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J. Differential Equations 257 (2014) 1814–1839

www.elsevier.com/locate/jde

On a nonclassical fractional boundary-value problem

for the Laplace operator

Mykola Krasnoschok, Nataliya Vasylyeva ∗

Institute of Applied Mathematics and Mechanics of NAS of Ukraine, R. Luxemburg, str. 74, 83114 Donetsk, Ukraine

Received 29 August 2013; revised 6 May 2014

Available online 17 June 2014

Abstract

In this paper we consider a boundary-value problem for the Poisson equation with a boundary condition comprising the fractional derivative in time and the right-hand sides dependent on time. We prove the one-valued solvability of this problem, and provide the coercive estimates of the solution.© 2014 Elsevier Inc. All rights reserved.

MSC: primary 35R11, 35C15; secondary 35B65, 35R35

Keywords: Elliptic equations; Fractional dynamic boundary condition; Caputo derivative; Coercive estimates

1. Introduction

Let a and b be some positive constants, ν ∈ (0, 1),

R2+ = {(x1, x2) : x1 ∈ (−∞,+∞), x2 > 0

}, R2+,T = R2+ × (0, T ); R1

T = R1 × (0, T ).

In this paper we analyze the following problem with a fractional temporal derivative in a boundary condition:

* Corresponding author.E-mail addresses: krasnoschok@iamm.ac.donetsk.ua (M. Krasnoschok), vasylyeva@iamm.ac.donetsk.ua

(N. Vasylyeva).

http://dx.doi.org/10.1016/j.jde.2014.05.0220022-0396/© 2014 Elsevier Inc. All rights reserved.

M. Krasnoschok, N. Vasylyeva / J. Differential Equations 257 (2014) 1814–1839 1815

∂2u

∂x21

+ ∂2u

∂x22

= f0(x1, x2, t) in R2+,T ; (1.1)

u(x1,0, t) = aρ(x1, t) + f1(x1, t) on R1T ; (1.2)

Dνt ρ(x1, t) − b

∂u

∂x2= f2(x1, t) on R1

T ; (1.3)

ρ(x1,0) = 0 in R1, (1.4)

where fi , i = 0,2, are some given functions, Dνt denotes the Caputo fractional derivative with

respect to t and is defined by (see, for example, (2.4.6) in [22])

Dνt w(·, t) = 1

Γ (1 − ν)

∂

∂t

t∫0

w(·, τ )dτ

(t − τ)ν− w(·,0)

Γ (1 − ν)tν, ν ∈ (0,1), (1.5)

where Γ (ν) is the Gamma function.Analysis of problem (1.1)–(1.4) is important in order to study the free boundary problem

for the Laplace equation in the case of subdiffusion (the fractional quasistationary Stefan prob-lem or the fractional Hele-Shaw problem [45]). We recall that the anomalous diffusion means that the diffusive motion cannot be modeled as standard Brownian motion [8,33], and the mean square displacement of the diffusing species 〈(�x)2〉 scales as a nonlinear power law in time, i.e. 〈(�x)2〉 ∼ tν for some real number ν. If ν ∈ (0, 1), this is referred as a subdiffusion.

The fractional Hele-Shaw problem describes the evolution of fluid which is subjected to the “fractional” Darcy law [35], i.e. the “fractional” fluid velocity is proportional to the pressure gradient. This problem arises in controlled drug release system [30]; in studying of materials with memory [46]; in the transport processes at the Earth’s surface [45]. The mathematical model of the fractional Hele-Shaw problem is the following [46].

Let Ω(t) be a bounded domain in R2 for every t ∈ [0, T ], with the boundary ∂Ω = Γ ∪Γ (t), Γ ∩ Γ (t) = ∅. Here Γ is a given fixed curve and Γ (t) is an unknown boundary (free boundary). Assume that the equation of the free boundary is described as follows

Φ(y, t) = 0,

where Φ(y, t) is an unknown function.We look for the fluid pressure p(y, t), y ∈ Ω(t), t ∈ [0, T ], and the free boundary Γ (t) by the

following conditions:

�yp = 0 in Ω(t); p = g(y, t) on Γ ; (1.6)

p = 0, and Dνt Φ(y, t) = −μ(∇yp,∇yΦ) on Γ (t), ν ∈ (0,1); (1.7)

Ω(0) is given, (1.8)

where g(y, t) is a given function, Dνt Φ(y, t) is the “fractional” velocity of the free boundary

Γ (t) in the direction of the outward normal to Ω(t), μ is some positive constant.If ν = 1, problem (1.6)–(1.8) is interpreted as the mathematical model of the classical Hele-

Shaw problem which has been intensively studied for more than half a century. The literature on this and related problems is vast. A bibliography containing references up to the late nineties has

1816 M. Krasnoschok, N. Vasylyeva / J. Differential Equations 257 (2014) 1814–1839

been compiled by Howison [19]. In the case of ν ∈ (0, 1), presence of the fractional derivative in time in condition (1.7) complicates essentially investigation of problem (1.6)–(1.8) because some useful properties of ordinary derivative (e.g., product rule, chain rule) are not carried over to the case of a fractional derivative operator. To the best of our knowledge, some exact solutions of the fractional Hele-Shaw problem and analogous ones were constructed in [30,45,46] if Ω(t) ⊂ R1.

The general approach to the analysis of free boundary problems in smooth classes consists of the following. First, the problem in an unknown domain is reduced to a nonlinear problem in a fixed domain. Then the one-to-one solvability of the corresponding linearized problem al-lows one to reduce the investigation of the free boundary problem to a fixed point problem. This method has been proposed by Bazaliy in [3] to study the classical Stefan problem, next this ap-proach has been applied to investigate the classical Hele-Shaw problem in [4,5,49]. On this route the main analytical difficulties deal with the research of a model nonclassical boundary-value problem. In the case of the fractional Hele-Shaw problem, such a model problem is (1.1)–(1.4), where the functions ρ and u are the deviations from the initial pressure and initial shape of free boundary in (1.6)–(1.8).

Moreover, the mathematical interests to the problem studied in this paper is due to its pecu-liarities. The first is a fractional derivative in time of the function ρ(x1, t). In recent years the additional motivation for the studies of initial or initial–boundary value problems with fractional derivatives has been stimulated by experimental measurements of subdiffusion in porous me-dia [9], glass forming materials [47], biological media [40]. The review paper by Klafter et al. [25] provides numerous references to physical phenomena in which anomalous diffusion occurs. Here we refer to several works on the mathematical treatments for such problems. Kochubei [26], Pskhu [39,38] constructed the fundamental solution in Rd and proved the maximum principle for the Cauchy problem in the case of linear subdiffusion equation. Gejji and Jafari [17] solved a non-homogeneous fractional diffusion-wave equation in a one-dimensional bounded domain. Metzler and Klafter [33], using the method of images and the Fourier–Laplace transformation technique, obtained the solutions of different boundary value problems for the homogenous fractional dif-fusion equation in a half-space and in a box. Mophou and Guérékata [34]; and Sakamoto and Yamamoto [41] proved the one-valued solvability of the initial–boundary value problem for the fractional diffusion equation with variable coefficients which is t -independent with homogenous Dirichlet conditions in the Sobolev space. As source books related with fractional derivatives, see Samko, Kilbas and Marichev [42] which is an encyclopedic treatment of the fractional calculus and also Kilbas, Srivastava and Trujillo [22], Mainardi [32], Podlubny [37], Pskhu [38].

The last complication in (1.1)–(1.4) is stipulated by the presence of the fractional dynamic boundary condition. Indeed, due to relations (1.2) and (1.4) problem (1.1)–(1.4) can be reduced to searching of the function u(x, t) by the following conditions:

∂2u

∂x21

+ ∂2u

∂x22

= f0(x1, x2, t) in R2+,T ; (1.9)

Dνt u(x1,0, t) − ba

∂u

∂x2= af2(x1, t) + Dν

t f1(x1, t) ≡ f2(x1, t) on R1T ; (1.10)

u(x1,0,0) = f1(x1,0) in R1. (1.11)

If ν = 1, problem (1.9)–(1.11) is the simple example of a boundary value problem for the Poisson equation with dynamic boundary condition (1.10).

M. Krasnoschok, N. Vasylyeva / J. Differential Equations 257 (2014) 1814–1839 1817

Problems like (1.9)–(1.11) with ν = 1 appear in the investigations of diverse processes in physics and chemistry [7,14,16]. Here we make no pretense to provide a complete survey on the results related to a problem of the type (1.9)–(1.11), if ν = 1, and present only some of them. In the case the right-hand sides f2 and f0 are nonlinear, problem (1.9)–(1.11) has been studied recently in [1,2,20,23,44], where the global existence and uniqueness of positive solutions or a nonexistence of solutions have been proved. As for the case of the bounded domain with a smooth boundary and nonlinear right hand sides in (1.9)–(1.11), these problems have been investigated in [4,6,11–13,15,49]. Paper [13] is devoted to the study of the boundedness and a priori bounds of the global solutions. Problems of the regularity of classical solutions have been researched with the semigroup approach in [11,12], and with the potential technique in [4,6,15,49].

As for the investigation of problems like (1.9)–(1.11) with ν ∈ (0, 1), Kirane and Tatar [24]have analyzed the issue of nonexistence of local and global solutions for elliptic systems with the fractional dynamic boundary condition; and Krasnoschok and Vasylyeva [27] have studied local solvability of the initial–boundary value problems for linear and nonlinear subdiffusion equations with the fractional dynamic boundary conditions.

In the present paper we establish the existence and uniqueness of the solution (u, ρ) of prob-lem (1.1)–(1.4), and provide sharp estimates for it. In the subsequent work we will use these results to study the fractional Hele-Shaw problem (1.6)–(1.8) and to deduce the one-to-one solv-ability in a small time interval.

The paper is organized as follows: In Section 2, we define the functional spaces, which co-incide with parabolic Hölder classes if ν = 1, and formulate the main results of our work, Theorems 2.1, 2.2. Section 3 is devoted to research of problem (1.1)–(1.4) in the case of f0, f1 ≡ 0, and to the proof of Theorem 2.2. To deduce this result, we first derive, in Section 3.1, integral representations for the functions u and ρ. Then, in Section 3.2, we estimate u and ρtogether with their corresponding derivative. Next, using technique from [29] and results of Sec-tion 3.2, we complete the proof of Theorem 2.2 in Section 3.3. Investigations from Section 4 deal with the proof of Theorem 2.1 which is based on the results of Section 3. Appendix A contains the proofs of some auxiliary assertions which are applied in Section 3.

2. Functional spaces and the main results

Let Ω be a given domain in Rn, n ≥ 2, ΩT = Ω × (0, T ); x, x be any points in Ω ; t, τ ∈[0, T ]; α, β ∈ (0, 1). Denote

〈v〉(α)x,ΩT

= sup(x,t),(x,t)∈ΩT , x �=x

|v(x, t) − v(x, t)||x − x|α ;

〈v〉(β)t,ΩT

= sup(x,t),(x,τ )∈ΩT , t �=τ

|v(x, t) − v(x, τ )||t − τ |β ;

〈v〉(α)x,Ω = sup

x,x∈Ω, x �=x

|v(x) − v(x)||x − x|α .

In this paper we will use the two types of the functional spaces C([0, T ], Ck+α(Ω)) and Ck+α, k+α

2 ν(ΩT ) where k is integer, k = 0, 1, 2, ν ∈ (0, 1). Recall that the spaces C([0, T ],Ck+α(Ω)) used by many authors (see, e.g., [31] and references there), here we only remindthe definition of these classes.

1818 M. Krasnoschok, N. Vasylyeva / J. Differential Equations 257 (2014) 1814–1839

Definition 2.1. The spaces C([0, T ], Ck+α(Ω)) consist of k times continuously differentiable in x functions u(x, t), which are α-Hölder continuous in x, uniformly with respect to t . The norm in C([0, T ], Ck+α(Ω)) is

‖u‖C([0,T ],Ck+α(Ω)) =k∑

|j |=0

(supΩT

∣∣Djxu(x, t)

∣∣ + ⟨D

jxu

⟩(α)

x,ΩT

).

As for the spaces Ck+α, k+α2 ν(ΩT ), k = 0, 1, 2, if ν = 1, then these classes coincide with ordi-

nary Hölder spaces (see (1.10)–(1.12) in Chapter 3 of [28]). In the case of Ω ⊂ R1, and ν ∈ (0, 1), the definition of Ck+α, k+α

2 ν(ΩT ), k = 0, 1, 2, is represented by (2.1) and (2.2) in [27].

Definition 2.2. We will say that the function v(x, t) ∈ Ck+α, k+α2 ν(ΩT ), k = 0, 1, 2, iff the fol-

lowing norms are finite:

‖v‖C

k+α, k+α2 ν

(ΩT )= ‖v‖C([0,T ],Ck+α(Ω)) +

k∑|j |=0

⟨D

jxv

⟩( k+α−|j |2 ν)

t,ΩT, if k = 0,1;

‖v‖C

2+α, 2+α2 ν

(ΩT )= ‖v‖C([0,T ],C2+α(Ω)) + sup

ΩT

∣∣Dνt v(x, t)

∣∣ +2∑

|j |=0

⟨D

jxv

⟩( 2+α−|j |2 ν)

t,ΩT

+ ⟨Dν

t v⟩( α

2 ν)

t,ΩT+ ⟨

Dνt v

⟩(α)

x,ΩT.

In a similar way we introduce the spaces Ck+α, k+α2 ν(∂ΩT ) and C([0, T ], Ck+α(∂Ω)), where

∂ΩT = ∂Ω ×(0, T ). Moreover, we will use the normal Hölder classes Ck+α(Ω) and Ck+α(∂Ω), their definitions can be found, for instance, in [29].

The main result of our paper is the following:

Theorem 2.1. Let α, ν ∈ (0, 1), f0 ∈ C([0, T ], Cα(R2+)), f1 ∈ C([0, T ], C2+α(R1)), f2 ∈C([0, T ], C1+α(R1)), and

f0, f1, f2 = 0 if either t = 0 or |x| > R0 for any t ∈ [0, T ], (2.1)

for some positive number R0. Then there exists a unique solution (u(x, t), ρ(x1, t)) of(1.1)–(1.4), u ∈ C([0, T ], C2+α(R2+)), ρ ∈ C([0, T ], C2+α(R1)); and the estimate holds:

‖u‖C([0,T ],C2+α(R2+)) + ∥∥Dνt ρ

∥∥C([0,T ],C1+α(R1))

+ ‖ρ‖C([0,T ],C2+α(R1))

≤ C[‖f0‖C([0,T ],Cα(R2+)) + ‖f1‖C([0,T ],C2+α(R1)) + ‖f2‖C([0,T ],C1+α(R1))

], (2.2)

where C is a positive constant independent of the right-hand sides.

Note that in the case of the more smooth right hand sides of problem (1.1)–(1.4) we get the following result:

M. Krasnoschok, N. Vasylyeva / J. Differential Equations 257 (2014) 1814–1839 1819

Theorem 2.2. Let α, ν ∈ (0, 1), the function f2 ∈ C1+α, 1+α2 ν(R1

T ) and satisfy (2.1),

f0, f1 ≡ 0. (2.3)

Then there exists a unique solution (u(x1, x2, t), ρ(x1, t)) of problem (1.1)–(1.4), u ∈C2+α, 2+α

2 ν(R2+T ), ρ ∈ C2+α, 2+α2 ν(R1

T ); and the estimate holds:

‖u‖C

2+α, 2+α2 ν

(R2+T )+ ∥∥Dν

t ρ∥∥

C1+α, 1+α

2 ν(R1

T )+ ‖ρ‖

C2+α, 2+α

2 ν(R1

T )

≤ C‖f2‖C

1+α, 1+α2 ν

(R1T )

, (2.4)

where C is a positive constant independent of the right-hand sides.

Henceforward the letter C will be used to denote different constants encountered in our for-mulas. For simplicity, we shall first prove Theorem 2.2.

3. The solvability of problem (1.1)–(1.4) in the case of (2.3)

3.1. Construction of a solution

We denote by u(ξ, x2, t) the Fourier transform of u(x1, x2, t), and by u(·, p) the Laplace transform of u(·, t), and use the notation “∗” instead of “”.

Note that condition (2.1) allows us to extend the function f2(x1, t) by 0 for t ∈ (−∞, 0). Thus, we can apply, at least, formally the Fourier and Laplace transformations to problem (1.1)–(1.4)under conditions (2.1) and (2.3). Then, we get

∂2u∗

∂x22

− ξ2u∗ = 0, x2 ∈ (0,+∞), ρ∗(ξ,0) = 0; (3.1)

u∗(ξ,0,p) − aρ∗(ξ,p) = 0; (3.2)

pνρ∗(ξ,p) − b∂u∗

∂x2(ξ,0,p) = f ∗

2 (ξ,p). (3.3)

To obtain Eq. (3.3) we used formula (2.25.3) of [37]:

Dνt w(·, t) = pνw(·,p) − pν−1w(·,0). (3.4)

Using Eqs. (3.1) and (3.2), we have

u∗(ξ, x2,p) = a exp(−|ξ |x2

)ρ∗(ξ,p), (3.5)

and then condition (3.3) leads to

ρ∗(ξ,p) = f ∗2 (ξ,p)

ν. (3.6)

p + ab|ξ |

1820 M. Krasnoschok, N. Vasylyeva / J. Differential Equations 257 (2014) 1814–1839

Denote

G∗(ξ,p) := (pν + ab|ξ |)−1

, (3.7)

and one can easily check that

G∗(ξ,p) =∞∫

0

exp[−(

pν + ab|ξ |)σ ]dσ. (3.8)

Thus, relations (3.5)–(3.7) deduce to

ρ∗(ξ,p) = G∗(ξ,p)f ∗2 (ξ,p), (3.9)

u∗(ξ, x2,p) = a exp(−|ξ |x2

)G∗(ξ,p)f ∗

2 (ξ,p). (3.10)

After that we take advantage of the equality

+∞∫−∞

exp[−A|ξ | − iξx1

]dξ = 2A

(x21 + A2)

, A > 0, (3.11)

to calculate the inverse Laplace and Fourier transformations of the function u∗(ξ, x2, p) and get

u(x1, x2, t) =+∞∫

−∞K(x1 − y, x2)ρ(y, t)dy, (3.12)

where

K(x1, x2) = ax2

2π(x21 + x2

2). (3.13)

To get the representation for the function ρ(x1, t) we need formula (2.30) from [36]:

e−cpν = t−1W(−ct−ν;−ν,0

), c > 0. (3.14)

Here W(z; β, γ ) is the Wright function which is defined for z, β , γ ∈ C as (see, e.g., (1.8.1(27)) in v. 3 of [10])

W(z;β,γ ) =∞∑

k=0

zk

k!Γ (βk + γ ). (3.15)

Note that the principal properties of the Wright functions are described in Chapter 4.1, v. 1, and Chapter 18.1, v. 3 of [10]; Chapter 1.11 in [22]; Chapter 1.3 in [36]; Chapter 2 in [38]; [21].

M. Krasnoschok, N. Vasylyeva / J. Differential Equations 257 (2014) 1814–1839 1821

Equalities (3.11) with A := abσ and (3.14) with c := σ together with the inverse Laplace and Fourier transformations lead to the following representation

ρ(x1, t) =t∫

0

dτ

+∞∫−∞

G(x1 − y, t − τ)f2(y, τ )dy, (3.16)

where

G(x1, t) ={

12π

∫ +∞0 t−1W(−σ t−ν;−ν,0) abσ

(abσ)2+x21dσ, t > 0;

0, t < 0.(3.17)

We remark that the representation for the function G(x1, t) in the case of t < 0 has been obtained like (1.3) from Chapter 4 of [28] for the heat kernel Γ (x, t).

For simplicity, we assume that a = b = 1.

3.2. Estimates of the solution (u(x, t), ρ(x1, t))

At the beginning, we define the fractional Riemann–Liouville integral of a function g(·, t)with respect to t as (see, e.g., (2.1.1) in [22]):

I θt g(·, t) := 1

Γ (θ)

t∫0

g(·, τ )dτ

(t − τ)1−θ, θ > 0, t > 0. (3.18)

Let k be an integer number, and θ ∈ R1. Furthermore, we use the following notation

∂θt g(·, t) =

⎧⎪⎨⎪⎩g(·, t), θ = 0;∂k

∂tkg(·, t), θ = k;

∂k

∂tkI k−θt g(·, t), k − 1 < θ ≤ k.

(3.19)

Note that if θ ≥ 0, ∂θt g(·, t) is the fractional Riemann–Liouville derivative in t (see (2.1.8)

in [22]), and in the case of θ ∈ (0; 1] it can be rewritten ∂θt g(·, t) as

∂θt g(·, t) = 1

Γ (1 − θ)

∂

∂t

t∫0

g(·, τ )dτ

(t − τ)θ. (3.20)

The next lemma describes the main properties of the kernel G(x1, t) which will be necessary to estimate the function ρ(x1, t). Its proof is given in Appendix A.1.

Lemma 3.1. Let k, m and n be integer numbers, m + k > 0, m, n = 0, 1; k = 0, 1, 2, ... , α, θ be real numbers, α ∈ (0, 1), t > 0. Then

1822 M. Krasnoschok, N. Vasylyeva / J. Differential Equations 257 (2014) 1814–1839

i:

0 ≤ G(x1, t) ≤ Ct2ν−1

x21

, x1 ∈ R1; (3.21)

ii:

+∞∫−∞

∂k

∂xk1

G(x1, t)dx1 ={

tν−1

Γ (ν), k = 0,

0, k = 1; (3.22)

iii:

+∞∫−∞

I 1−νt G(x1, t)dx1 = 1; (3.23)

iv:

+∞∫0

∣∣∣∣∂νmt

∂k

∂xk1

G(x1, t)

∣∣∣∣dt ≤ C

|x1|k+m; (3.24)

v:

+∞∫0

∣∣∣∣∂θt

∂n

∂xn1G(x1, t)

∣∣∣∣xα1 dx1 ≤ Ctν(α−n+1)+1−θ , (3.25)

where θ ∈ [0, +∞) if n = 1, and if n = 0 then either θ = 0 or θ > (α + 1)ν − 1;

vi:

t∫0

∣∣∣∣∣+∞∫ε

∂ντ G(y, τ )dy

∣∣∣∣∣dτ ≤ C, (3.26)

for some positive number ε.

Proposition 3.1. Let α, ν ∈ (0, 1), condition (2.1) hold and f2(x1, t) ∈ C([0, T ], C1+α(R1)). Then the following inequalities hold:∣∣ρ(x1, t)

∣∣ ≤ Ctν supR1

T

∣∣f2(x1, t)∣∣; (3.27)

supR1

T

∣∣∣∣∂kρ

∂xk1

∣∣∣∣ +⟨∂kρ

∂xk1

⟩(α)

x1,R1T

≤ C

⟨∂k−1f2

∂xk−11

⟩(α)

x1,R1T

, k = 1,2. (3.28)

M. Krasnoschok, N. Vasylyeva / J. Differential Equations 257 (2014) 1814–1839 1823

Proof. Inequality (3.27) is a simple consequence of the nonnegativity of G(x1, t) (see (3.21))

and estimate (3.22) with k = 0. As for the proof of (3.28), we represent ∂kρ

∂xk1

, k = 1, 2, as

∂kρ

∂xk1

=t∫

0

dτ

+∞∫−∞

∂G

∂(x1 − y)(x1 − y, t − τ)

[∂k−1f2

∂yk−1(y, τ ) − ∂k−1f2

∂xk−11

(x1, τ )

]dy. (3.29)

Here we took into account property (3.22) with k = 1.

Note that, the estimate of supR1T

| ∂kρ

∂xk1|, k = 1, 2, follows from (3.29) and inequality (3.25) with

θ = 0, n = 1.Let x1 > x1 and denote

�xρ(k) := ∂kρ

∂xk1

(x1, t) − ∂kρ

∂xk1

(x1, t), �x1 = x1 − x1.

To obtain estimate (3.28), we use the following representation for �xρ(k) (the analogous repre-

sentation was Chapter 4 in [28]):

�xρ(k) =

t∫0

dτ

∫|x1−y|≤2|�x1|

[∂k−1f2

∂yk−1(y, τ ) − ∂k−1f2

∂xk−11

(x1, τ )

]∂G

∂(x1 − y)(x1 − y, t − τ)dy

+t∫

0

dτ

∫|x1−y|≤2|�x1|

[∂k−1f2

∂xk−11

(x1, τ ) − ∂k−1f2

∂yk−1(y, τ )

]∂G

∂(x1 − y)(x1 − y, t − τ)dy

+t∫

0

dτ

∫|x1−y|≥2|�x1|

[∂k−1f2

∂yk−1(y, τ ) − ∂k−1f2

∂xk−11

(x1, τ )

]

×[

∂G

∂(x1 − y)(x1 − y, t − τ) − ∂G

∂(x1 − y)(x1 − y, t − τ)

]dy

+t∫

0

dτ

[∂k−1f2

∂xk−11

(x1, τ ) − ∂k−1f2

∂xk−11

(x1, τ )

] ∫|x1−y|≥2|�x1|

∂G

∂(x1 − y)(x1 − y, t − τ)dy

≡4∑

j=1

Ij . (3.30)

Note that the estimates of I1 and I2 are the simplest and follow from (3.24) where k = 1 and m = 0:

|I1| + |I2| ≤ C

⟨∂k−1f2

∂xk−1

⟩(α)

1

∣∣�xα1

∣∣. (3.31)

1 x1,RT

1824 M. Krasnoschok, N. Vasylyeva / J. Differential Equations 257 (2014) 1814–1839

As for the term I3, we rewrite the difference [ ∂G∂(x1−y)

(x1 − y, t − τ) − ∂G∂(x1−y)

(x1 − y, t − τ)]as:[

∂G

∂(x1 − y)(x1 − y, t − τ) − ∂G

∂(x1 − y)(x1 − y, t − τ)

]=

x1∫x1

∂2G

∂(η − y)2(η − y, t − τ)dη,

and evaluate I3 as

|I3| ≤∫

|x1−y|≥2|�x1|dy|x1 − y|α

∣∣∣∣∣x1∫

x1

dη

+∞∫0

dτ

∣∣∣∣ ∂2G

∂(η − y)2(η − y, t − τ)

∣∣∣∣∣∣∣∣∣

×⟨∂k−1f2

∂xk−11

⟩(α)

x1,R1T

. (3.32)

Then, we apply (3.24) with m = 0, k = 2, to the inner integral in (3.32) and get

|I3| ≤ const.

⟨∂k−1f2

∂xk−11

⟩(α)

x1,R1T

∫|x1−y|≥2|�x1|

dy|x1 − y|α∣∣∣∣∣

x1∫x1

dη

|η − y|2∣∣∣∣∣

≤ const.

⟨∂k−1f2

∂xk−11

⟩(α)

x1,R1T

∫|x1−y|≥2|�x1|

|x1 − y|α|�x1||x1 − y||x1 − y|dy. (3.33)

The change of variable: z = x1−y|�x1| , in the last integral in (3.33) leads to

|I3| ≤ const.

⟨∂k−1f2

∂xk−11

⟩(α)

x1,R1T

|�x1|α+∞∫2

dz

z(1 − z)1−α≤ C

⟨∂k−1f2

∂xk−11

⟩(α)

x1,R1T

|�x1|α. (3.34)

At last, one can easily check that ∂G(x1,t)∂x1

is the odd function in x1. Thus,

I4 = 0. (3.35)

Hence, estimates (3.31), (3.33) and (3.35) together with representation (3.30) prove inequal-ity (3.28). �Proposition 3.2. Let the conditions of Proposition 3.1 hold. Then the function Dν

t ρ (x1, t) satis-fies the following inequalities:

1∑k=0

supR1

T

∣∣∣∣Dνt

∂kρ(x1, t)

∂xk1

∣∣∣∣ ≤ C‖f2‖C([0,T ],C1+α(R1)), (3.36)

⟨Dν

t

∂kρ

∂xk1

⟩(α)

x1,R1T

≤ C

⟨∂kf2

∂xk1

⟩(α)

x1,R1T

, k = 0,1. (3.37)

M. Krasnoschok, N. Vasylyeva / J. Differential Equations 257 (2014) 1814–1839 1825

Proof. First, we need the next representation of Dνt

∂kρ(x1,t)

∂xk1

, k = 0, 1:

Dνt

∂kρ(x1, t)

∂xk1

=t∫

0

dτ

−∞∫+∞

∂νt G(x1 − y, t − τ)

[∂kf2(y, τ )

∂yk− ∂kf2(x1, τ )

∂xk1

]dy

+ ∂kf2(x1, t)

∂xk1

. (3.38)

The derivation of this formula is technically tedious so we give it in Appendix A.2.Then, inequality (3.36) follows immediately from representation (3.38) and estimate (3.25)

where θ := ν, n := 0.Let x1 > x1 and �x1 := x1 − x1. To prove (3.37) it is enough to evaluate the difference

�Dνt ρ

(k) :=[

Dνt

∂kρ(x1, t)

∂xk1

− ∂kf2(x1, t)

∂xk1

]−

[Dν

t

∂kρ(x1, t)

∂xk1

− ∂kf2(x1, t)

∂xk1

]. (3.39)

Let us represent �Dνt ρ

(k) like (3.30) as

�Dνt ρ

(k) =t∫

0

dτ

∫|x1−y|≤2|�x1|

∂νt G(x1 − y, t − τ)

[∂kf2(y, τ )

∂yk− ∂kf2(x1, τ )

∂xk1

]dy

+t∫

0

dτ

∫|x1−y|≤2|�x1|

∂νt G(x1 − y, t − τ)

[∂kf2(x1, τ )

∂xk1

− ∂kf2(y, τ )

∂yk

]dy

+t∫

0

dτ

∫|x1−y|≥2|�x1|

[∂kf2(y, τ )

∂yk− ∂kf2(x1, τ )

∂xk1

][∂νt G(x1 − y, t − τ)

− ∂νt G(x1 − y, t − τ)

]dy

+t∫

0

dτ

[∂kf2(x1, τ )

∂xk1

− ∂kf2(x1, τ )

∂xk1

] ∫|x1−y|≥2|�x1|

∂νt G(x1 − y, t − τ)dy

≡4∑

i=1

Ji. (3.40)

Note that the terms Ji , i = 1,3 are estimated like Ij , j = 1,3 from (3.30). Estimates of J1 and J2 follow from inequality (3.24) where m = 1, k = 0:

|J1| + |J2| ≤ C|�x1|α⟨∂kf2

∂xk

⟩(α)

1. (3.41)

1 x1,RT

1826 M. Krasnoschok, N. Vasylyeva / J. Differential Equations 257 (2014) 1814–1839

As for J3, we represent it as

J3 =t∫

0

dτ

∫|x1−y|≥2|�x1|

dy

[∂kf2(y, τ )

∂yk− ∂kf2(x1, τ )

∂xk1

] x1∫x1

∂νt

∂G(η − y, t − τ)

∂(η − y)dη.

Then simple calculations together with inequality (3.24) with m = 1, k = 1 lead to

|J3| ≤ C|�x1|α⟨∂kf2

∂xk1

⟩(α)

x1,R1T

. (3.42)

At last, estimate of J4 is a simple consequence of inequality (3.26) with ε = |�x1|:

|J4| ≤ C|�x1|α⟨∂kf2

∂xk1

⟩(α)

x1,R1T

. (3.43)

Thus inequality (3.37) is deduced from representation (3.40) and estimates (3.41)–(3.43). �Now we obtain the corresponding estimates of the function u(x1, x2, t) which is given

by (3.12). Due to representation (3.12), it is obvious that

Dνt u(x1, x2, t) =

+∞∫−∞

K(x1 − y, x2)Dνt ρ(y, t)dy. (3.44)

Next, based on representations (3.12) and (3.44) and results of Chapter 3 in [29] we can concludethe following result.

Proposition 3.3. Let the conditions of Proposition 3.1 be satisfied. Then the function u(x1, x2, t)given by (3.12) satisfies the following inequality:

‖u‖C([0,T ],C2+α(R2+)) + ∥∥Dνt u

∥∥C([0,T ],C1+α(R2+))

≤ C‖f2‖C([0,T ],C1+α(R1)). (3.45)

Moreover, the simple implication of Propositions 3.1 and 3.2 is the next estimate of the func-

tions ∂kρ

∂xk1

with respect to t .

Proposition 3.4. Let the conditions of Proposition 3.1 be satisfied. Then the following estimate holds:

2∑k=1

⟨∂kρ

∂xk1

⟩( 2−k+α2 ν)

t,R1T

≤ C‖f2‖C([0,T ],C1+α(R1)). (3.46)

Proof. First, following the arguments of Theorem 3.1 in [42], one can prove the next statements:

M. Krasnoschok, N. Vasylyeva / J. Differential Equations 257 (2014) 1814–1839 1827

i: Let ψ, Dνt ψ ∈ C[0, T ], then∣∣ψ(t1) − ψ(t2)

∣∣ ≤ C max[0,T ]

∣∣Dνt ψ

∣∣|t1 − t2|ν, ∀t1, t2 ∈ [0, T ]; (3.47)

ii: Let ϕ(x, t), Dνt ϕ(x, t) ∈ C([0, T ], Cα(Q)), then⟨

ϕ(·, t1) − ϕ(·, t2)⟩(α)

x,Q≤ C

⟨Dν

t ϕ⟩(α)

x,QT|t1 − t2|ν . (3.48)

Based on (3.47), we can deduce that∣∣∣∣ ∂ρ

∂x1(x, t1) − ∂ρ

∂x1(x, t2)

∣∣∣∣ ≤ C|t1 − t2|ν supR1

T

∣∣∣∣Dνt

∂ρ

∂x1

∣∣∣∣. (3.49)

This inequality implies that ⟨∂ρ

∂x1

⟩(ν)

t,R1T

≤ C supR1

T

∣∣∣∣Dνt

∂ρ

∂x1

∣∣∣∣. (3.50)

Thus, due to 1+α2 ν < ν, we assert from (3.50) and (3.36) that

⟨∂ρ

∂x1

⟩( 1+α2 ν)

t,R1T

≤ C‖f2‖C([0,T ],C1+α(R1)). (3.51)

To evaluate 〈 ∂2ρ

∂x21〉(

α2 ν)

t,R1T

, we use the next interpolation inequality (see Lemma 3.1 of [43] and

Corollary 1.2.18 of [31]):

‖V ‖Cl0 (Q) ≤ C‖V ‖σ

Cl2 (Q)‖V ‖1−σ

Cl1 (Q), (3.52)

where σ ∈ (0, 1), 0 ≤ l1 < l2, l0 = l1 + σ(l2 − l1), ∂Q ∈ C2+α . If Q is an unbounded domain then the norms in (3.52) should be exchanged by the corresponding seminorms (see [43]).

Let V := ρ(x1, t1) − ρ(x1, t2), l0 := 2, l2 := 2 + α, l1 := α, σ := 1 − α2 in (3.52), then

supR1

∣∣∣∣∂2ρ

∂x21

(x1, t1) − ∂2ρ

∂x21

(x1, t2)

∣∣∣∣ ≤ const.∥∥ρ(·, t1) − ρ(·, t2)

∥∥1− α2

C2+α(R1)

× ∥∥ρ(·, t1) − ρ(·, t2)∥∥ α

2Cα(R1)

. (3.53)

Next, we apply inequalities (3.47) and (3.48) to evaluate ‖ρ(·, t1) − ρ(·, t2)‖Cα(R1) and have∥∥ρ(·, t1) − ρ(·, t2)∥∥

Cα(R1)≤ const.|t1 − t2|ν

∥∥Dνt ρ

∥∥C([0,T ],Cα(R1))

. (3.54)

Note that ∥∥ρ(·, t1) − ρ(·, t2)∥∥

2+α 1 ≤ const.‖ρ‖C([0,T ],C2+α(R1)). (3.55)

C (R )

1828 M. Krasnoschok, N. Vasylyeva / J. Differential Equations 257 (2014) 1814–1839

Thus, inequalities (3.53)–(3.55) allow us to assert

⟨∂2ρ

∂x21

⟩( α2 ν)

t,R1T

≤ C‖ρ‖1− α2

C([0,T ],C2+α(R1))

∥∥Dνt ρ

∥∥α/2C([0,T ],Cα(R1))

. (3.56)

At last, application of Propositions 3.1 and 3.2 to the right-hand side of (3.57) leads to

⟨∂2ρ

∂x21

⟩( α2 ν)

t,R1T

≤ C‖f2‖C([0,T ],C1+α(R1)). (3.57)

Hence, inequalities (3.51) and (3.57) deduce the statement of Proposition 3.4. �3.3. Proof of Theorem 2.2

First we get estimate of (2.4). Due to inequality

‖f2‖C([0,T ],C1+α(R1)) ≤ ‖f2‖C

1+α, 1+α2 ν

(R1T )

, (3.58)

and the results of Propositions 3.1–3.3, we have

‖u‖C([0,T ],C2+α(R2+)) + ∥∥Dνt u

∥∥C([0,T ],C1+α(R2+))

+ ‖ρ‖C([0,T ],C2+α(R1))

+ ∥∥Dνt ρ

∥∥C([0,T ],C1+α(R1))

≤ C‖f2‖C

1+α, 1+α2 ν

(R1T )

. (3.59)

Thus, to get (2.4) it is necessary to obtain the estimates of the Hölder seminorms in t of the functions u(x1, x2, t), ρ(x1, t) and their corresponding derivatives.

The results of Chapter 3 in [29] together with representation (3.12) and estimate (3.59) allow us to infer ∥∥u(·, t)∥∥

C2+α(R2+)≤ C

∥∥ρ(·, t)∥∥C2+α(R1)

,∥∥Dνt u(·, t)∥∥

C1+α(R2+)≤ C

∥∥Dνt ρ(·, t)∥∥

C1+α(R1), ∀t ∈ [0, T ]. (3.60)

Then estimates (3.59), (3.60) and (3.46), (3.47) lead to

2∑|k|=1

⟨Dk

xu⟩( 2−|k|+α

2 ν)

t,R2+T

≤ C‖f2‖C([0,T ],C1+α(R1)). (3.61)

After that we evaluate 〈Dνt ρ〉(

1+α2 ν)

t,R1T

. To this end, we need the following properties of the functions

ρ(x1, t) and u(x1, x2, t).

Corollary 3.1. Let the conditions of Proposition 3.1 hold. Then the functions ρ(x1, t) and u(x1, x2, t) given by formulas (3.12) and (3.16) satisfy boundary condition (1.3).

M. Krasnoschok, N. Vasylyeva / J. Differential Equations 257 (2014) 1814–1839 1829

The proof of this statement is given in Appendix A.3.Due to Corollary 3.1 and estimate (3.61), we can conclude that

⟨Dν

t ρ⟩( 1+α

2 ν)

t,R1T

≤⟨

∂u

∂x2

⟩( 1+α2 ν)

t,R2+T

+ 〈f2〉(1+α

2 ν)

t,R1T

≤ C‖f2‖C

1+α, 1+α2 ν

(R1T )

. (3.62)

The estimate of 〈Dνt ρx1〉(

α2 ν)

t,R1T

is proved with the arguments like them from Proposition 3.4. Indeed,

interpolation inequality (3.52) with V := Dνt ρ(x1, t1) − Dν

t ρ(x1, t2), l0 := 1, l2 := 1 +α, l1 := 0, σ := 1

1+αgives

supR1

∣∣Dνt ρx1(x1, t1) − Dν

t ρx1(x1, t2)∣∣ ≤ const.

∥∥Dνt ρ

∥∥ 11+α

C([0,T ],C1+α(R1))

× (⟨Dν

t ρ⟩( 1+α

2 ν)

t,R1T

|t1 − t2| 1+α2 ν

) α1+α . (3.63)

Then, applying inequalities (3.58), (3.59) and (3.62) to the right-hand side of (3.63), we have⟨Dν

t ρx1

⟩( α2 ν)

t,R1T

≤ C‖f2‖C

1+α, 1+α2 ν

(R1T )

. (3.64)

Thus, based on (3.59), (3.61), (3.46), (3.62) and (3.64) we deduce estimate (2.4).Note that, the uniqueness of the constructed solution (u(x1, x2, t), ρ(x1, t)) follows from (2.4).

Moreover, representation (3.12) and estimate (2.4) mean that u = K � ρ, where ρ ∈ C2+α(R1)

for ∀t ∈ [0, T ]. Thus, arguments from Chapter 3 in [29] allow us to show that u(x1, x2, t) satis-fies conditions (1.1) and (1.2) with f0, f1 ≡ 0. Inequality (3.27) ensures that ρ(x1, t) meets the requirement (1.4). Hence, all the written above together with Corollary 3.1 allow us to conclude that the functions u(x1, x2, t) and ρ(x1, t) given by (3.12) and (3.16) satisfy (1.1)–(1.4). That completes the proof of Theorem 2.2.

4. Proof of Theorem 2.1

At the beginning we prove Theorem 2.1 in the case of (2.3). Note that, estimate (2.2) follows immediately from the results of Propositions 3.1–3.3. This inequality ensures the uniqueness of the solution.

Thus, to prove Theorem 2.1 in the case of (2.3), it is necessary to show continuity of the functions u(x1, x2, t) and ρ(x1, t), Dν

t ρ together with their derivatives with respect to xi , i = 1, 2. To this end, we consider the following sequence of the functions {f2n}∞n=1:

f2n ∈ C1+α, 1+α2 ν

(R1

T

), f2n −→

n→∞f2 in C([0, T ],C1+α

(R1)), (4.1)

and find the functions un(x1, x2, t) and ρn(x1, t) as a solution of the problem:

�un = 0 in R2+T ; ρn(x1,0) = 0 in R1;

un = aρn and Dνt ρn − b

∂un = f2n on R1T . (4.2)

∂x2

1830 M. Krasnoschok, N. Vasylyeva / J. Differential Equations 257 (2014) 1814–1839

Then we apply Theorem 2.2 to problem (4.2) and get

un ∈ C2+α, 2+α2 ν

(R2+T

), ρn ∈ C2+α, 2+α

2 ν(R1

T

), Dν

t ρn ∈ C1+α, 1+α2 ν

(R1

T

), (4.3)

and the estimate like (2.4) holds. Embeddings (4.3) mean that

un ∈ C([0, T ],C2+α

(R2+

)), ρn ∈ C

([0, T ],C2+α(R1)),

Dνt ρn ∈ C

([0, T ],C1+α(R1)).

Summing up all the written above and using the results of Propositions 3.1–3.3, we can assert the following:

i: the sequences: {un}∞n=1, {ρn}∞n=1, {Dνt un}∞n=1, {Dν

t ρn}∞n=1 are fundamental in the spaces: C([0, T ], C2+α(R2+)), C([0, T ], C2+α(R1)), C([0, T ], C1+α(R2+)), C([0, T ], C1+α(R1)), correspondingly;

ii: as the spaces C([0, T ], Ck+α(R2+)) and C([0, T ], Ck+α(R1)), k = 1, 2, are Banach, there exist the functions u ∈ C([0, T ], C2+α(R2+)) and ρ ∈ C([0, T ], C2+α(R1)) such that

un → u in C([0, T ],C2+α

(R2+

)), ρn → ρ in C

([0, T ],C2+α(R1)),

Dνt ρn → Dν

t ρ in C([0, T ],C1+α

(R1)).

Moreover, these functions (u, ρ) satisfy (1.1)–(1.4) under condition (2.3); and estimate (2.2). Thus, Theorem 2.1 in the case of (2.3) has been proved.

To get Theorem 2.1 in the general case, we look for the solution of problem (1.1)–(1.4) in the form

u(x1, x2, t) = u1(x1, x2, t) + u2(x1, x2, t), (4.4)

where u1(x1, x2, t) is a solution of the following Dirichlet problem:

�u1 = f0(x, t) in R2+T ; u1 = f1(x1, t) on R1T ; (4.5)

and the function u2 satisfies the conditions

�u2 = 0 in R2+T ; ρ(x1,0) = 0 in R1;

u2 = aρ and Dνt ρ − b

∂u2

∂x2= f2 + b

∂u1

∂x2:= f2(x1, t) on R1

T . (4.6)

The results of §2, Chapter 3 in [29] give the one-valued solvability of (4.5) in C([0, T ],C2+α(R2+)), and

‖u1‖C([0,T ],C2+α(R2+)) ≤ C(‖f0‖C([0,T ],C2+α(R2+)) + ‖f1‖C([0,T ],C2+α(R1))

). (4.7)

Inequality (4.7) leads to estimate of f2:

M. Krasnoschok, N. Vasylyeva / J. Differential Equations 257 (2014) 1814–1839 1831

‖f2‖C([0,T ],C1+α(R1)) ≤ C(‖f0‖C([0,T ],C2+α(R2+)) + ‖f1‖C([0,T ],C2+α(R1))

+ ‖f2‖C([0,T ],C1+α(R1))

).

This inequality allows us to apply Theorem 2.1 in the case of (2.3) to problem (4.6) and to get one-to-one solvability of (4.6): u2 ∈ C([0, T ], C2+α(R2+)), ρ ∈ C([0, T ], C2+α(R1)), and the estimate like (2.4).

All the written above completes the proof of Theorem 2.1.

Appendix A

A.1. Proof of Lemma 3.1

To prove the statement (i) of Lemma 3.1, we need the next equality for the Wright functions (see, e.g., (2.2.5) in [38] or (F7) in [32]):

W(−z;−β, δ − 1) + (1 − δ)W(−z;−β, δ) = βzW(−z;−β, δ − β), (A.1)

which can be rewritten in the case of δ = 1 as

W(−z;−β,0) = βzW(−z;−β,1 − β). (A.2)

Using (A.2) with β := ν, z := σ t−ν , we can represent the function G(x1, t) as

G(x1, t) = 1

2π

+∞∫0

νσ 2t−1−ν

σ 2 + x21

W(−σ t−ν;−ν,1 − ν

)dσ, t > 0. (A.3)

After the change of variable: y = σ t−ν , we have

G(x1, t) = ν

2π

+∞∫0

y2t2ν−1

y2t2ν + x21

W(−y;−ν,1 − ν)dy

≤ const.t2ν−1

x21

+∞∫0

y2W(−y;−ν,1 − ν)dy. (A.4)

To evaluate the integral in the last inequality in (A.4), we apply equality (10) from [39]:

+∞∫0

zl−1W(−z;−β, δ)dz = Γ (l)

Γ (βl + δ), either l > 0,

if β > 0, or l > −1 if β = 0; δ ∈ (0,1), (A.5)

1832 M. Krasnoschok, N. Vasylyeva / J. Differential Equations 257 (2014) 1814–1839

where we put l := 3, β := ν, δ := 1 − ν and z := −y. Thus, we have got

G(x1, t) ≤ const.Γ (3)

Γ (1 + 2ν)

t2ν−1

x21

,

which means that the right inequality in (3.21) has been proved.To get the next statements of Lemma 3.1, we use essentially the following estimates of the

Wright functions:

W(−z;−β, δ) > 0 for ∀δ, z ≥ 0, (A.6)∣∣W(−z;−β, δ)∣∣ ≤ C

1 + |z|− δ−1β

if δ < 1, β > 0, (A.7)

W(−z;−β, δ) ≤ C1 exp(−C2z

11−β

) ∀z > 0, δ ∈ R1, (A.8)

where the constant C2 := C2(β) is bounded and can be chosen positive if β ∈ (0, 1) and either δ ≥ 1 or 0 < β ≤ δ < 1. The proof of inequalities (A.6) and (A.8) in the case of δ ≥ 1 is given in Lemmas 2.2.4 and 2.2.6 [38]. As for proving (A.8) if 0 < β ≤ δ < 1, it has been represented in Appendix A.1 of [27]. Inequality (A.7) follows from Lemma 3 of [39].

Note that the left inequality in (3.21) follows immediately from (A.6) and representa-tion (A.3).

To deduce statement (ii) of Lemma 3.1 in the case of k = 0, we do sequentially the change of variables:

x1 = σy, and σ = λtν; (A.9)

and have

+∞∫−∞

G(x1, t)dx1 = 1

π

+∞∫0

dσ t−1W(−σ t−ν;−ν,0

) +∞∫0

dy

1 + y2

= tν−1

+∞∫0

W(−λ;−ν,0)dλ = tν−1ν

+∞∫0

W(−λ;−ν,1 − ν)λdλ. (A.10)

Here we used (A.2) to get the last equality in (A.10). Then, the application of equality (A.5) with l = 2, z = λ, β = ν, δ = 1 − ν to (A.10) leads to (3.22) if k = 0.

It is easy to check that the function ∂G(x1,t)∂x1

is odd with respect to x1 and that is why ∫ +∞−∞

∂G(x1,t)∂x1

dx1 = 0.Next, one can verify that

I 1−νt

tν−1

Γ (ν)= 1. (A.11)

Thus, equality (3.23) follows immediately from (3.22) with k = 0 and (A.11).

M. Krasnoschok, N. Vasylyeva / J. Differential Equations 257 (2014) 1814–1839 1833

To infer statements (iv), (v), we need the well known formulas for a differentiation of the Wright functions (see, e.g. (8) and (9) in [39] or (11) in [26]; [48]):

d

dzW(z;−β, δ) = W(z;−β, δ − β), (A.12)

∂γt

[tδ−1W

(−ct−β;−β, δ)] = tδ−γ−1W

(−ct−β;−β, δ − γ), γ, δ ∈ R1. (A.13)

Thus, formula (A.13) with γ = νm allows us to calculate that

∂νmt

∂kG(x1, t)

∂xk1

= 1

2π

+∞∫0

t−1−νmW(−σ t−ν;−ν,−νm

) ∂k

∂xk1

σ

σ 2 + x21

dσ. (A.14)

Applying formula (0.432(1)) of [18] to ∂k

∂xk1

1σ 2+x2

1, one can get that

∣∣∣∣ ∂k

∂xk1

1

σ 2 + x21

∣∣∣∣ ≤ c(k)

[σ 2 + x21 ]1+ k

2

, (A.15)

where c(k) is a positive constant and depends on k. Thus, inequality (A.15) together with equal-ity (A.14) lead to

+∞∫0

∣∣∣∣∂νmt

∂kG(x1, t)

∂xk1

∣∣∣∣dt ≤ const.

+∞∫0

dσ

+∞∫0

t−1−νmW(−σ t−ν;−ν,−νm

)× σ

[σ 2 + x21 ]1+ k

2

dt.

Then the change of variable: t = ( λσ)−1/ν , in the inner integral allows us to deduce that

+∞∫0

∣∣∣∣∂νmt

∂kG(x1, t)

∂xk1

∣∣∣∣dt ≤ const.

+∞∫0

dσσ 1−m

[σ 2 + x21 ]1+ k

2

+∞∫0

W(−λ;−ν,−νm)λm−1dλ.

(A.16)

Due to the Wright formula (A.2) we can write that

W(−λ;−ν,−νm)λm−1 = νW(−λ;−ν,1 − ν) if m = 0. (A.17)

Then equalities (A.5), (A.17) lead to

+∞∫W(−λ;−ν,−νm)λm−1dλ ≤ const. m = 0. (A.18)

0

1834 M. Krasnoschok, N. Vasylyeva / J. Differential Equations 257 (2014) 1814–1839

To prove (A.18) in the case of m = 1, it is enough to use (A.7). Hence, applying the change of variables: σ = |x1|y, we get

+∞∫0

∣∣∣∣∂νmt

∂kG(x1, t)

∂xk1

∣∣∣∣dt ≤ const.

+∞∫0

y1−mdy

|x1|m+k(y2 + 1)1+ k2

. (A.19)

The integral on the right-hand side of (A.19) is bounded if m + k > 0, m = 0, 1, k = 0, 1, 2, ... .That proves inequality (3.24) in Lemma 3.1.

Next, we obtain inequality (3.25) from Lemma 3.1. To this end, we use representation (A.14)with νm := θ and estimate (A.15) and have

+∞∫0

∣∣∣∣∂θt

∂nG(x1, t)

∂xn1

∣∣∣∣xα1 dx1 ≤ const.

+∞∫0

dσ

+∞∫0

t−1−θ∣∣W (−σ t−ν;−ν,−θ

)∣∣ xα1 σdx1

(σ 2 + x21)

n+22

.

Then, the change of variables: x1 = ση, σ = tνλ, gives

+∞∫0

∣∣∣∣∂θt

∂nG(x1, t)

∂xn1

∣∣∣∣xα1 dx1 ≤ const.tν(α−n+1)−1−θ

+∞∫0

ηαdη

(1 + η2)n+2

2

×+∞∫0

λ(α+1−n)−1∣∣W(−λ;−ν,−θ)

∣∣dλ

≤ const.tν(α−n+1)−1−θ

+∞∫0

λ(α+1−n)−1∣∣W(−λ;−ν,−θ)

∣∣dλ

≡ const.tν(α−n+1)−1−θ I, if n ≥ 0, θ ∈ R1. (A.20)

Note that the restriction of the integral I in the case of θ = 0 and n = 0, 1, follows immediately from (A.5). If θ ≥ 1 and n = 0, 1, then estimate (A.6) ensures the boundedness of I .

At last, if either 0 < θ < 1, n = 1 or (α + 1)ν − 1 < θ < 1, n = 0, we use (A.7) to conclude that

I ≤ const. (A.21)

Hence, statement (v) of Lemma 3.1 follows from (A.21) and (A.20).Finally, we have to deduce inequality (3.26). First of all, we calculate ∂ν

t G. Equalities (A.13)and (A.12) give that

∂νt G(x1, t) = 1

2π

+∞∫t−ν−1W

(−σ t−ν;−ν,−ν) σdσ

σ 2 + x21

0

M. Krasnoschok, N. Vasylyeva / J. Differential Equations 257 (2014) 1814–1839 1835

= − 1

2π

+∞∫0

t−1 ∂

∂σW

(−σ t−ν;−ν,0) σdσ

σ 2 + x21

. (A.22)

Then, we integrate by parts (A.22) and take into account the equality ∂∂σ

σ

σ 2+x21

= − ∂∂x1

x1σ 2+x2

1and estimate (A.7). Thus, we get

∂νt G(x1, t) = − 1

2π

+∞∫0

t−1W(−σ t−ν;−ν,0

) ∂

∂x1

x1

σ 2 + x21

dσ. (A.23)

Using representation (A.23) and estimate (A.6), we can deduce that

t∫0

∣∣∣∣∣+∞∫ε

∂ντ G(y, τ )dy

∣∣∣∣∣dτ = 1

2π

t∫0

dττ−1

+∞∫0

W(−στ−ν;−ν,0

) ε

σ 2 + ε2dσ.

After the change of variables: λ = στ−ν and σ/ε = z, we have

t∫0

∣∣∣∣∣+∞∫ε

∂ντ G(y, τ )dy

∣∣∣∣∣dτ ≤ const.

+∞∫0

dz

1 + z2

+∞∫0

W(−λ;−ν,0)

λdλ ≤ C. (A.24)

To show the restriction of the integral ∫ +∞

0W(−λ;−ν,0)

λdλ it is necessary to apply (A.2) and

then (A.5). That finishes the proof of Lemma 3.1.

A.2. The getting of formula (3.38)

Here we give the proof of (3.38) in the case of k = 0, as the case of k = 1 is getting in the same way. To obtain representation (3.38), we need the following properties of the fractional derivative (see Lemma 2.10 and formula (2.4.10) in [22]):

i:

Dνt ϕ(t) = ∂ν

t ϕ(t) if ϕ(0) = 0, ν ∈ (0,1), (A.25)

where ∂νt ϕ(t) is given by (3.19).

ii: If the functions ϕi(t), i = 1, 2, are bounded in [0, T ], and ∂νt ϕ1(t) ∈ L1[0, T ], then

∂νt

t∫0

ϕ1(t − τ)ϕ2(τ )dτ =t∫

0

ϕ2(t − τ)∂νt ϕ1(τ )dτ + ϕ2(t) lim

z→0I 1−αz ϕ1(z), ∀t ∈ [0, T ],

(A.26)

where I νt ϕ1(t) is given by (3.18).

One can check that properties (A.25) and (A.26) hold if the functions ϕ, ϕi depend on x.

1836 M. Krasnoschok, N. Vasylyeva / J. Differential Equations 257 (2014) 1814–1839

Due to property (3.22) of the function G(x1, t), we can rewrite the function ρ(x1, t) as

ρ(x1, t) =t∫

0

dτ

+∞∫−∞

G(x1 − y, t − τ)[f2(y, τ ) − f2(x1, τ )

]dy + I ν

t f2(x1, t). (A.27)

Note that, inequality (3.27) means that ρ(x1, 0) = 0. Thus, equalities (A.25) and (A.27) allow us to represent Dν

t ρ(x1, t) as

Dνt ρ(x1, t) =

t∫0

dτ

+∞∫−∞

[f2(y, t − τ) − f2(x1, t − τ)

]∂ντ G(x1 − y, τ )dy + f2(x1, t)

++∞∫

−∞

[f2(y, t) − f2(x1, t)

]limz→0

I 1−νz G(x1 − y, z)dy. (A.28)

To get formula (3.38), it is enough to show that the last term in (A.28) vanishes.To this end, we use inequality (3.25) with n = 0, θ = ν − 1, and get

∣∣∣∣∣+∞∫

−∞

[f2(y, t) − f2(x1, t)

]limz→0

I 1−νz G(x1 − y, z)dy

∣∣∣∣∣≤ const.〈f2〉(α)

x1,R1T

limz→0

+∞∫0

yα∣∣∂ν−1

z G(y, z)∣∣dy

≤ const. limz→0

zν(α+1)+2−ν〈f2〉x1,R1T

= 0. (A.29)

A.3. The proof of Corollary 3.1

First, using (3.10) and (3.11), (3.14), we represent u(x1, x2, t) as

u(x1, x2, t) = 1

2π

t∫0

dτ

+∞∫−∞

dy

+∞∫0

σ + x2

(x1 − y)2 + (σ + x2)2τ−1W

(−στ−ν;−ν,0)

× [f2(y, t − τ) − f2(x1, t − τ)

]+ 1

2π

t∫0

dτf2(x1, t − τ)τ−1

+∞∫0

dσW(−στ−ν;−ν,0

)

×+∞∫

σ + x2

(x1 − y)2 + (σ + x2)2dy ≡ L1 +L2. (A.30)

−∞

M. Krasnoschok, N. Vasylyeva / J. Differential Equations 257 (2014) 1814–1839 1837

Then we show that

∂L2

∂x2= 0. (A.31)

To this end, it is enough to do the change of variables: x1−yσ+x2

= z, and rewrite L2 as

L2 = 1

π

t∫0

dτf2(x1, t − τ)

+∞∫0

dστ−1W(−στ−ν;−ν,0

) +∞∫0

dz

1 + z2. (A.32)

Due to (A.31), we can calculate ∂u∂x2

|x2=0:

∂u

∂x2

∣∣∣∣x2=0

= 1

2π

t∫0

dτ

+∞∫−∞

dy[f2(y, t − τ) − f2(x1, t − τ)

] +∞∫0

τ−1W(−στ−ν;−ν,0

)

× (x1 − y)2 − σ 2

[(x1 − y)2 + σ 2]2dσ. (A.33)

Next, we use formula (A.23) for ∂νt G(x1, t) and representation (3.38) for Dν

t ρ(x1, t), so that

Dνt ρ(x1, t) = f2(x1, t) + 1

2π

t∫0

dτ

+∞∫−∞

dy[f2(y, t − τ) − f2(x1, t − τ)

]

×+∞∫0

τ−1W(−στ−ν;−ν,0

) (x1 − y)2 − σ 2

[(x1 − y)2 + σ 2]2dσ. (A.34)

Thus, the statement of Corollary 3.1 follows from (A.33) and (A.34).

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