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:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ1_HTML.gif
(1.1)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq1_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq2_HTML.gif , and http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ2_HTML.gif
(1.2)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq4_HTML.gif is the Laplace transform for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq5_HTML.gif with respect to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq6_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq7_HTML.gif is the Heaviside function:
http://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 http://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 http://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 http://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 http://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 http://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 http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq16_HTML.gif , where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq18_HTML.gif , where http://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, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq20_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq21_HTML.gif (a sectionally analytic function http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq23_HTML.gif must satisfy the H http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq27_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq28_HTML.gif . Here and below http://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  http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq30_HTML.gif , so that http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq33_HTML.gif for all Re  http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq35_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq36_HTML.gif . Direct Laplace transformation http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq37_HTML.gif is
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq38_HTML.gif is defined by
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ5_HTML.gif
(1.5)
Weighted Lebesgue space is http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq39_HTML.gif where
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ6_HTML.gif
(1.6)
for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq40_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq41_HTML.gif and
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ8_HTML.gif
(1.8)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq42_HTML.gif , with the norm
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ9_HTML.gif
(1.9)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq43_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq44_HTML.gif , with the norm
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ10_HTML.gif
(1.10)

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq45_HTML.gif , http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq47_HTML.gif by formula
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ11_HTML.gif
(1.11)
where
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq48_HTML.gif :
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq49_HTML.gif the initial data http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq50_HTML.gif are such that the norm http://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 http://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:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ14_HTML.gif
(1.14)
for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq53_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq54_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq55_HTML.gif and
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq56_HTML.gif be smooth contour and http://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 http://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 http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ16_HTML.gif
(2.1)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq60_HTML.gif and http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq63_HTML.gif and tends to a definite limit http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq64_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq65_HTML.gif , such that for large http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq66_HTML.gif the following inequality holds:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ17_HTML.gif
(2.2)
Then Cauchy type integral
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq67_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq69_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq70_HTML.gif at all points of imaginary axis http://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:
http://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:
http://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:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ21_HTML.gif
(2.6)
Setting http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq72_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq73_HTML.gif , we define
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ22_HTML.gif
(2.7)
where the function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq74_HTML.gif is given by
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ23_HTML.gif
(2.8)
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ24_HTML.gif
(2.9)
for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq75_HTML.gif . Here and below http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq76_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq77_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq79_HTML.gif given by
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq80_HTML.gif ,
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq81_HTML.gif . Then there exists a unique solution http://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:
http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq84_HTML.gif :
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ28_HTML.gif
(2.13)
Let http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq87_HTML.gif , such that Re http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq88_HTML.gif . We define the operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq89_HTML.gif by
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ30_HTML.gif
(2.15)
Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq90_HTML.gif is analytic for all Re  http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq91_HTML.gif , we have
http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq93_HTML.gif
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ33_HTML.gif
(2.18)
with some function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq94_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ34_HTML.gif
(2.19)
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ36_HTML.gif
(2.21)
where Re http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq95_HTML.gif and Re http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq96_HTML.gif . Here, the functions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq97_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq98_HTML.gif are the Laplace transforms for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq99_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq101_HTML.gif . We will find the function http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq104_HTML.gif and Re http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq106_HTML.gif
http://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
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq107_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq108_HTML.gif are given by Cauchy type integrals:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ39_HTML.gif
(2.24)
http://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:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ41_HTML.gif
(2.26)
where
http://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
http://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,
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq109_HTML.gif
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ45_HTML.gif
(2.30)
where
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq110_HTML.gif , Re http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq111_HTML.gif : http://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  http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq113_HTML.gif and http://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  http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq115_HTML.gif , which satisfy on the contour Re  http://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 http://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
http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq119_HTML.gif given on the contour http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ48_HTML.gif
(2.33)
where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq122_HTML.gif and the condition http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq123_HTML.gif holds. These functions are determined by
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq124_HTML.gif given on the contour http://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,
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq126_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq127_HTML.gif , constituting the boundary values of functions, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq128_HTML.gif and http://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
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq130_HTML.gif and the free term http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq132_HTML.gif lder condition on the contour Re  http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq134_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq135_HTML.gif do not have limiting value as http://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 http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ52_HTML.gif
(2.37)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq138_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq139_HTML.gif , and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq141_HTML.gif from point http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq142_HTML.gif to point http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq143_HTML.gif through http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq145_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq146_HTML.gif are analytic for Re http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq147_HTML.gif and the function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq148_HTML.gif is analytic for Re http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq149_HTML.gif .

We observe that the function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq150_HTML.gif , given on the contour Re http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq151_HTML.gif , satisfies the Hölder condition and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq152_HTML.gif does not vanish for any Re http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq153_HTML.gif . Also we have
http://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 http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ54_HTML.gif
(2.39)
where
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ56_HTML.gif
(2.41)
where http://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
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq156_HTML.gif in both sides to get
http://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), http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq157_HTML.gif . Now, we substitute http://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
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq159_HTML.gif is a smooth closed contour and http://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 http://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
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq162_HTML.gif satisfies on Re http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq165_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq167_HTML.gif , given by
http://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
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq168_HTML.gif , analytic in Re  http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq169_HTML.gif , and the function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq170_HTML.gif , analytic in Re  http://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 http://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 http://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):
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ63_HTML.gif
(2.48)
Since http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq175_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq176_HTML.gif , as http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq177_HTML.gif for http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq179_HTML.gif and the limiting values for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq180_HTML.gif are given by
http://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 http://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 http://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 http://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
http://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
http://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),
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ67_HTML.gif
(2.52)
We observe that http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq186_HTML.gif :
http://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 http://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  http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ69_HTML.gif
(2.54)

Thus, the function http://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  http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq191_HTML.gif constitutes an analytic function in Re http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq193_HTML.gif . We now return to solution http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq195_HTML.gif where the function http://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 http://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:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ70_HTML.gif
(2.55)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq198_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq199_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq200_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq201_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq202_HTML.gif and http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ71_HTML.gif
(2.56)
We note that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq204_HTML.gif , as http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq205_HTML.gif , and write http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq206_HTML.gif in the form
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ73_HTML.gif
(2.58)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq207_HTML.gif , and for second integral we have
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq208_HTML.gif
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ75_HTML.gif
(2.60)
Now, we estimate function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq209_HTML.gif defined by
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq210_HTML.gif the estimate
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ77_HTML.gif
(2.62)
where Re  http://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,
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq212_HTML.gif in the form
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ79_HTML.gif
(2.64)
where
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ80_HTML.gif
(2.65)
Thus, for Re  http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq213_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ81_HTML.gif
(2.66)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq214_HTML.gif satisfies
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq215_HTML.gif , where Re  http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq216_HTML.gif Re  http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq217_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq218_HTML.gif , to obtain
http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq220_HTML.gif , we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ84_HTML.gif
(2.69)
where
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ85_HTML.gif
(2.70)
The function http://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
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ87_HTML.gif
(2.72)
Here,
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ88_HTML.gif
(2.73)
We have used inequality http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq222_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq223_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq224_HTML.gif is some positive constant. Taking http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq225_HTML.gif and making the change of variables: http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq226_HTML.gif and http://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):
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ90_HTML.gif
(2.75)
Then,
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ91_HTML.gif
(2.76)
where
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ92_HTML.gif
(2.77)
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq228_HTML.gif defined by (2.78) satisfies
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq229_HTML.gif , where Re http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq230_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq231_HTML.gif , we get
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ95_HTML.gif
(2.80)
Then, taking http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq232_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq233_HTML.gif and making the change of variable http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq234_HTML.gif and http://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 http://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:
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq237_HTML.gif , where Re http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq238_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq239_HTML.gif , we get
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq240_HTML.gif , we arrive to
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ99_HTML.gif
(2.84)
where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq242_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq243_HTML.gif , and choosing http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq244_HTML.gif , we get
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ100_HTML.gif
(2.85)
where
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq245_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq246_HTML.gif in equation for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq247_HTML.gif we obtain
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq248_HTML.gif :
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ103_HTML.gif
(2.88)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq249_HTML.gif is given by (1.11) and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq250_HTML.gif . By last inequality
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ104_HTML.gif
(2.89)
Therefore,
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ105_HTML.gif
(2.90)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq251_HTML.gif and http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq255_HTML.gif , defined in (2.61),
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ107_HTML.gif
(2.92)
where
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq256_HTML.gif we get for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq257_HTML.gif
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ109_HTML.gif
(2.94)
To estimate http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq259_HTML.gif :
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ111_HTML.gif
(2.96)
to obtain for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq260_HTML.gif
http://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 http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ113_HTML.gif
(2.98)
where Re  http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq262_HTML.gif . Then, by Cauchy Theorem, for http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq264_HTML.gif
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ115_HTML.gif
(2.100)
Taking http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq265_HTML.gif and making the change of variables http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq266_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq267_HTML.gif , in the last inequality, we obtain
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq268_HTML.gif . Thus,
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ117_HTML.gif
(2.102)
Now, for small http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ118_HTML.gif
(2.103)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq270_HTML.gif . First, we rewrite the Green function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq271_HTML.gif in the form
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ119_HTML.gif
(2.104)
where
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ120_HTML.gif
(2.105)
The contours http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq272_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq273_HTML.gif are defined in (2.96) and
http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq275_HTML.gif and using the inequality http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq276_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq277_HTML.gif , we obtain the estimate http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq278_HTML.gif or
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ122_HTML.gif
(2.107)
for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq279_HTML.gif . Now, we estimate http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq280_HTML.gif . Using http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq281_HTML.gif , for Re  http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq282_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq283_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq284_HTML.gif , we get
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq285_HTML.gif    http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq286_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq287_HTML.gif , into the last inequality, we obtain
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ125_HTML.gif
(2.110)
Thus,
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ127_HTML.gif
(2.112)
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ128_HTML.gif
(2.113)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq288_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq290_HTML.gif can be written in the form
http://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,
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq291_HTML.gif . Making the change of variable http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq292_HTML.gif , we get for the function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq293_HTML.gif
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq294_HTML.gif :
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ133_HTML.gif
(2.118)
Then,
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ134_HTML.gif
(2.119)
for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq295_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq296_HTML.gif are defined as above. Thus, for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq297_HTML.gif ,
http://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
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq298_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq299_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq300_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq301_HTML.gif . Then, the convolution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq302_HTML.gif belongs to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq303_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq304_HTML.gif and Young's inequality
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq305_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq306_HTML.gif ; then
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ138_HTML.gif
(2.123)
(iii)Interpolation Inequality. Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq307_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq308_HTML.gif ; then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq309_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq310_HTML.gif , and the interpolation inequality holds:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ139_HTML.gif
(2.124)

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

(iv)Arithmetic-Geometric Mean Inequality. If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq313_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq314_HTML.gif are nonnegative, then
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ141_HTML.gif
(2.126)
since
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ143_HTML.gif
(2.128)
Therefore,
http://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 http://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  http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq316_HTML.gif
http://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
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq317_HTML.gif , where Re  http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq318_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq319_HTML.gif , and
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ147_HTML.gif
(2.132)
we obtain
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq320_HTML.gif we obtain for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq321_HTML.gif
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ149_HTML.gif
(2.134)
Therefore,
http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq323_HTML.gif , where http://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
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ152_HTML.gif
(2.137)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq325_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq326_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq327_HTML.gif , and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq329_HTML.gif , we get
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ153_HTML.gif
(2.138)
Thus, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq330_HTML.gif , provided http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq331_HTML.gif . Using http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq332_HTML.gif , it follows that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ154_HTML.gif
(2.139)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq333_HTML.gif . We note that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq334_HTML.gif , since http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ155_HTML.gif
(2.140)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq336_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq337_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq338_HTML.gif . Now, we estimate the operator http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq340_HTML.gif satisfies the following inequality:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ156_HTML.gif
(2.141)
Then, by the inequality http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq341_HTML.gif , we obtain
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ158_HTML.gif
(2.143)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq342_HTML.gif , we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ159_HTML.gif
(2.144)
Then, using http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq343_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq344_HTML.gif , we get
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ160_HTML.gif
(2.145)
Therefore,
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ161_HTML.gif
(2.146)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq345_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq346_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq348_HTML.gif , with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq349_HTML.gif . Then, for some http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq350_HTML.gif there exists a unique solution
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq351_HTML.gif can be chosen as follows: http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq352_HTML.gif , where http://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 http://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 http://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 http://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 http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq360_HTML.gif the inequality http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq361_HTML.gif Now, we show the estimate
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ163_HTML.gif
(3.1)
for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq362_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq363_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq364_HTML.gif . In fact, using the inequality
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ164_HTML.gif
(3.2)
we get
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ165_HTML.gif
(3.3)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq365_HTML.gif , and
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ167_HTML.gif
(3.5)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq366_HTML.gif , and
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq367_HTML.gif , on the interval http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq369_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ169_HTML.gif
(3.7)
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ170_HTML.gif
(3.8)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq370_HTML.gif , and
http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq372_HTML.gif in the space http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq373_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq374_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq375_HTML.gif . Here, the constant http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ172_HTML.gif
(3.10)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq377_HTML.gif . Indeed, from the integral formula
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq378_HTML.gif ), we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ174_HTML.gif
(3.12)
since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq379_HTML.gif is sufficient small. Therefore, the operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq380_HTML.gif transforms a ball of ratio http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq381_HTML.gif into itself, in the space http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq383_HTML.gif :
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ175_HTML.gif
(3.13)
since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq384_HTML.gif is sufficient small. Thus, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq385_HTML.gif is a contraction mapping in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq386_HTML.gif . Therefore, there exists a unique solution http://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:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ176_HTML.gif
(3.14)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq388_HTML.gif . We denote http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq389_HTML.gif . From Lemma 2.7 we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ177_HTML.gif
(3.15)
for all http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq391_HTML.gif we have
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq392_HTML.gif
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ179_HTML.gif
(3.17)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq393_HTML.gif , provided that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq394_HTML.gif , and in the same way
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ180_HTML.gif
(3.18)
provided that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_IEq395_HTML.gif . Also we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ181_HTML.gif
(3.19)
for all http://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
http://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 http://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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