Open Access

On Coupled Klein-Gordon-Schrödinger Equations with Acoustic Boundary Conditions

Boundary Value Problems20102010:132751

DOI: 10.1155/2010/132751

Received: 8 July 2010

Accepted: 10 September 2010

Published: 15 September 2010

Abstract

We are concerned with the existence and energy decay of solution to the initial boundary value problem for the coupled Klein-Gordon-Schrödinger equations with acoustic boundary conditions.

1. Introduction

In this paper, we are concerned with global existence and uniform decay for the energy of solutions of Klein-Gordon-Schrödinger equations:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ1_HTML.gif
(1.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq1_HTML.gif is a bounded domain, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq2_HTML.gif , with boundary https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq3_HTML.gif of class https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq4_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq5_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq6_HTML.gif are two disjoint pieces of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq7_HTML.gif each having nonempty interior and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq8_HTML.gif are given functions. We will denote by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq9_HTML.gif the unit outward normal vector to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq10_HTML.gif . https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq11_HTML.gif stands for the Laplacian with respect to the spatial variables; https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq12_HTML.gif denotes the derivative with respect to time https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq13_HTML.gif . Here https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq14_HTML.gif is the normal displacement to the boundary at time https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq15_HTML.gif with the boundary point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq16_HTML.gif .

The above equations describe a generalization of the classical model of the Yukawa interaction of conserved complex nucleon field with neutral real meson field. Here, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq17_HTML.gif is complex scalar nucleon field while https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq18_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq19_HTML.gif are real scalar meson one.

In three dimension, [15] studied the global existence for the Cauchy problem to
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ2_HTML.gif
(1.2)

Klein-Gordon-Schrödinger equations have been studied as many as ever by many authors (cf. [611], and a list of references therein). However, they did not have treated acoustic boundary conditions.

Boundary conditions of the fifth and sixth equations are called acoustic boundary conditions. Equation (1.1)5 (the fifth equation of (1.1)) does not contain the second derivative https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq20_HTML.gif , which physically means that the material of the surface is much more lighter than a liquid flowing along it. As far as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq21_HTML.gif in (1.1)6 (the sixth equation of (1.1)) is concerned to the case of a nonporous boundary, (1.1)6 simulates a porous boundary when a function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq22_HTML.gif is nonnegative. When general acoustic boundary conditions, which had the presence of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq23_HTML.gif in (1.1)5, are prescribed on the whole boundary, Beale [1214] proved the global existence and regularity of solutions in a Hilbert space of data with finite energy by means of semigroup methods. The asymptotic behavior was obtained in [13], but no decay rate was given there. Recently, the acoustic boundary conditions have been treated by many authors (cf. [1521] and a list of references therein). However, energy decay problem with acoustic boundary conditions was studied by a few authors. For instance, Rivera and Qin [22] proved the polynomial decay for the energy of the wave motion using the Lyapunov functional technique in the case of general acoustic boundary conditions and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq24_HTML.gif . Frota and Larkin [23] considered global solvability and the exponential decay of the energy for the wave equation with acoustic boundary conditions, which eliminated the second derivative term for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq25_HTML.gif . However, it is not simple to apply the semigroup theory as well as Galerkin's method in [23] because a system of corresponding ordinary equations is not normal and one cannot apply directly the Carathéodory's theorem. So they overcame this problem using the degenerated second order equation. And Park and Ha [20] studied the existence and uniqueness of solutions and uniform decay rates for the Klein-Gordon-type equation by using the multiplier technique. Moreover, [20] proved the exponential and polynomial decay rates of solutions for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq26_HTML.gif .

In this paper, we prove the existence and uniqueness of solutions and uniform decay rates for the Klein-Gordon-Schrödinger equations with acoustic boundary conditions and allow to apply the method developed in [23]. However, [23] did not treat the Klein-Gordon-Schrödinger equations.

This paper is organized as follows. In Section 2, we recall the notation and hypotheses and introduce our main result. In Section 3, using Galerkin's method, we prove the existence and uniqueness of solutions to problem (1.1). In Section 4, we prove the exponential energy decay rate for the solutions obtained in Section 3.

2. Notations and Main Results

We begin this section introducing some notations and our main results. Throughout this paper we define the Hilbert space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq27_HTML.gif with the norm https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq28_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq29_HTML.gif is an https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq30_HTML.gif -norm and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq31_HTML.gif ; without loss of generality we denote https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq32_HTML.gif . Moreover, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq33_HTML.gif -norm and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq34_HTML.gif -norm are denoted by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq35_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq36_HTML.gif , respectively. Denoting by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq37_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq38_HTML.gif the trace map of order zero and the Neumann trace map on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq39_HTML.gif , respectively, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ3_HTML.gif
(2.1)
and the generalized Green formula
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ4_HTML.gif
(2.2)

holds for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq40_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq41_HTML.gif . We denoted https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq42_HTML.gif . By the Poincare's inequality, the norm https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq43_HTML.gif is equivalent to the usual norm from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq44_HTML.gif . Now we give the hypotheses for the main results.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq45_HTML.gif Hypotheses on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq46_HTML.gif

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq47_HTML.gif be a bounded domain in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq48_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq49_HTML.gif with boundary https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq50_HTML.gif of class https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq51_HTML.gif . Here https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq52_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq53_HTML.gif are two disjoint pieces of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq54_HTML.gif , each having nonempty interior and satisfying the following conditions:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ5_HTML.gif
(2.3)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq55_HTML.gif represents the unit outward normal vector to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq56_HTML.gif .

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq57_HTML.gif Hypotheses on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq58_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq59_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq60_HTML.gif

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq61_HTML.gif are continuous functions such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ6_HTML.gif
(2.4)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ7_HTML.gif
(2.5)

Moreover, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq62_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq63_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq64_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq65_HTML.gif is a real constant.

In physical situation, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq66_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq67_HTML.gif are parameters representing the gratitude of diffusion and dissipation effects. Also, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq68_HTML.gif is a fluid density and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq69_HTML.gif describes the mass of a meson. Boundary condition (1.1)6 (the sixth equation of (1.1)) simulates a porous boundary because of the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq70_HTML.gif .

We define the energy of system (1.1) by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ8_HTML.gif
(2.6)

Now, we are in a position to state our main result.

Theorem 2.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq71_HTML.gif satisfy the inequality
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ9_HTML.gif
(2.7)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq72_HTML.gif is a positive constant. Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq73_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq74_HTML.gif hold. Then problem (1.1) has a unique strong solution verifying
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ10_HTML.gif
(2.8)
Moreover, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq75_HTML.gif satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ11_HTML.gif
(2.9)
then one has the following energy decay:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ12_HTML.gif
(2.10)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq76_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq77_HTML.gif are positive constants.

Note

By the hypothesis in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq78_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq79_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq80_HTML.gif , so we can assume (2.9).

3. Existence of Solutions

In this section, we prove the existence and uniqueness of solutions to problem (1.1). Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq81_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq82_HTML.gif be orthonormal bases of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq83_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq84_HTML.gif , respectively, and define https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq85_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq86_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq87_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq88_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq89_HTML.gif be sequences of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq90_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq91_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq92_HTML.gif strongly in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq93_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq94_HTML.gif strongly in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq95_HTML.gif . For each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq96_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq97_HTML.gif , we consider
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ13_HTML.gif
(3.1)
satisfying the approximate perturbed equations
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ14_HTML.gif
(3.2)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq98_HTML.gif and for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq99_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq100_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq101_HTML.gif . The local existence of regular functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq102_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq103_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq104_HTML.gif is standard, because (3.2) is a normal system of ordinary differential equation. A solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq105_HTML.gif to the problem (1.1) on some interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq106_HTML.gif will be obtained as the limit of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq107_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq108_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq109_HTML.gif . Then, this solution can be extended to the whole interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq110_HTML.gif , for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq111_HTML.gif , as a consequence of the a priori estimates that will be proved in the next step.

3.1. The First Estimate

Replacing https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq112_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq113_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq114_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq115_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq116_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq117_HTML.gif in (3.2), respectively, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ15_HTML.gif
(3.3)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ16_HTML.gif
(3.4)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ17_HTML.gif
(3.5)
Taking the real part in (3.3), we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ18_HTML.gif
(3.6)
On the other hand, by Young's inequality we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ19_HTML.gif
(3.7)
Substituting the above inequality in (3.6), and then integrating (3.6) over https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq118_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq119_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ20_HTML.gif
(3.8)
Using the fact that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq120_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq121_HTML.gif , (2.7) and Gronwall's lemma, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ21_HTML.gif
(3.9)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq122_HTML.gif is a positive constant which is independent of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq123_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq124_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq125_HTML.gif .

3.2. The Second Estimate

First of all, we are going to estimate https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq126_HTML.gif . By taking https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq127_HTML.gif in (3.2)3 (the third equation of (3.2)), we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ22_HTML.gif
(3.10)
By considering https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq128_HTML.gif and hypotheses on the initial data, for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq129_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq130_HTML.gif , we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ23_HTML.gif
(3.11)
Now, by replacing https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq131_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq132_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq133_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq134_HTML.gif in (3.2), respectively, also differentiating (3.2)3 with respect to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq135_HTML.gif , and then substituting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq136_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ24_HTML.gif
(3.12)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ25_HTML.gif
(3.13)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ26_HTML.gif
(3.14)
We now estimate the last term on the left-hand side of (3.12) and the term on the right-hand side of (3.12). Applying Green's formula, we deduce
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ27_HTML.gif
(3.15)
Considering the equality
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ28_HTML.gif
(3.16)
for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq137_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ29_HTML.gif
(3.17)
Hence,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ30_HTML.gif
(3.18)
Also,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ31_HTML.gif
(3.19)
Hence,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ32_HTML.gif
(3.20)
Replacing the above calculations in (3.12) and then taking the real part, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ33_HTML.gif
(3.21)
On the other hand, we can easily check that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ34_HTML.gif
(3.22)
Therefore,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ35_HTML.gif
(3.23)
Replacing (3.23) in (3.13) and using the imbedding https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq138_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ36_HTML.gif
(3.24)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq139_HTML.gif is an imbedding constant.

Adding (3.14), (3.21), and (3.24), we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ37_HTML.gif
(3.25)
By choosing https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq140_HTML.gif and integrating (3.25) from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq141_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq142_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ38_HTML.gif
(3.26)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq143_HTML.gif is a positive constant. Using the hypotheses on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq144_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq145_HTML.gif , (2.7), (3.11), and Gronwall's lemma, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ39_HTML.gif
(3.27)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq146_HTML.gif is a positive constant which is independent of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq147_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq148_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq149_HTML.gif .

3.3. The Third Estimate

First of all, we are going to estimate https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq150_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq151_HTML.gif . By taking https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq152_HTML.gif in (3.2), we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ40_HTML.gif
(3.28)
By considering https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq153_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq154_HTML.gif and hypotheses on the initial data, for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq155_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq156_HTML.gif , we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ41_HTML.gif
(3.29)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ42_HTML.gif
(3.30)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq157_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq158_HTML.gif are positive constants.

Now by differentiating (3.2) with respect to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq159_HTML.gif and substituting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq160_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq161_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq162_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ43_HTML.gif
(3.31)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ44_HTML.gif
(3.32)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ45_HTML.gif
(3.33)
Taking the real part in (3.31), we infer
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ46_HTML.gif
(3.34)
Considering the equality
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ47_HTML.gif
(3.35)
for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq163_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ48_HTML.gif
(3.36)
Also,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ49_HTML.gif
(3.37)
From (3.34)–(3.37), we conclude
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ50_HTML.gif
(3.38)
On the other hand, we can easily check that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ51_HTML.gif
(3.39)
Therefore,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ52_HTML.gif
(3.40)
Replacing (3.40) in (3.32), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ53_HTML.gif
(3.41)
Combining (3.33), (3.38), and (3.41), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ54_HTML.gif
(3.42)
Integrating (3.42) from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq164_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq165_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ55_HTML.gif
(3.43)
Therefore, using the hypotheses on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq166_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq167_HTML.gif , (2.7), (3.11), (3.29), (3.30), and Gronwall's lemma, we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ56_HTML.gif
(3.44)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq168_HTML.gif is a positive constant which is independent of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq169_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq170_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq171_HTML.gif .

According to (3.9), (3.27), and (3.44), we obtain that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ57_HTML.gif
(3.45)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ58_HTML.gif
(3.46)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ59_HTML.gif
(3.47)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ60_HTML.gif
(3.48)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ61_HTML.gif
(3.49)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ62_HTML.gif
(3.50)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ63_HTML.gif
(3.51)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ64_HTML.gif
(3.52)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ65_HTML.gif
(3.53)
From (3.45)–(3.52), there exist subsequences https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq172_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq173_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq174_HTML.gif , which we still denote by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq175_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq176_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq177_HTML.gif , respectively, such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ66_HTML.gif
(3.54)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ67_HTML.gif
(3.55)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ68_HTML.gif
(3.56)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ69_HTML.gif
(3.57)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ70_HTML.gif
(3.58)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ71_HTML.gif
(3.59)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ72_HTML.gif
(3.60)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ73_HTML.gif
(3.61)
We can see that (3.9), (3.27), and (3.44) are also independent of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq178_HTML.gif . Therefore, by the same argument as (3.45)–(3.61) used to obtain https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq179_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq180_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq181_HTML.gif from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq182_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq183_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq184_HTML.gif , respectively, we can pass to the limit when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq185_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq186_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq187_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq188_HTML.gif , obtaining functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq189_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq190_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq191_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ74_HTML.gif
(3.62)

Thus, by the above convergences and (3.53), we can prove the existence of solutions to (1.1) satisfying (2.8).

3.4. Uniqueness

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq192_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq193_HTML.gif be two-solution pair to problem (1.1). Then we put
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ75_HTML.gif
(3.63)
From (3.2), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ76_HTML.gif
(3.64)
for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq194_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq195_HTML.gif . By replacing https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq196_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq197_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq198_HTML.gif in (3.64), it holds that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ77_HTML.gif
(3.65)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ78_HTML.gif
(3.66)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ79_HTML.gif
(3.67)
Taking the real part in (3.65), we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ80_HTML.gif
(3.68)
We now estimate the last term on the left-hand side of (3.68) and the term on the right-hand side of (3.68). We can easily check that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ81_HTML.gif
(3.69)
By using the fact that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq199_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq200_HTML.gif , we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ82_HTML.gif
(3.70)
Also,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ83_HTML.gif
(3.71)
Hence by Hölder's inequality, (3.45), and (3.47), we deduce
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ84_HTML.gif
(3.72)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq201_HTML.gif is a positive constant. Replacing (3.70) and (3.72) in (3.68), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ85_HTML.gif
(3.73)
On the other hand, we can easily check that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ86_HTML.gif
(3.74)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq202_HTML.gif is a positive constant. Therefore, we can rewrite (3.66) as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ87_HTML.gif
(3.75)
Adding (3.67), (3.73), and (3.75), we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ88_HTML.gif
(3.76)

Applying Gronwall's lemma, we conclude that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq203_HTML.gif . This completes the proof of existence and uniqueness of solutions for problem (1.1).

4. Uniform Decay

Multiplying the first equation of (1.1) by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq204_HTML.gif and integrating over https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq205_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ89_HTML.gif
(4.1)
Taking the real part in the above equality, it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ90_HTML.gif
(4.2)
Now, multiplying the second equation of (1.1) by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq206_HTML.gif and integrating over https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq207_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ91_HTML.gif
(4.3)
Taking into account (1.1)5 and (1.1)6 (the fifth and sixth equations of (1.1)), we can see that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ92_HTML.gif
(4.4)
Therefore (4.3) can be rewritten as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ93_HTML.gif
(4.5)
Adding (4.2) and (4.5), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ94_HTML.gif
(4.6)
By choosing https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq208_HTML.gif and the hypotheses on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq209_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ95_HTML.gif
(4.7)

So we conclude that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq210_HTML.gif is a nonincreasing function.

Now we consider a perturbation of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq211_HTML.gif . For each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq212_HTML.gif , we define
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ96_HTML.gif
(4.8)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ97_HTML.gif
(4.9)
By definition of the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq213_HTML.gif , Poincare's inequality, and imbedding theorem, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ98_HTML.gif
(4.10)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq214_HTML.gif is a Poincare constant. Hence, from (4.8) and (4.10), there exists a positive constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq215_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ99_HTML.gif
(4.11)
for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq216_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq217_HTML.gif . This means that there exist positive constants https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq218_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq219_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ100_HTML.gif
(4.12)
On the other hand, differentiating https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq220_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ101_HTML.gif
(4.13)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ102_HTML.gif
(4.14)

Now, we will estimate the terms on the right-hand side of (4.14).

Estimates for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq221_HTML.gif

Using the first equation of (1.1), we can easily check that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ103_HTML.gif
(4.15)

Estimates for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq222_HTML.gif

Applying Green's formula, we deduce
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ104_HTML.gif
(4.16)

Estimates for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq223_HTML.gif

Using the second equation of (1.1) and Young's inequality, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ105_HTML.gif
(4.17)

Estimates for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq224_HTML.gif

Similar to estimates for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq225_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ106_HTML.gif
(4.18)
By replacing (4.15)–(4.18) in (4.14) and choosing https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq226_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq227_HTML.gif , we conclude that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ107_HTML.gif
(4.19)
From (2.9) and (4.19), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ108_HTML.gif
(4.20)

We now estimate the last term on the right-hand side of (4.20).

Estimates for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq228_HTML.gif

From the fifth equation of (1.1), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ109_HTML.gif
(4.21)

Estimates for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq229_HTML.gif

By Young's inequality, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ110_HTML.gif
(4.22)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq230_HTML.gif is an arbitrary positive constant. By replacing (4.21) and (4.22) in (4.20), we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ111_HTML.gif
(4.23)
We note that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ112_HTML.gif
(4.24)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq231_HTML.gif . By the above inequality and choosing https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq232_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ113_HTML.gif
(4.25)
we conclude that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ114_HTML.gif
(4.26)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq233_HTML.gif is a positive constant. Now choosing https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq234_HTML.gif sufficiently small, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ115_HTML.gif
(4.27)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq235_HTML.gif is a positive constant. Therefore,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ116_HTML.gif
(4.28)
From (4.12), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ117_HTML.gif
(4.29)

This implies the proof of Theorem 2.1 is completed.

Authors’ Affiliations

(1)
Department of Mathematics, Iowa State University
(2)
Department of Mathematics, Pusan National University

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© Tae Gab Ha and Jong Yeoul Park. 2010

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