say thatu ∈ W 1,1

loc ([0,+∞), V ) ∩W 2,1loc ([0,+∞), H) is a strong solution to (1) if (1) holds in V ∗ for

almost every t > 0.

Theorem 2.1. Let E and G satisfy (E) and (G). Let u be a strong solution to (1) andthere exists t1 > 0 such that u(t) ∈ B for all t ≥ t1. Then there exist ϕ ∈ B and t0 ≥ 0such that

E(u(t)) ≤ 2Ψ−1(t− t0), (6)

‖u(t)− ϕ‖ ≤ Φ(Ψ−1(t− t0)), (7)

‖u(t)‖ ≤√

Ψ−1(t− t0)) (8)

hold for all t > t0, some CΦ, CΨ > 0 and

Φ(t) = CΦ

∫ t

0

1

Θ(s)h(Θ(s))ds and Ψ(t) = CΨ

∫ 12

t

1

Θ2(s)h(Θ(s))ds. (9)

If we take Θ(s) = s1−θ and h(s) = sα in Theorem 2.1, we obtain the same convergencerate as in [5, Theorem 2.2].

5

The next result combines the method from [6] (resp. [3]) and [5] to obtain decay esti-mates for relatively compact solutions with (KLS) satisfied only on a small neighborhoodof some ϕ ∈ ωV (u), where

ωV (u) = ϕ ∈ V : ∃ tn +∞, s.t. ‖u(tn)− ϕ‖V → 0.Theorem 2.2. Let u be a strong solution to (1) with UT := (u(t), u(t)), t ≥ T relativelycompact in V ×H and ϕ ∈ ωV (u) with E(ϕ) = 0. Let (E) and (G) hold with the followingchanges.

• (KLS), (4) hold with B replaced by BV (ϕ, δ) for some δ > 0,

• (e2) holds with B replaced by ‘any compact subset of V with CM depending on thesubset’,

• h is admissible with cA = 1,

Then limt→+∞ ‖u(t) − ϕ‖V = 0 and there exists t0 ≥ 0 such that the decay estimates (6),(7) and (8) hold for all t > t0, some CΦ, CΨ > 0 and Φ, Ψ defined in (9).

Theorem 2.3. Theorems 2.1 and 2.2 remain valid if we replace (g1) by

(g1’) for every R > 0 there exists a convex function γ : R+ → R+ with property (K) andsuch that γ(0) = 0, lims→+∞ γ(s) = +∞, γ(s) ≥ cs2 for some c > 0 and all s smallenough, and γ(‖g(v)‖∗) ≤ 〈g(v), v〉V ∗,V on V ∩B(0, R).

Let us mention, that condition (g1) implies boundedness of ‖g(v(t))‖∗, while condition(g1’) does not. We show in Section 5 that (g1’) is useful in many examples.

Proof. The proofs of Theorems 2.1 and 2.2 remain valid except that we have to be morecareful by estimating the term ‖M(u)‖∗‖g(v)‖∗. Take R > 0 such that ‖v(t)‖ ≤ R for allt ≥ 0 and γ corresponding to this R. Let γ∗ be the convex conjugate to γ. By [3, Lemma3.2] we have γ∗(s) ≤ Cs2 for all s small enough. Then using Young’s inequality we obtain

‖M(u)‖∗‖g(v)‖∗ ≤ γ∗(

1

K‖M(u)‖∗

)+ γ (K‖g(v)‖∗) . (10)

Since we know that ‖M(u)‖∗ is bounded, taking K large enough yields

‖M(u)‖∗‖g(v)‖∗ ≤C

K2‖M(u)‖2

∗ + C(K)〈g(v), v〉V ∗,V ,

where we also used property (K) for function γ. The rests of the proofs remain unchanged.

It was mentioned in [2] and also in [5] that estimating ‖u(t)− ϕ‖ by the lenght of thetrajectory

∫ +∞t‖u(s)‖ds often does not yield an optimal result. In fact, the trajectory can

be much longer than the distance ‖u(t)−ϕ‖ if it has a shape of a spiral (which is typicallythe case for second order equations with weak damping). In many aplications, one canobtain a better estimate by estimating ‖u− ϕ‖ by E(u) directly.

6

Corollary 2.4. Let the assumptions of Theorems 2.1, 2.2 or 2.3 are satisfied and α : R+ →R+ be a nondecreasing function such that α(E(u) − E(ϕ)) ≥ ‖u − ϕ‖ on a neighborhoodof ϕ. Then

‖u(t)− ϕ‖ ≤ α(2Ψ−1(t− t0))

holds for some t0 and all t > t0.

Proof. We have ‖u(t)− ϕ‖ ≤ α(E(u(t))− E(ϕ)) ≤ α(2Ψ−1(t− t0)).

3 Proof of Theorem 2.1

For the strong solution u from the Theorem let us denote v(t) := u(t) and

E1(t) :=1

2‖v(t)‖2 + E(u(t)).

ThenE ′1(t) = 〈v(t), v(t)〉V,V ∗ + 〈M(u(t)), u(t)〉V ∗,V = −〈v(t), g(v(t))〉V,V ∗ (11)

It follows from (g2) that E1 is nonincreasing, so it is either positive for all t ≥ 0 or v(t) = 0for all t ≥ t0. In the latter case, u(t) = ϕ for t ≥ t0 and there is nothing to prove. So, wemay assume that E1(t) > 0 for all t ≥ 0. Moreover, it follows that ‖v(t)‖ and E(u(t)) arebounded and by (e3) also ‖M(u)‖∗ is bounded.

Further, we define for s, t ≥ 0

B(s) := h(Θ(s)), H(t) = E1(t) + εB(E1(t))〈M(u(t)), v(t)〉∗,

where ε > 0 will be specified later. We first show that for all t ≥ t1 the inequality

1

2E1(t) ≤ H(t) ≤ 2E1(t) (12)

holds if ε > 0 is small enough. Both inequalities follow immediately from the estimate

|εB(E1(t))〈M(u(t)), v(t)〉∗| ≤ εCB(E1(t))G(E1(t))√

2E1(t) ≤ εCE1(t) ≤ 1

2E1(t), (13)

where the first inequality is a consequence of definition of E1 and (e3) if applied the Cauchy–Schwarz inequality and H → V ∗, the second inequality is due to (g4) and definition ofB(·) and in the third inequality we take ε < 1

2C.

We now derive some estimates for H ′(t). Let us fix t > t1 and write (u, v) instead of(u(t), v(t)) and also E, E1 instead of E(t), E1(t). We start with

H ′(t) = E ′1 + εB′(E1)E ′1〈M(u), v〉∗ + εB(E1)〈M ′(u)v, v〉∗ + εB(E1)〈M(u), v〉∗= −〈g(v), v〉V ∗,V − εB′(E1)〈g(v), v〉V ∗,V 〈M(u), v〉∗ + εB(E1)〈M ′(u)v, v〉∗− εB(E1)〈M(u), g(v)〉∗ − εB(E1)〈M(u),M(u)〉∗

= −〈g(v), v〉V ∗,V − εB(E1)‖M(u)‖2∗ + εB(E1)〈M ′(u)v, v〉∗

− εB′(E1)〈v, g(v)〉V,V ∗〈M(u), v〉∗ − εB(E1)〈M(u), g(v)〉∗

(14)

7

In the last expression we keep the first two terms and estimate the other terms from above.By admissibility of h and Θ we have B′(s) = h′(Θ(s))Θ′(s) ≤ C h(Θ(s))

Θ(s)· Θ(s)

s= C B(s)

s. So,

B(·) is admissible. Then the fourth term on the right-hand side in (14) can be estimated(with help of (13)) by

|εB′(E1)〈v, g(v)〉V,V ∗〈M(u), v〉∗| ≤1

E1

|εB(E1)〈v, g(v)〉V,V ∗〈M(u), v〉∗| ≤1

2〈v, g(v)〉V,V ∗ .

The third term on the right-hand side in (14) is estimated as follows (ψ∗ being the convexconjugate to the function ψ from condition (g5))

|εB(E1)〈M ′(u)v, v〉∗| ≤ εB(E1)C‖v‖2

≤ εC

(1

Kψ∗(B(E1)) + C(K)ψ(‖v‖2)

)

≤ εC

(C

Kψ(Θ2(E1)) + C(K)ψ(‖v‖2)

)

≤ εC

(C

Kψ(Θ2(E)) +

C

Kψ(Θ2(‖v‖2)) + C(K)ψ(‖v‖2)

)

≤ εC

(C

KΘ2(E)h(Θ(E)) + 2C(K)‖v‖2h(‖v‖)

)

≤ εC

(C

K‖M(u)‖2

∗h(Θ(E1)) + 2C(K)‖v‖2h(‖v‖))

≤ 1

4εB(E1)‖M(u)‖2

∗ + εC〈v, g(v)〉∗.

(15)

Here we used (e2) (first inequality), Young inequality (second), Lemma 6.4 (third), C-sublinearity of ψ(Θ2(·)) (fourth), definition of ψ and Θ(s) ≤ √s (fifth), (KLS) inequalityand E ≤ E1 (sixth) and we have taken K = 1

4C2 and used (g2) in the last inequality.The fifth term on the right-hand side of (14) is estimated by

ε|B(E1)〈M(u), g(v)〉∗| ≤ εB(E1)(1

4‖M(u)‖2

∗ + C‖g(v)‖2∗)

≤ 1

4εB(E1)‖M(u)‖2

∗ + εCB(E1)‖v‖2

≤ 2

4εB(E1)‖M(u)‖2

∗ + εC〈v, g(v)〉∗,

where we used the Cauchy-Schwarz and Young inequalities (first step), (g1) (second step)and (15) (last step).

Altogether, we have

H ′(t) ≤ −(1− 1

2− 2εC)〈v, g(v)〉∗ −

1

4εB(E1)‖M(u)‖2

∗

≤ −c(h(‖v‖))‖v‖2 +B(E1)‖M(u)‖2∗).

(16)

8

Denoting χ(s) := B(s)Θ2(s) we obtain

−H ′(t) ≥ cB(E)‖M(u)‖2∗

≥ cB(E)Θ(E)2

= cχ(E)

= cχ(E1 −1

2‖v‖2)

≥ C1χ(E1)− Cχ(1/2‖v‖2))

= C1χ(E1)− CΘ2(1/2‖v‖2)h(Θ(1/2‖v‖2))

≥ C1χ(E1)− C‖v‖2h(‖v‖)≥ C1χ(E1) + CH ′(t).

Here we used (16) (in the first step), (KLS) inequality (second step), definition of χ (third),definition of E1 (fourth), C-sublinearity of χ (fifth), definition of χ and B (sixth), Θ(s) ≤C√s and property (K) for h (seventh) and (16) (last step). It follows that

−(C + 1)H ′(t) ≥ C1χ(E1(t)) ≥ 1

2C1χ(H(t)).

Take CΨ = 2(C + 1)/C1. Then

d

dtΨ(H(t)) = CΨ

−1

χ(H(t))H ′(t) ≥ 1

and we have

Ψ(H(t))−Ψ(H(t0))) ≥ t− t0.

It follows that limt→+∞Ψ(H(t)) = +∞, so we can take t0 such that Ψ(H(t0)) ≥ 0 and weget Ψ(H(t)) ≥ t− t0. Since Ψ is decreasing (by definition) we obtain

H(t) ≤ Ψ−1(t− t0).

Now, (6) and (8) follow immediately. To show the estimate (7), let us compute

− 1

CΦ

d

dtΦ(H(t)) ≥ C · h(‖v‖)‖v‖2 +B(E1)‖M(u)‖2

∗Θ(H(t))B(H(t))

≥ C · h(‖v‖)‖v‖2 +B(E1)‖M(u)‖2∗

(Θ(‖v‖2) + ‖M(u)‖∗)B(E1)

≥ C‖v‖ · h(‖v‖)‖v‖2 +B(E1)‖M(u)‖2∗

B(E1)‖v‖2 +B(E1)‖v‖‖M(u)‖∗.

(17)

In the first inequality we used the definition of Φ and (16). In the second inequality weused H ≤ 2E1, C-sublinearity of Θ, (KLS) inequality and C-sublinearity of B. In the last

9

inequality we used Θ(s) ≤ c√s only. We estimate the two terms in the last denominator

by the nominator. Using (15) we obtain

B(E1)‖v‖2 ≤ C(B(E1)‖M(u)‖2∗ + ‖v‖2h(‖v‖)) (18)

and (using Young inequality and (18))

B(E1(t))‖v‖‖M(u)‖∗ ≤ B(E1)‖M(u)‖2∗ +B(E1)‖v‖2

≤ (1 + C)B(E1)‖M(u)‖2∗ + C‖v‖2h(‖v‖). (19)

From (17), (18) and (19) we obtain − ddt

Φ(H(t)) ≥ CCΦ‖v‖ = ‖v‖ (choosing CΦ = C) and

integrating from t to +∞ we conclude that

∫ +∞

t

‖v(s)‖ds ≤ Φ(H(t))− lims→+∞

Φ(H(s)) ≤ Φ(Ψ−1(t− t0)).

Hence u ∈ L1([0,+∞)), so u has a limit ϕ and (7) holds since ‖u(t)−ϕ‖ ≤∫ +∞t‖v(s)‖ds.

4 Proof of Theorem 2.2

We may assume ϕ = 0 and denote v(t) := u(t). We show below that ‖u(t)‖V → 0 by thesame method as in [3]. So, we know that there exists t1 such that u(t) ∈ BV (ϕ, δ) for allt > t1 and the assumptions of Theorem 2.1 are satisfied with B = BV (ϕ, δ). So, we applyTheorem 2.1 and obtain the desired decay estimates.

So, it only remains to show ‖u(t)‖V → 0. By [1, Theorem 2.6], it is sufficient to finda function E ∈ C(V × H,R), such that t 7→ E(u(t), v(t)) is nondecreasing for t ≥ 0 andsatisfies

− d

dtE(u(t), v(t)) ≥ c‖u(t)‖∗ (20)

whenever u(t) ∈ BV (0, η) for some fixed η > 0. We show that these conditions are satisfiedby the function

E(u, v) := Φ(H(u, v)),

where

H(u, v) =1

2‖v‖2 + E(u) + εh(‖v‖∗)〈M(u), v〉∗, u ∈ V, v ∈ H

with ε small enough.Let us write for short E(t) (resp. H(t)) for E(u(t), v(t)) (resp. H(u(t), v(t))) and u,

v instead of u(t), v(t). By relative compactness of UT , quantities ‖v‖ and ‖M(u)‖∗ are

10

bounded, so we can use (g1), resp. (g1’). We have

H ′(t) = 〈v, v〉V,V ∗ + 〈M(u), v〉V ∗,V + εh′(‖v‖∗)〈v, vt〉∗‖v‖∗

〈M(u), v〉∗

+ εh(‖v‖∗)〈M ′(u)v, v〉∗ + εh(‖v‖∗)〈M(u), v〉∗= −〈g(v), v〉V ∗,V − εh′(‖v‖∗)

1

‖v‖∗〈M(u), v〉2∗

− εh′(‖v‖∗)1

‖v‖∗〈g(v), v〉∗〈M(u), v〉∗ + εh(‖v‖∗)〈M ′(u)v, v〉∗

− εh(‖v‖∗)〈M(u),M(u)〉∗ − εh(‖v‖∗)〈g(v),M(u)〉∗

and by positivity of the second term on the right

H ′(t) ≤ −〈g(v), v〉V ∗,V − εh(‖v‖∗)‖M(u)‖2∗ − εh(‖v‖∗)〈g(v),M(u)〉∗

− εh′(‖v‖∗)1

‖v‖∗〈g(v), v〉∗〈M(u), v〉∗ + εh(‖v‖∗)〈M ′(u)v, v〉∗

(21)

(here and in what follows, if v = 0 then any term containing 1‖v‖∗ has to be replaced by 0).

We show that the third, fourth and fifth terms in the last expression are dominated by thefirst and second terms.

The last term in (21) is estimated (with help of (e2) and (g2)) by

|εh(‖v‖∗)〈M ′(u)v, v〉∗| ≤ εh(‖v‖∗)C‖v‖2 ≤ εC〈g(v), v〉V ∗,V ≤1

4〈g(v), v〉V ∗,V

if ε is small enough. The third term on the right-hand side of (21) is estimated by

|εh(‖v‖∗)〈g(v),M(u)〉∗| ≤ εh(‖v‖∗)‖M(u)‖∗‖g(v)‖∗.

and the fourth term (applying the Cauchy–Schwarz inequality and admissibility of h) by

∣∣∣∣εh′(‖v‖∗)1

‖v‖∗〈g(v), v〉∗〈M(u), v〉∗

∣∣∣∣ ≤ εcAh(‖v‖∗)‖M(u)‖∗‖g(v)‖∗.

By Young’s inequality and (g1) we have

‖M(u)‖∗‖g(v)‖∗ ≤1

K‖M(u)‖2

∗ + C(K)‖g(v)‖2∗ ≤

1

K‖M(u)‖2

∗ + C(K)‖v‖2.

So, the third and fourth terms from (21) are estimated by

ε(1 + cA)h(‖v‖∗)(

1

K‖M(u)‖2

∗ + C(K)‖v‖2

)≤ 1

2εh(‖v‖∗)‖M(u)‖2

∗ + εCh(‖v‖∗)‖v‖2

≤ 1

2εh(‖v‖∗)‖M(u)‖2

∗ +1

4〈g(v), v〉V ∗,V

11

(we first took K large enough and then ε small enough). Altogether, we have

−H ′(t) ≥ 1

2〈g(v), v〉V ∗,V + ε

1

2h(‖v‖∗)‖M(u)‖2

∗ ≥ ch(‖v‖∗)(‖v‖2 + ‖M(u)‖2

∗)

(22)

where we used (g2) in the second inequality. Now we compute

E ′(t) =CΦH

′(t)

Θ(H(t))h(Θ(H(t)))≤ −Ch(‖v‖∗) (‖v‖2 + ‖M(u)‖2

∗)

Θ(H(t))h(Θ(H(t)))(23)

and see that E is nonincreasing along solutions for t > 0.Now, we assume that ‖u‖V is small and apply (e1) to obtain (20). We compute

Θ(H(u, v)) ≤ C

(Θ(

1

2‖v‖2) + Θ(E(u)) + Θ(‖M(u)‖∗‖v‖∗)

)

≤ C(Θ(‖v‖2) + ‖M(u)‖∗ + Θ(‖M(u)‖2

∗) + Θ(‖v‖2))

≤ C(‖v‖+ ‖M(u)‖∗) ,

where we used C-sublinearity and monotonicity of Θ, boundedness of h on compact in-tervals and property (K) for Θ and the Cauchy–Schwarz inequality (first step), Young’sinequality, (KLS), H → V ∗ and again C-sublinearity and property (K) (second step), andΘ(s) ≤ C

√s (third step). Since h is nondecreasing and has property (K) we have

Θ(H(u, v))h(Θ(H(u, v))) ≤ C(‖v‖+ ‖M(u)‖∗)h(‖v‖+ ‖M(u)‖∗). (24)

Since h is admissible with cA = 1 we have(

s

h(s)

)′=h(s)− sh′(s)

h2(s)≥ 0,

i. e., sh(s)

is nondecreasing. From ‖v‖+ ‖M(u)‖∗ ≥ c∗‖v‖∗ we obtain

‖v‖+ ‖M(u)‖∗h(‖v‖+ ‖M(u)‖∗)

≥ c∗‖v‖∗h(c∗‖v‖∗)

≥ c∗‖v‖∗C(c∗)h(‖v‖∗)

. (25)

Altogether, inserting the estimates (24) and (25) into (23) we obtain

−E ′(t) ≥ C · h(‖v‖∗)(‖v‖+ ‖M(u)‖∗)2

(‖v‖+ ‖M(u)‖∗)h(‖v‖+ ‖M(u)‖∗)≥ C‖v(t)‖∗

for all t where ‖u(t)‖V < η and the proof is complete.

5 Applications

In this section we show that Theorem 2.3 applies to the damping functions from [3], i.e., weconsider a bounded open set Ω ⊂ Rn, H = L2(Ω,RN), V = H1

0 (Ω,RN) (or V = H1(Ω,RN),Ω with Lipschitz boundary) and a function G : Rn → Rn satisfying the following conditions

12

(GG) There exist τ > 0 and an admissible function h : R+ → R+ satisfying (g3), (g4), (g5)such that

(gg1) there exists C2 > 0 such that |G(z)| ≤ C2|z| for all z ∈ B(0, τ),

(gg2) there exists C3 > 0 such that C3|z| ≤ |G(z)| for all z ∈ Rn \B(0, τ),

(gg3) if n = 2 then there exist C4 > 0, α > 0 such that |G(z)| ≤ C4|z|α+1 for allz ∈ Rn \B(0, τ); if n > 2 then the inequality holds with α = 4

n−2,

(gg4) there exists C5 > 0 such that 〈G(z), z〉 ≥ C5|G(z)||z| for all z ∈ Rn.

(gg5) |G(z)| ≥ h(|z|)|z| for all z ∈ B(0, τ).

Proposition 5.1. Let G : Rn → Rn satisfy (GG) and define (g(v))(x) := G(v(x)) forv ∈ V . Then g(V ) ⊂ V ∗ and g satisfies (G) with (g1) replaced by (g1’).

Proof. We first show that g(v) ∈ V ∗. Since Lp(Ω,RN) → V ∗ for p = α+2α+1

it is enough to

show that g(v) ∈ Lp(Ω,RN). We have

∫

Ω

|G(v(x))|p =

∫

|v(x)|≥τ|G(v(x))|p +

∫

|v(x)|<τ|G(v(x))|p

≤∫

|v(x)|≥τCp

4 |v(x)|p(α+1) +

∫

|v(x)|<τCp

2 |v(x)|p

≤ Cp4

∫

Ω

|v(x)|α+2 + |Ω|Cp2τ

p

≤ C‖v‖1

α+2

V + |Ω|Cp2τ

p,

where the second inequality follows from (gg3) and (gg1) and the last inequality fromV → Lα+2(Ω).

We show (g2). We define

h(s) :=

h(s)

2for s ∈ [0, δ)

h(δ)2

+ (1δ− 1

s)h′(δ)δ2

2for s ∈ [δ,+∞)

as in [3, proof of Proposition 3.3]. It is easy to show that h is admissible and |G(z)| ≥h(|z|)|z| holds for all z ∈ Rn if δ > 0 is small enough and such that h′(δ) > 0. Moreover, his bounded and ψ defined by ψ(s) = sh(

√s) is convex on R+ (see [3, proof of Proposition

13

3.3]). Then we have

〈g(v), v〉V ∗,V =

∫

Ω

〈G(v(x)), v(x)〉

≥∫

Ω

C5h(|v(x)|)|v(x)|2

= C5|Ω|∫

Ω

ψ(|v(x)|2)dx

|Ω|

≥ C5|Ω|ψ(∫

Ω

|v(x)|2 dx|Ω|

)

≥ Cψ(‖v‖2)

= Ch(‖v‖)‖v‖2

≥ Ch(‖v‖)‖v‖2,

where we used Jensen’s inequality in the fourth step, property (K) in the fifth step andinequality h(s) ≤ Ch(s) on compact intervals [0, K] in the sixth step.

We show (g1’). By [3, Proposition 3.3] there exists a function γ : R+ → R+ such thatγ(G(s)) ≤ CG(s)s and s 7→ γ(s1/p) is convex for s ≥ 0 and γ(s) ≥ Cs2 for small s ≥ 0.Then we have

γ(‖g(v)‖∗) ≤ Cγ

((∫

Ω

|G(v(x))|p)1/p

)

≤ C

∫

Ω

γ(|G(v(x))|)

≤ C

∫

Ω

|G(v(x))||v(x)|

≤ C

∫

Ω

〈G(v(x)), v(x)〉

= C〈g(v), v〉V ∗,V .

The first inequality follows from Lp → V ∗, monotonicity and property (K) of γ, the secondinequality is Jensen’s inequality applied to s 7→ γ(s1/p) together with property (K), thethird follows from γ(G(s)) ≤ CG(s)s and the fourth from (gg4).

Let us consider the following examples taken from [5].

A critical semilinear wave equation. Let Ω ⊂ Rn be bounded open and connectedand consider the following Dirichlet problem

utt + g(ut)−∆u− λ1u+ |u|p−1u = 0 in R+ × Ω,

u(t, x) = 0 on R+ × ∂Ω,(26)

14

where λ1 is the first eigenvalue of −∆ and p > 1 with (N − 2)p < N + 2. It correspondsto (1) with H = L2(Ω), V = H1

0 (Ω) and

E(u) =1

2

∫

Ω

(|∇u|2 − λ1|u|2)dx+1

p+ 1

∫

Ω

|u|p+1dx.

According to [5], (e1)-(e3) hold with Θ(s) = Cs1−θ, θ = 1p+1

and G(s) = C√s on any

bounded subset of V and any strong solution to (26) is bounded in V . Moreover, E(u) ≥c‖u‖p+1

V .

A semilinear wave equation with Neumann boundary conditions. Let Ω ⊂ Rn

be bounded open and connected and consider the following Neumann problem

utt + g(ut)−∆u+ |u|p−1u = 0 in R+ × Ω,∂∂nu(t, x) = 0 on R+ × ∂Ω,

(27)

where p > 1 with (n− 2)p < n+ 2. We have H = L2(Ω), V = H1(Ω) and

E(u) =1

2

∫

Ω

|∇u|2dx+1

p+ 1

∫

Ω

|u|p+1dx.

According to [5], (e1)-(e3) hold with Θ(s) = Cs1−θ, θ = 1p+1

and G(s) = C√s on any

bounded subset of V and any strong solution to (26) is bounded in V .

Now, we present some examples of damping functions g and obtain convergence toequilibrium and decay estimates for solutions of (26) and (27).

Example 5.2. Let us consider (g(v)) = G(v(x)) with G having different growth/decay fors < 0, s > 0, |s| large, |s| small, e.g.

G(s) =

|s|b1s, s > 1,

|s|a1s, s ∈ [0, 1],

|s|a2s, s ∈ [−1, 0),

|s|b2s, s < −1,

with 0 ≤ a1 < a2 <1p, b1, b2 ≤ 4

n−2. Then we have by Theorem 2.3

‖u(t)− ϕ‖ ≤ Ct− 1−a2p

(a2+1)p−1

and for equation (26) even

‖u(t)− ϕ‖V ≤ Ct− 1

(a2+1)p−1

by Corollary 2.4.

15

Example 5.3. In this example we show more delicate decay estimates in the logarithmicscale. Let

G(s) =

|s|as lnr(1/|s|) |s| ≤ 1,

c|s|bs |s| > 1,

with b < 4n−2

, 0 < a < 1p, r ∈ R or a = 1

p, r > 1.

If a > 1p

and r ≥ 0 then one can apply Theorem 2.3 with h(s) = sa to obtain

‖u(t)− ϕ‖ ≤ Ct−1−ap

(a+1)p−1

as in the previous example. If a < 1p, r < 0, we can apply Theorem 2.3 with h(s) = sa+ε

(for ε > 0 small enough) to obtain

‖u(t)− ϕ‖ ≤ Ct−1−(a+ε)p

(a+ε+1)p−1 .

If a = 1p, we cannot estimate G by any power such that (g3) holds. However, in all cases,

one can take h(s) = sa lnr(1/s) and obtain better decay estimates if a < 1p

and obtain some

decay estimates even for a = 1p. In fact, we have Θ2(s)h(Θ(s)) = s(1−θ)(2+a)(1−θ)r lnr(1/s)

and by Lemma 6.5

Ψ(t) = C

∫ 1/2

t

1

s(1−θ)(2+a) lnr(1/s)ds ∼ t1−(1−θ)(2+a) ln−r(1/t), t→ 0+, (28)

where f ∼ g means f = O(g) and g = O(f). Then by Lemma 6.6

Ψ−1(t) ∼ t1

1−(1−θ)(2+a) lnr

1−(1−θ)(2+a) (t), t→ +∞. (29)

For equation (26) we have by Corollary 2.4

‖u(t)− ϕ‖V ≤ C(Ψ−1(t− t0)

) 1p+1 ≤ Ct−

1(a+1)p−1 ln−

r(a+1)p−1 (t).

For equation (27) we have in case a < 1p

by Lemma 6.5

Φ(t) = C

∫ t

0

1

s(1−θ)(1+a) lnr(1/s)ds ∼ t1−(1−θ)(1+a) ln−r(1/t), t→ 0+, (30)

which yields for large t

‖u(t)− ϕ‖ ≤ Φ(Ψ−1(t− t0)) ≤ Ct−1−ap

(a+1)p−1 ln−pr

(a+1)p−1 (t). (31)

If a = 1p, then we have

Φ(t) = C

∫ t

0

1

s(1−θ)(1+a) lnr(1/s)ds = C

∫ t

0

1

s lnr(1/s)ds ∼ ln1−r(1/t) (32)

for t→ 0+ and therefore for large t

‖u(t)− ϕ‖ ≤ Φ(Ψ−1(t− t0)) ≤ C ln1−r(t). (33)

16

In fact, by similar computations as above with help of Lemmas 6.5, 6.6, we have: if

G(s) ≥ |s|a lnr1(1/|s|) . . . lnrk(ln . . . ln(1/|s|))

on a neighborhood of zero, then for large t we obtain

‖u(t)− ϕ‖ ≤ Ct−1−ap

(a+1)p−1 ln−pr1

(a+1)p−1 (t) ln−pr2

(a+1)p−1 (ln(t)) . . . ln−prk

(a+1)p−1 (ln . . . ln(t))

provided a > 1p

and

‖u(t)− ϕ‖ ≤ C ln1−rj(ln . . . ln(t)) ln−rj+1(ln . . . ln(t)) . . . ln−rk(ln . . . ln(t))

provided a = 1p, r1 = · · · = rj−1 = 1, rj > 1, rj+1, . . . , rk ∈ R.

6 Appendix

Lemma 6.1. If f is admissible, then it has property (K).

Proof. For K ≤ 1 it is sufficient to take C(K) = 1 since f is nondecreasing. Now, let us

fix t ≥ 0. Then for s > t we have f ′(s)f(s)≤ cA

sand integrating from t to T > t we obtain

ln(f(T ))− ln(f(t)) = lnf(T )

f(t)≤ cA ln

T

t,

so f(T ) ≤ f(t)(Tt

)cA and taking T = Kt for K > 1 we have property (K) with C(K) =KcA .

Lemma 6.2. Let f be nonnegative, nondecreasing and f , g have property (K). Then thecomposition f(g(·)) has property (K).

Proof. We have f(g(Kx)) ≤ f(C(K)g(x)) ≤ C(C(K))f(g(x)).

Lemma 6.3. Let f be nonnegative, nondecreasing and has property (K). Then it is C-sublinear, i.e., there exists C > 0 such that

f(x+ y) ≤ C(x+ y) for all x, y ≥ 0.

Proof. We have

f(x+ y) ≤ f(2 maxx, y) ≤ C(2)f(maxx, y) ≤ C maxf(x), f(y) ≤ C(f(x) + f(y)).

Lemma 6.4. Let ψ∗ be convex conjugate to the function ψ from (h3). Then ψ∗(h(√s)) ≤

cψ(s) for all s ≥ 0.

17

Proof. It holds that

ψ∗(h(√s)) = ψ∗(ψ(s)/s) ≤ ψ∗(ψ′(s)) = sψ′(s)− ψ(s).

Further,

ψ(2s)− ψ(s) =

∫ 2s

s

ψ′(r)dr ≥ s · ψ′(s).

So,ψ∗(h(

√s)) ≤ ψ(2s)− 2ψ(s) ≤ (K − 2)ψ(s)

since ψ has property (K).

Lemma 6.5. Let F be a primitive function to

f(t) = ta lnr1(1/t) lnr2(ln(1/t)) . . . lnrk(ln . . . ln(1/t))

on (0, ε), a 6= −1. Moreover, if a > −1, we assume limt→0+ F (t) = 0. Then

|F (t)| ∼ t1+a lnr1(1/t) lnr2(ln(1/t)) . . . lnrk(ln . . . ln(1/t)) as t→ 0+, (34)

where F ∼ g means F = O(g) and g = O(F ). If a = −1, r1 = · · · = rj−1 = −1, rj < −1,then

|F (t)| ∼ lnrj+1(ln . . . ln(1/t)) lnrj+1(ln . . . ln(1/t)) . . . lnrk(ln . . . ln(1/t)) as t→ 0+. (35)

Proof. Let us denote the right-hand side of (34) by G(t) and differentiate

G′(t) = (a+ 1)f(t) +k∑

i=1

tf(t)ri

ln(. . . ln(1/t)) . . . ln(1/t)1t

· −1

t2= f(t)(1 + a+ o(1)).

If a > −1, then 1CG′(s) ≤ f(s) ≤ CG′(s) on (0, ε) for some C > 1 and

F (t) =

∫ t

0

f(s) ≤ C

∫ t

0

G′(s)ds = CG(t)

and similarly F (t) ≥ 1CG(t). If a < −1, then 1

CG′(s) ≤ f(s) ≤ CG′(s) on (0, ε) for some

C < −1.

|F (t)| =∫ c

t

f(s)ds+ d ≤ C

∫ c

t

G′(s)ds+ d = CG(c)− CG(t) + d ≤ CG(t),

where the last inequality holds since G(t) → +∞ as t → 0+ and C < 0. Analogously wecan estimate |F (t)| from below. So, (34) is proven and (35) can be proven by the samemethod.

18

Lemma 6.6. Let

f(t) = ta lnr1(1/t) lnr2(ln(1/t)) . . . lnrk(ln . . . ln(1/t))

on (0, ε), a < 0. Then

f−1(t) ∼ t1a ln−

r1a (t) ln−

r2a (ln(t)) . . . ln−

rka (ln . . . ln(t)) as t→ +∞. (36)

Proof. Let us denote by g(t) the right-hand side of (36) and let us assume that ri ≥ 0 forall i = 1, 2, . . . , k. We show that f(g(t)) ≤ Ct for large t. Since

1

g(t)= t−

1a o(t−

1a ), as t→ +∞,

we have for t large enough

ln

(1

g(t)

)≤ ln

(t−

2a

)= −2

aln(t).

Further, if h(t)→ +∞, then for c > 0 and large t it holds that ln(ch(t)) = ln c+ lnh(t) ≤2 lnh(t). Therefore,

lnri(

ln . . . ln

(1

g(t)

))≤ lnri

(ln . . .

−2

aln (t)

)≤ 2ri lnri (ln . . . ln (t)) .

Now, we can compute

f(g(t)) = g(t)ak∏

i=1

lnri(

ln . . . ln

(1

g(t)

))

= t ln−r1(t) . . . ln−rk(ln . . . ln(t)) ·k∏

i=1

lnri(

ln . . . ln

(1

g(t)

))

≤ t ln−r1(t) . . . ln−rk(ln . . . ln(t)) ·(−2

a

)r1 k∏

i=2

2ri lnri (ln . . . ln (t))

≤ t ·(−1

a

)r1 k∏

i=1

2ri .

We can easily modify the estimates above to obtain f(g(t)) ≥ t ·(− 1a

)r1 ∏ki=1 2−ri and

similarly if we omit the assumption that ri are positive, we get

t

K≤ f(g(t)) ≤ Kt with K := Cr1

k∏

i=1

2|ri|, C := max

−1

a,−a

.

19

Applying f−1 (which is decreasing for large t) to these inequalities with s = t/K, we obtain

f−1(s) ≥ f−1(f(g(Ks))) = g(sK) ≥ K1/a

Cg(s),

resp. with s = Kt

f−1(s) ≤ f−1(f(g(s/K))) = g(s/K) ≤ C

K1/ag(s).

References

[1] T. Barta, Convergence to equilibrium of relatively compact solutions to evolution equa-tions, Electron. J. Differential Equations 2014 (2014), No. 81, 1–9.

[2] T. Barta, Rate of convergence and Lojasiewicz type estimates, preprint 2015.

[3] T. Barta and E. Fasangova, Convergence to equilibrium for solutions of an abstractwave equation with general damping function, J. Differential Equations 260 (2016),no. 3, 2259–2274.

[4] P. Begout, J. Bolte, M.A. Jendoubi, On damped second-order gradient systems, J.Differential Equations 259 (2015), no. 7, 3115–3143.

[5] I. Ben Hassen, A. Haraux, Convergence and decay estimates for a class of secondorder dissipative equations involving a non-negative potential energy, J. Funct. Anal.260 (2011), no. 10, 2933–2963.

[6] L. Chergui, Convergence of global and bounded solutions of the wave equation withnonlinear dissipation and analytic nonlinearity, J. Evol. Equ. 9 (2009), 405–418.

[7] R. Chill, A. Fiorenza, Convergence and decay rate to equilibrium of bounded solutionsof quasilinear parabolic equations, J. Differential Equations 228 (2006), no. 2, 611–632.

[8] R. Chill, A. Haraux, and M. A. Jendoubi, Applications of the Lojasiewicz-Simongradient inequality to gradient-like evolution equations, Anal. Appl. 7 (2009), 351–372.

[9] J.K. Hale and G. Raugel, Convergence in gradient-like systems with applications toPDE, Z. Angew. Math. Phys. 43 (1992), no. 1, 63–124.

[10] A. Haraux, Asymptotics for some nonlinear hyperbolic equations with a one-dimensional set of rest points, Bol. Soc. Brasil. Mat. 17 (1986), no. 2, 51–65.

20

[11] A. Haraux and M.A. Jendoubi, Convergence of bounded weak solutions of the waveequation with dissipation and analytic nonlinearity, Calc. Var. Partial DifferentialEquations 9 (1999), no. 2, 95–124.

[12] A. Haraux and M.A. Jendoubi, Decay estimates to equilibrium for some evolutionequations with an analytic nonlinearity, Asymptot. Anal. 26 (2001), no. 1, 21–36.

[13] A. Haraux, E. Zuazua, Decay estimates for some semilinear damped hyperbolic prob-lems, Arch. Rational Mech. Anal. 100 (1988), no. 2, 191–206.

[14] M.A. Jendoubi, Convergence of global and bounded solutions of the wave equation withlinear dissipation and analytic nonlinearity, J. Differential Equations 144 (1998), no.2, 302–312.

[15] M. A. Jendoubi and P. Polacik, Non-stabilizing solutions of semilinear hyperbolic andelliptic equations with damping, Proc. Royal Soc. Edinburgh Sect A 133 (2003), no. 5,1137–1153.

[16] K. Kurdyka, On gradients of functions definable in o-minimal structures, Ann. Inst.Fourier (Grenoble) 48 (1998), no. 3, 769–783.

[17] S. Lojasiewicz, Une propriete topologique des sous-ensembles analytiques reels, Collo-ques internationaux du C.N.R.S.: Les equations aux derivees partielles, Paris (1962),Editions du C.N.R.S., Paris, 1963.

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Documents

is.cuni.cz 3 This habilitation thesis is based on the following research articles: [B1] T. B arta, R. Chill, and E. Fa sangov a, Every ordinary di erential equa-tion with a strict