Open Access

Asymptotics for Nonlinear Evolution Equation with Module-Fractional Derivative on a Half-Line

Boundary Value Problems20102011:946143

DOI: 10.1155/2011/946143

Received: 22 April 2010

Accepted: 16 June 2010

Published: 29 June 2010

Abstract

We consider the initial-boundary value problem for a nonlinear partial differential equation with module-fractional derivative on a half-line. We study the local and global in time existence of solutions to the initial-boundary value problem and the asymptotic behavior of solutions for large time.

1. Introduction

We study the local and global existence and asymptotic behavior for solutions to the initial-boundary value problem:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ1_HTML.gif
(1.1)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq1_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq2_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq3_HTML.gif is the module-fractional derivative operator defined by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ2_HTML.gif
(1.2)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq4_HTML.gif is the Laplace transform for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq5_HTML.gif with respect to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq6_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq7_HTML.gif is the Heaviside function:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ3_HTML.gif
(1.3)

The Cauchy problem for a wide class of nonlinear nonlocal dissipative equations has been studied extensively. In particular, the general approach for the study of the large time asymptotics to the Cauchy problem for different nonlinear equations was investigated in the book [1] and the references therein.

The boundary value problems are more natural for applications and play an important role in the contemporary mathematical physics. However, their mathematical investigations are more complicated even in the case of the differential equations, with more reason to the case of nonlocal equations. We need to answer such basic question as how many boundary values should be given in the problem for its solvability and the uniqueness of the solution? Also it is interesting to study the influence of the boundary data on the qualitative properties of the solution. For examples and details see [212] and references therein.

The general theory of nonlinear nonlocal equations on a half-line was developed in book [13], where the pseudodifferential operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq8_HTML.gif on a half-line was introduced by virtue of the inverse Laplace transformation. In this definition it was important that the symbol https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq9_HTML.gif must be analytic in the complex right half-plane. We emphasize that the pseudodifferential operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq10_HTML.gif in (1.1) has a nonanalytic nonhomogeneous symbol https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq11_HTML.gif and the general theory from book [13] cannot be applied to the problem (1.1) directly. As far as we know there are few results on the initial-boundary value problems with pseudodifferential equations having a nonanalytic symbol. The case of rational symbol https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq12_HTML.gif which has some poles in the complex right half-plane was studied in [14, 15], where it was proposed a new method for constructing the Green operator based on the introduction of some necessary condition at the singular points of the symbol https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq13_HTML.gif . In [16] there was considered the initial-boundary value problem for a pseudodifferential equation with a nonanalytic homogeneous symbol https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq14_HTML.gif , where the theory of sectionally analytic functions was implemented for proving that the initial-boundary value problem is well posed. Since the symbol https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq15_HTML.gif does not grow fast at infinity, so there were no boundary data in the corresponding problem.

In the present paper we consider the same problem as in [16] but with symbol https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq16_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq17_HTML.gif . The approach used in this paper is more general and simple than the one used in [16]; however to get the same result are necessary more accurate estimates than the ones obtained here for the Green operator.

To construct Green operator we proposed a new method based on the integral representation for a sectionally analytic function and the theory of singular integro-differential equations with Hilbert kernel (see [16, 17]). We arrive to a boundary condition of type https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq18_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq19_HTML.gif . The aim is to find two analytic functions, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq20_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq21_HTML.gif (a sectionally analytic function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq22_HTML.gif ), in the left and right complex semi-planes, respectively, such that the boundary condition is satisfied. Two conditions are necessary to solve the problem: first, the function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq23_HTML.gif must satisfy the H https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq24_HTML.gif lder condition both in the finite points and in the vicinity of the infinite point of the contour and, second, the index of function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq25_HTML.gif must be zero. In our case both conditions do fail. To overcome this difficulty, we introduce an auxiliary function such that the H https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq26_HTML.gif lder and zero-index conditions are fulfilled.

To state precisely the results of the present paper we give some notations. We denote https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq27_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq28_HTML.gif . Here and below https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq29_HTML.gif is the main branch of the complex analytic function in the complex half-plane Re  https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq30_HTML.gif , so that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq31_HTML.gif (we make a cut along the negative real axis https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq32_HTML.gif ). Note that due to the analyticity of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq33_HTML.gif for all Re  https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq34_HTML.gif the inverse Laplace transform gives us the function which is equal to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq35_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq36_HTML.gif . Direct Laplace transformation https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq37_HTML.gif is
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ4_HTML.gif
(1.4)
and the inverse Laplace transformation https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq38_HTML.gif is defined by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ5_HTML.gif
(1.5)
Weighted Lebesgue space is https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq39_HTML.gif where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ6_HTML.gif
(1.6)
for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq40_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq41_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ7_HTML.gif
(1.7)
Now, we define the metric spaces
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ8_HTML.gif
(1.8)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq42_HTML.gif , with the norm
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ9_HTML.gif
(1.9)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq43_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq44_HTML.gif , with the norm
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ10_HTML.gif
(1.10)

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq45_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq46_HTML.gif .

Now we state the main results. We introduce https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq47_HTML.gif by formula
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ11_HTML.gif
(1.11)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ12_HTML.gif
(1.12)
We define the linear functional https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq48_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ13_HTML.gif
(1.13)

Theorem 1.1.

Suppose that for small https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq49_HTML.gif the initial data https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq50_HTML.gif are such that the norm https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq51_HTML.gif is sufficiently small. Then, there exists a unique global solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq52_HTML.gif to the initial-boundary value problem (1.1). Moreover the following asymptotic is valid:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ14_HTML.gif
(1.14)
for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq53_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq54_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq55_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ15_HTML.gif
(1.15)

2. Preliminaries

In subsequent consideration we shall have frequently to use certain theorems of the theory of functions of complex variable, the statements of which we now quote. The proofs may be found in all text-book of the theory. Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq56_HTML.gif be smooth contour and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq57_HTML.gif a function of position on it.

Definition 2.1.

The function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq58_HTML.gif is said to satisfy on the curve https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq59_HTML.gif the Hölder condition, if for two arbitrary points of this curve
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ16_HTML.gif
(2.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq60_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq61_HTML.gif are positive numbers.

Theorem 2.2.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq62_HTML.gif be a complex function, which obeys the Hölder condition for all finite https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq63_HTML.gif and tends to a definite limit https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq64_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq65_HTML.gif , such that for large https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq66_HTML.gif the following inequality holds:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ17_HTML.gif
(2.2)
Then Cauchy type integral
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ18_HTML.gif
(2.3)
constitutes a function analytic in the left and right semiplanes. Here and below these functions will be denoted by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq67_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq68_HTML.gif , respectively. These functions have the limiting values https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq69_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq70_HTML.gif at all points of imaginary axis https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq71_HTML.gif , on approaching the contour from the left and from the right, respectively. These limiting values are expressed by Sokhotzki-Plemelj formulae:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ19_HTML.gif
(2.4)
Subtracting and adding the formula (2.4) we obtain the following two equivalent formulae:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ20_HTML.gif
(2.5)

which will be frequently employed hereafter.

We consider the following linear initial-boundary value problem on half-line:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ21_HTML.gif
(2.6)
Setting https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq72_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq73_HTML.gif , we define
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ22_HTML.gif
(2.7)
where the function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq74_HTML.gif is given by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ23_HTML.gif
(2.8)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ24_HTML.gif
(2.9)
for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq75_HTML.gif . Here and below https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq76_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq77_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq78_HTML.gif are a left and right limiting values of sectionally analytic function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq79_HTML.gif given by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ25_HTML.gif
(2.10)
where for some fixed real point https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq80_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ26_HTML.gif
(2.11)

All the integrals are understood in the sense of the principal values.

Proposition 2.3.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq81_HTML.gif . Then there exists a unique solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq82_HTML.gif for the initial-boundary value problem (2.6), which has an integral representation:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ27_HTML.gif
(2.12)

Proof.

In order to obtain an integral representation for solutions of the problem (2.6) we suppose that there exist a solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq83_HTML.gif , which is continued by zero outside of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq84_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ28_HTML.gif
(2.13)
Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq85_HTML.gif be a function of the complex variable https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq86_HTML.gif , which obeys the Hölder condition for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq87_HTML.gif , such that Re https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq88_HTML.gif . We define the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq89_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ29_HTML.gif
(2.14)
Using the Laplace transform we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ30_HTML.gif
(2.15)
Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq90_HTML.gif is analytic for all Re  https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq91_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ31_HTML.gif
(2.16)
Therefore, applying the Laplace transform with respect to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq92_HTML.gif to problem (2.6) and using (2.15) and (2.16), we obtain for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq93_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ32_HTML.gif
(2.17)
We rewrite (2.17) in the form
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ33_HTML.gif
(2.18)
with some function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq94_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ34_HTML.gif
(2.19)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ35_HTML.gif
(2.20)
Applying the Laplace transform with respect to time variable to (2.18), we find
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ36_HTML.gif
(2.21)
where Re https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq95_HTML.gif and Re https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq96_HTML.gif . Here, the functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq97_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq98_HTML.gif are the Laplace transforms for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq99_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq100_HTML.gif with respect to time, respectively. In order to obtain an integral formula for solutions to the problem (2.6) it is necessary to know the function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq101_HTML.gif . We will find the function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq102_HTML.gif using the analytic properties of the function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq103_HTML.gif in the right-half complex planes Re https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq104_HTML.gif and Re https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq105_HTML.gif . Equation (2.16) and the Sokhotzki-Plemelj formulae imply for Re https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq106_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ37_HTML.gif
(2.22)
In view of Sokhotzki-Plemelj formulae via (2.21) the condition (2.22) can be written as
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ38_HTML.gif
(2.23)
where the sectionally analytic functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq107_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq108_HTML.gif are given by Cauchy type integrals:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ39_HTML.gif
(2.24)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ40_HTML.gif
(2.25)
To perform the condition (2.23) in the form of a nonhomogeneous Riemann-Hilbert problem we introduce the sectionally analytic function:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ41_HTML.gif
(2.26)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ42_HTML.gif
(2.27)
Taking into account the assumed condition (2.19), we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ43_HTML.gif
(2.28)
Also observe that from (2.24) and (2.26) by Sokhotzki-Plemelj formulae,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ44_HTML.gif
(2.29)
Substituting (2.23) and (2.28) into this equation we obtain for Re https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq109_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ45_HTML.gif
(2.30)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ46_HTML.gif
(2.31)

Equation (2.30) is the boundary condition for a nonhomogeneous Riemann-Hilbert problem. It is required to find two functions for some fixed point https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq110_HTML.gif , Re https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq111_HTML.gif : https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq112_HTML.gif , analytic in the left-half complex plane Re  https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq113_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq114_HTML.gif , analytic in the right-half complex plane Re  https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq115_HTML.gif , which satisfy on the contour Re  https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq116_HTML.gif the relation (2.30).

Note that bearing in mind formula (2.27) we can find the unknown function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq117_HTML.gif , which involved in the formula (2.21), by the relation
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ47_HTML.gif
(2.32)

The method for solving the Riemann problem https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq118_HTML.gif is based on the following results. The proofs may be found in [17].

Lemma 2.4.

An arbitrary function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq119_HTML.gif given on the contour https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq120_HTML.gif , satisfying the Hölder condition, can be uniquely represented in the form
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ48_HTML.gif
(2.33)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq121_HTML.gif are the boundary values of the analytic functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq122_HTML.gif and the condition https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq123_HTML.gif holds. These functions are determined by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ49_HTML.gif
(2.34)

Lemma 2.5.

An arbitrary function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq124_HTML.gif given on the contour https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq125_HTML.gif , satisfying the Hölder condition, and having zero index,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ50_HTML.gif
(2.35)
is uniquely representable as the ratio of the functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq126_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq127_HTML.gif , constituting the boundary values of functions, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq128_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq129_HTML.gif , analytic in the left and right complex semiplane and having in these domains no zero. These functions are determined to within an arbitrary constant factor and given by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ51_HTML.gif
(2.36)
In the formulations of Lemmas 2.4 and 2.5 the coefficient https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq130_HTML.gif and the free term https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq131_HTML.gif of the Riemann problem are required to satisfy the H https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq132_HTML.gif lder condition on the contour Re  https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq133_HTML.gif . This restriction is essential. On the other hand, it is easy to observe that both functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq134_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq135_HTML.gif do not have limiting value as https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq136_HTML.gif . So we cannot find the solution using https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq137_HTML.gif . The principal task now is to get an expression equivalent to the boundary value problem (2.30), such that the conditions of lemmas are satisfied. First, we introduce the function
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ52_HTML.gif
(2.37)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq138_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq139_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq140_HTML.gif . We make a cut in the plane https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq141_HTML.gif from point https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq142_HTML.gif to point https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq143_HTML.gif through https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq144_HTML.gif . Owing to the manner of performing the cut the functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq145_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq146_HTML.gif are analytic for Re https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq147_HTML.gif and the function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq148_HTML.gif is analytic for Re https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq149_HTML.gif .

We observe that the function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq150_HTML.gif , given on the contour Re https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq151_HTML.gif , satisfies the Hölder condition and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq152_HTML.gif does not vanish for any Re https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq153_HTML.gif . Also we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ53_HTML.gif
(2.38)
Therefore in accordance with Lemma 2.5 the function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq154_HTML.gif can be represented in the form of the ratio
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ54_HTML.gif
(2.39)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ55_HTML.gif
(2.40)
From (2.37) and (2.39) we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ56_HTML.gif
(2.41)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq155_HTML.gif . We note that (2.41) is equivalent to
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ57_HTML.gif
(2.42)
Now, we return to the nonhomogeneous Riemann-Hilbert problem defined by the boundary condition (2.30). We substitute the above equation in (2.30) and add https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq156_HTML.gif in both sides to get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ58_HTML.gif
(2.43)
On the other hand, by Sokhotzki-Plemelj formulae and (2.25), https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq157_HTML.gif . Now, we substitute https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq158_HTML.gif from this equation in formula (2.43); then by (2.41) we arrive to
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ59_HTML.gif
(2.44)

In subsequent consideration we shall have to use the following property of the limiting values of a Cauchy type integral, the statement of which we now quote. The proofs may be found in [17].

Lemma 2.6.

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq159_HTML.gif is a smooth closed contour and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq160_HTML.gif a function that satisfies the Hölder condition on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq161_HTML.gif , then the limiting values of the Cauchy type integral
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ60_HTML.gif
(2.45)

also satisfy this condition.

Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq162_HTML.gif satisfies on Re https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq163_HTML.gif the Hölder condition, on basis of Lemma 2.6 the function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq164_HTML.gif also satisfies this condition. Therefore, in accordance with Lemma 2.4 it can be uniquely represented in the form of the difference of the functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq165_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq166_HTML.gif , constituting the boundary values of the analytic function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq167_HTML.gif , given by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ61_HTML.gif
(2.46)
Therefore, (2.44) takes the form
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ62_HTML.gif
(2.47)
The last relation indicates that the function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq168_HTML.gif , analytic in Re  https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq169_HTML.gif , and the function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq170_HTML.gif , analytic in Re  https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq171_HTML.gif , constitute the analytic continuation of each other through the contour Re https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq172_HTML.gif . Consequently, they are branches of a unique analytic function in the entire plane. According to Liouville theorem this function is some arbitrary constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq173_HTML.gif . Thus, we obtain the solution of the Riemann-Hilbert problem defined by the boundary condition (2.30):
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ63_HTML.gif
(2.48)
Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq174_HTML.gif is defined by a Cauchy type integral, with density https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq175_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq176_HTML.gif , as https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq177_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq178_HTML.gif . Using this property in (2.48) we get https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq179_HTML.gif and the limiting values for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq180_HTML.gif are given by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ64_HTML.gif
(2.49)
Now, we proceed to find the unknown function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq181_HTML.gif involved in the formula (2.21) for the solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq182_HTML.gif of the problem (2.6). First, we represent https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq183_HTML.gif as the limiting value of analytic functions on the left-hand side complex semiplane. From (2.41) and Sokhotzki-Plemelj formulae we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ65_HTML.gif
(2.50)
Now, making use of (2.49) and the above equation, we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ66_HTML.gif
(2.51)
Thus, by formula (2.32),
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ67_HTML.gif
(2.52)
We observe that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq184_HTML.gif is boundary value of a function analytic in the left-hand side complex semi-plane and therefore satisfies our basic assumption (2.19). Having determined the function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq185_HTML.gif , bearing in mind formula (2.21) we determine the required function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq186_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ68_HTML.gif
(2.53)
Now we prove that, in accordance with last relation, the function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq187_HTML.gif constitutes the limiting value of an analytic function in Re  https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq188_HTML.gif . In fact, making use of Sokhotzki-Plemelj formulae and using (2.41), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ69_HTML.gif
(2.54)

Thus, the function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq189_HTML.gif is the limiting value of an analytic function in Re  https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq190_HTML.gif . We note the fundamental importance of the proven fact, the solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq191_HTML.gif constitutes an analytic function in Re https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq192_HTML.gif , and, as a consequence, its inverse Laplace transform vanishes for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq193_HTML.gif . We now return to solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq194_HTML.gif of the problem (2.6). Taking inverse Laplace transform with respect to time and space variables, we obtain https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq195_HTML.gif where the function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq196_HTML.gif is defined by formula (2.8). Thus, Proposition 2.3 has been proved.

Now we collect some preliminary estimates of the Green operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq197_HTML.gif .

Lemma 2.7.

The following estimates are true, provided that the right-hand sides are finite:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ70_HTML.gif
(2.55)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq198_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq199_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq200_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq201_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq202_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq203_HTML.gif are given by (1.13) and (1.11), respectively.

Proof.

First, we estimate the function
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ71_HTML.gif
(2.56)
We note that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq204_HTML.gif , as https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq205_HTML.gif , and write https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq206_HTML.gif in the form
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ72_HTML.gif
(2.57)
For first integral in (2.57), we obtain the estimate
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ73_HTML.gif
(2.58)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq207_HTML.gif , and for second integral we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ74_HTML.gif
(2.59)
Therefore, substituting (2.58) and (2.59) in (2.57), we get for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq208_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ75_HTML.gif
(2.60)
Now, we estimate function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq209_HTML.gif defined by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ76_HTML.gif
(2.61)
Using (2.60), we get for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq210_HTML.gif the estimate
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ77_HTML.gif
(2.62)
where Re  https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq211_HTML.gif . Then, by (2.62) and Cauchy Theorem,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ78_HTML.gif
(2.63)
Equations (2.63) imply that we can write https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq212_HTML.gif in the form
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ79_HTML.gif
(2.64)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ80_HTML.gif
(2.65)
Thus, for Re  https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq213_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ81_HTML.gif
(2.66)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq214_HTML.gif satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ82_HTML.gif
(2.67)
In fact, we use (2.62) and inequality https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq215_HTML.gif , where Re  https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq216_HTML.gif Re  https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq217_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq218_HTML.gif , to obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ83_HTML.gif
(2.68)
Making the change of variable https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq219_HTML.gif , (2.67) follows. Now, substituting (2.66) in (2.8), for Green function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq220_HTML.gif , we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ84_HTML.gif
(2.69)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ85_HTML.gif
(2.70)
The function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq221_HTML.gif defined in (2.70) satisfies the estimate
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ86_HTML.gif
(2.71)
In fact, using (2.67) we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ87_HTML.gif
(2.72)
Here,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ88_HTML.gif
(2.73)
We have used inequality https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq222_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq223_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq224_HTML.gif is some positive constant. Taking https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq225_HTML.gif and making the change of variables: https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq226_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq227_HTML.gif , we obtain (2.71). Now, let us split (2.69):
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ89_HTML.gif
(2.74)
By Fubini's theorem and Cauchy's theorem, from the first and fourth summands we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ90_HTML.gif
(2.75)
Then,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ91_HTML.gif
(2.76)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ92_HTML.gif
(2.77)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ93_HTML.gif
(2.78)
Now, we show that function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq228_HTML.gif defined by (2.78) satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ94_HTML.gif
(2.79)
In fact, using (2.62) and the inequality https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq229_HTML.gif , where Re https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq230_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq231_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ95_HTML.gif
(2.80)
Then, taking https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq232_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq233_HTML.gif and making the change of variable https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq234_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq235_HTML.gif , we obtain (2.79). In the same way, we show that function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq236_HTML.gif defined in (2.78) satisfies the inequality:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ96_HTML.gif
(2.81)
In fact, using the inequality https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq237_HTML.gif , where Re https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq238_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq239_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ97_HTML.gif
(2.82)
Making the change of variable https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq240_HTML.gif , we arrive to
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ98_HTML.gif
(2.83)
Thus, (2.81) follows. Finally, we show that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ99_HTML.gif
(2.84)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq241_HTML.gif is given by (1.11). Making the change of variable https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq242_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq243_HTML.gif , and choosing https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq244_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ100_HTML.gif
(2.85)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ101_HTML.gif
(2.86)
Now, making the change of variable https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq245_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq246_HTML.gif in equation for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq247_HTML.gif we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ102_HTML.gif
(2.87)
Therefore, (2.84) follows. Finally, using estimates (2.71), (2.79), (2.81), and (2.84), we get the asymptotic for the Green function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq248_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ103_HTML.gif
(2.88)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq249_HTML.gif is given by (1.11) and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq250_HTML.gif . By last inequality
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ104_HTML.gif
(2.89)
Therefore,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ105_HTML.gif
(2.90)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq251_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq252_HTML.gif are given by (1.13) and (1.11), respectively, and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq253_HTML.gif . Thus, the first estimate in Lemma 2.7 has been proved.

Now, we are going to prove the second estimate in Lemma 2.7. First, for large https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq254_HTML.gif , using Sokhotzki-Plemelj formulae, we have for function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq255_HTML.gif , defined in (2.61),
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ106_HTML.gif
(2.91)
Substituting last equation in (2.8), we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ107_HTML.gif
(2.92)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ108_HTML.gif
(2.93)
Making the change of variable https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq256_HTML.gif we get for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq257_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ109_HTML.gif
(2.94)
To estimate https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq258_HTML.gif , we consider an extension to the function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq259_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ110_HTML.gif
(2.95)
and we use the contours
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ111_HTML.gif
(2.96)
to obtain for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq260_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ112_HTML.gif
(2.97)
Let us write the function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq261_HTML.gif , defined in (2.61), in the form
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ113_HTML.gif
(2.98)
where Re  https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq262_HTML.gif . Then, by Cauchy Theorem, for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq263_HTML.gif the second summand in last equation is zero. Thus, using (2.62) we obtain for Re https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq264_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ114_HTML.gif
(2.99)
From the last inequality and (2.97) we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ115_HTML.gif
(2.100)
Taking https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq265_HTML.gif and making the change of variables https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq266_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq267_HTML.gif , in the last inequality, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ116_HTML.gif
(2.101)
From (2.92) and the estimates (2.94) and (2.101) we get the estimate https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq268_HTML.gif . Thus,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ117_HTML.gif
(2.102)
Now, for small https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq269_HTML.gif , we are going to prove the estimate
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ118_HTML.gif
(2.103)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq270_HTML.gif . First, we rewrite the Green function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq271_HTML.gif in the form
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ119_HTML.gif
(2.104)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ120_HTML.gif
(2.105)
The contours https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq272_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq273_HTML.gif are defined in (2.96) and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ121_HTML.gif
(2.106)
Moreover, we have extended the function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq274_HTML.gif as in (2.95). Making the change of variable https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq275_HTML.gif and using the inequality https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq276_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq277_HTML.gif , we obtain the estimate https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq278_HTML.gif or
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ122_HTML.gif
(2.107)
for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq279_HTML.gif . Now, we estimate https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq280_HTML.gif . Using https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq281_HTML.gif , for Re  https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq282_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq283_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq284_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ123_HTML.gif
(2.108)
Making the change o variables https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq285_HTML.gif    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq286_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq287_HTML.gif , into the last inequality, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ124_HTML.gif
(2.109)
By (2.104) and the estimates (2.107) and (2.109) we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ125_HTML.gif
(2.110)
Thus,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ126_HTML.gif
(2.111)

Thus, we get (2.103) and the second estimate in Lemma 2.7 has been proved.

Let us introduce the operators
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ127_HTML.gif
(2.112)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ128_HTML.gif
(2.113)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq288_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq289_HTML.gif are defined in (2.104). Then, the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq290_HTML.gif can be written in the form
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ129_HTML.gif
(2.114)
Now, we are going to prove the third estimate in Lemma 2.7,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ130_HTML.gif
(2.115)
First, we estimate the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq291_HTML.gif . Making the change of variable https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq292_HTML.gif , we get for the function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq293_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ131_HTML.gif
(2.116)
Now, we make the change of variable https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq294_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ132_HTML.gif
(2.117)
Integrating by parts the last equation we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ133_HTML.gif
(2.118)
Then,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ134_HTML.gif
(2.119)
for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq295_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq296_HTML.gif are defined as above. Thus, for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq297_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ135_HTML.gif
(2.120)
Therefore, from the inequalities (2.116) and (2.120) we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ136_HTML.gif
(2.121)

We remember some well-known inequalities.

(i)Young's Inequality. Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq298_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq299_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq300_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq301_HTML.gif . Then, the convolution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq302_HTML.gif belongs to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq303_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq304_HTML.gif and Young's inequality
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ137_HTML.gif
(2.122)

holds.

(ii)Minkowski's Inequality. Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq305_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq306_HTML.gif ; then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ138_HTML.gif
(2.123)
(iii)Interpolation Inequality. Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq307_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq308_HTML.gif ; then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq309_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq310_HTML.gif , and the interpolation inequality holds:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ139_HTML.gif
(2.124)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq311_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq312_HTML.gif .

(iv)Arithmetic-Geometric Mean Inequality. If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq313_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq314_HTML.gif are nonnegative, then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ140_HTML.gif
(2.125)
Then, by (2.121) and Young's inequality (2.122), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ141_HTML.gif
(2.126)
since
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ142_HTML.gif
(2.127)
Finally, using the Interpolation Inequality (2.124) and the arithmetic-geometric mean inequality (2.125), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ143_HTML.gif
(2.128)
Therefore,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ144_HTML.gif
(2.129)
Now, we estimate the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq315_HTML.gif . First, by Cauchy Theorem we get for Re  https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq316_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ145_HTML.gif
(2.130)
By (2.41) we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ146_HTML.gif
(2.131)
Then, using (2.131) and the inequalities https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq317_HTML.gif , where Re  https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq318_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq319_HTML.gif , and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ147_HTML.gif
(2.132)
we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ148_HTML.gif
(2.133)
Then, using the inequalities (2.132) and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq320_HTML.gif we obtain for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq321_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ149_HTML.gif
(2.134)
Therefore,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ150_HTML.gif
(2.135)

Thus, the last estimate and (2.129) imply the third estimate in Lemma 2.7.

Now, we are going to prove the fourth estimate in Lemma 2.7. We use (2.114). First, we estimate the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq322_HTML.gif , defined in (2.112). Using the inequality https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq323_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq324_HTML.gif , and Minkowski's inequality (2.123), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ151_HTML.gif
(2.136)
Then, Young's inequality (2.122) implies
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ152_HTML.gif
(2.137)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq325_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq326_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq327_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq328_HTML.gif . Then, by the inequality (2.121) and the change of variables https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq329_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ153_HTML.gif
(2.138)
Thus, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq330_HTML.gif , provided https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq331_HTML.gif . Using https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq332_HTML.gif , it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ154_HTML.gif
(2.139)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq333_HTML.gif . We note that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq334_HTML.gif , since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq335_HTML.gif . Substituting (2.139) in (2.137), we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ155_HTML.gif
(2.140)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq336_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq337_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq338_HTML.gif . Now, we estimate the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq339_HTML.gif , defined in (2.113). First, we use that function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq340_HTML.gif satisfies the following inequality:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ156_HTML.gif
(2.141)
Then, by the inequality https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq341_HTML.gif , we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ157_HTML.gif
(2.142)
Substituting in the last inequality the estimate
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ158_HTML.gif
(2.143)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq342_HTML.gif , we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ159_HTML.gif
(2.144)
Then, using https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq343_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq344_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ160_HTML.gif
(2.145)
Therefore,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ161_HTML.gif
(2.146)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq345_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq346_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq347_HTML.gif . Finally, from estimates (2.140) and (2.146) we obtain the fourth estimate in Lemma 2.7. Then, we have proved Lemma 2.7.

Theorem 2.8.

Let the initial data be https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq348_HTML.gif , with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq349_HTML.gif . Then, for some https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq350_HTML.gif there exists a unique solution
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ162_HTML.gif
(2.147)

to the initial boundary-value problem (1.1). Moreover, the existence time https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq351_HTML.gif can be chosen as follows: https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq352_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq353_HTML.gif .

3. Proof of Theorem 1.1

By the Local Existence Theorem 2.8, it follows that the global solution (if it exist) is unique. Indeed, on the contrary, we suppose that there exist two global solutions with the same initial data. And these solutions are different at some time https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq354_HTML.gif . By virtue of the continuity of solutions with respect to time, we can find a maximal time segment https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq355_HTML.gif , where the solutions are equal, but for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq356_HTML.gif they are different. Now, we apply the local existence theorem taking the initial time https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq357_HTML.gif and obtain that these solutions coincide on some interval https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq358_HTML.gif , which give us a contradiction with the fact that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq359_HTML.gif is the maximal time of coincidence. So our main purpose in the proof of Theorem 1.1 is to show the global in time existence of solutions.

First, we note that Lemma 2.7 implies for the Green operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq360_HTML.gif the inequality https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq361_HTML.gif Now, we show the estimate
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ163_HTML.gif
(3.1)
for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq362_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq363_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq364_HTML.gif . In fact, using the inequality
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ164_HTML.gif
(3.2)
we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ165_HTML.gif
(3.3)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq365_HTML.gif , and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ166_HTML.gif
(3.4)
Then, the estimates (3.3), (3.4), and Lemma 2.7 imply
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ167_HTML.gif
(3.5)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq366_HTML.gif , and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ168_HTML.gif
(3.6)
Now, we integrate with respect to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq367_HTML.gif , on the interval https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq368_HTML.gif , the inequalities (3.5) and (3.6). Then, we get for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq369_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ169_HTML.gif
(3.7)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ170_HTML.gif
(3.8)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq370_HTML.gif , and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ171_HTML.gif
(3.9)
Then, the definition of the norm on the space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq371_HTML.gif and the estimates (3.7), (3.8), and (3.9) imply (3.1). Now, we apply the Contraction Mapping Principle on a ball with ratio https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq372_HTML.gif in the space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq373_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq374_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq375_HTML.gif . Here, the constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq376_HTML.gif coincides with the one that appears in estimate (3.1). First, we show that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ172_HTML.gif
(3.10)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq377_HTML.gif . Indeed, from the integral formula
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ173_HTML.gif
(3.11)
and the estimate (3.1) (with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq378_HTML.gif ), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ174_HTML.gif
(3.12)
since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq379_HTML.gif is sufficient small. Therefore, the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq380_HTML.gif transforms a ball of ratio https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq381_HTML.gif into itself, in the space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq382_HTML.gif . In the same way we estimate the difference of two functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq383_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ175_HTML.gif
(3.13)
since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq384_HTML.gif is sufficient small. Thus, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq385_HTML.gif is a contraction mapping in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq386_HTML.gif . Therefore, there exists a unique solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq387_HTML.gif to the Cauchy problem (1.1). Now we can prove asymptotic formula:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ176_HTML.gif
(3.14)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq388_HTML.gif . We denote https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq389_HTML.gif . From Lemma 2.7 we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ177_HTML.gif
(3.15)
for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq390_HTML.gif Also in view of the definition of the norm https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq391_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ178_HTML.gif
(3.16)
By a direct calculation we have for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq392_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ179_HTML.gif
(3.17)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq393_HTML.gif , provided that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq394_HTML.gif , and in the same way
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ180_HTML.gif
(3.18)
provided that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq395_HTML.gif . Also we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ181_HTML.gif
(3.19)
for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq396_HTML.gif By virtue of the integral equation (3.11) we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ182_HTML.gif
(3.20)

All summands in the right-hand side of (3.20) are estimated by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq397_HTML.gif via estimates (3.17)–(3.19). Thus by (3.20) the asymptotic (3.14) is valid. Theorem 1.1 is proved.

Authors’ Affiliations

(1)
Instituto de Matemáticas, UNAM

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© Martín P. Árciga A. 2011

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