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

  • TaeGab Ha1Email author and

    Affiliated with

    • JongYeoul Park2

      Affiliated with

      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:
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ1_HTML.gif
      (1.1)

      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq1_HTML.gif is a bounded domain, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq2_HTML.gif , with boundary http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq3_HTML.gif of class http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq4_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq5_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq6_HTML.gif are two disjoint pieces of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq7_HTML.gif each having nonempty interior and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq9_HTML.gif the unit outward normal vector to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq10_HTML.gif . http://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; http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq13_HTML.gif . Here http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq15_HTML.gif with the boundary point http://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, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq17_HTML.gif is complex scalar nucleon field while http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq18_HTML.gif and http://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
      http://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 http://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 http://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 http://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 http://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 http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq27_HTML.gif with the norm http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq28_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq29_HTML.gif is an http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq30_HTML.gif -norm and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq31_HTML.gif ; without loss of generality we denote http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq32_HTML.gif . Moreover, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq33_HTML.gif -norm and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq34_HTML.gif -norm are denoted by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq35_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq36_HTML.gif , respectively. Denoting by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq37_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq39_HTML.gif , respectively, we have
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ4_HTML.gif
      (2.2)

      holds for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq40_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq41_HTML.gif . We denoted http://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 http://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 http://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.

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

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

      where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq56_HTML.gif .

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

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

      Moreover, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq62_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq63_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq64_HTML.gif , and http://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, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq66_HTML.gif and http://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, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq68_HTML.gif is a fluid density and http://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 http://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
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq71_HTML.gif satisfy the inequality
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ9_HTML.gif
      (2.7)
      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq72_HTML.gif is a positive constant. Assume that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq73_HTML.gif and http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ10_HTML.gif
      (2.8)
      Moreover, if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq75_HTML.gif satisfies
      http://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:
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ12_HTML.gif
      (2.10)

      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq76_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq78_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq79_HTML.gif for all http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq81_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq82_HTML.gif be orthonormal bases of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq83_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq84_HTML.gif , respectively, and define http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq85_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq86_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq87_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq88_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq89_HTML.gif be sequences of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq90_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq91_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq92_HTML.gif strongly in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq93_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq94_HTML.gif strongly in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq95_HTML.gif . For each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq96_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq97_HTML.gif , we consider
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ14_HTML.gif
      (3.2)

      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq98_HTML.gif and for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq99_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq100_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq101_HTML.gif . The local existence of regular functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq102_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq103_HTML.gif , and http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq107_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq108_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq110_HTML.gif , for all http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq112_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq113_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq114_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq115_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq116_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq117_HTML.gif in (3.2), respectively, we obtain
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ15_HTML.gif
      (3.3)
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ16_HTML.gif
      (3.4)
      http://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
      http://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
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq118_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq119_HTML.gif , we get
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ20_HTML.gif
      (3.8)
      Using the fact that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq120_HTML.gif , http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ21_HTML.gif
      (3.9)

      where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq123_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq124_HTML.gif , and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq126_HTML.gif . By taking http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ22_HTML.gif
      (3.10)
      By considering http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq129_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq130_HTML.gif , we obtain
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ23_HTML.gif
      (3.11)
      Now, by replacing http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq131_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq132_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq133_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq135_HTML.gif , and then substituting http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq136_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ24_HTML.gif
      (3.12)
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ25_HTML.gif
      (3.13)
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ27_HTML.gif
      (3.15)
      Considering the equality
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ28_HTML.gif
      (3.16)
      for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq137_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ29_HTML.gif
      (3.17)
      Hence,
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ30_HTML.gif
      (3.18)
      Also,
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ31_HTML.gif
      (3.19)
      Hence,
      http://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
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ34_HTML.gif
      (3.22)
      Therefore,
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq138_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ36_HTML.gif
      (3.24)

      where http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ37_HTML.gif
      (3.25)
      By choosing http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq140_HTML.gif and integrating (3.25) from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq141_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq142_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ38_HTML.gif
      (3.26)
      where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq144_HTML.gif and http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ39_HTML.gif
      (3.27)

      where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq147_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq148_HTML.gif , and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq150_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq151_HTML.gif . By taking http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq152_HTML.gif in (3.2), we get
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ40_HTML.gif
      (3.28)
      By considering http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq153_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq155_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq156_HTML.gif , we obtain
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ41_HTML.gif
      (3.29)
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ42_HTML.gif
      (3.30)

      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq157_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq159_HTML.gif and substituting http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq160_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq161_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq162_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ43_HTML.gif
      (3.31)
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ44_HTML.gif
      (3.32)
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ46_HTML.gif
      (3.34)
      Considering the equality
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ47_HTML.gif
      (3.35)
      for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq163_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ48_HTML.gif
      (3.36)
      Also,
      http://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
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ51_HTML.gif
      (3.39)
      Therefore,
      http://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
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ54_HTML.gif
      (3.42)
      Integrating (3.42) from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq164_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq165_HTML.gif , we have
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq166_HTML.gif and http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ56_HTML.gif
      (3.44)

      where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq169_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq170_HTML.gif , and http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ57_HTML.gif
      (3.45)
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ58_HTML.gif
      (3.46)
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ59_HTML.gif
      (3.47)
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ60_HTML.gif
      (3.48)
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ61_HTML.gif
      (3.49)
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ62_HTML.gif
      (3.50)
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ63_HTML.gif
      (3.51)
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ64_HTML.gif
      (3.52)
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq172_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq173_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq174_HTML.gif , which we still denote by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq175_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq176_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq177_HTML.gif , respectively, such that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ66_HTML.gif
      (3.54)
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ67_HTML.gif
      (3.55)
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ68_HTML.gif
      (3.56)
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ69_HTML.gif
      (3.57)
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ70_HTML.gif
      (3.58)
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ71_HTML.gif
      (3.59)
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ72_HTML.gif
      (3.60)
      http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq179_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq180_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq181_HTML.gif from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq182_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq183_HTML.gif , and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq185_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq186_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq187_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq188_HTML.gif , obtaining functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq189_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq190_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq191_HTML.gif such that
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq192_HTML.gif and http://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
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ76_HTML.gif
      (3.64)
      for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq194_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq195_HTML.gif . By replacing http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq196_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq197_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq198_HTML.gif in (3.64), it holds that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ77_HTML.gif
      (3.65)
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ78_HTML.gif
      (3.66)
      http://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
      http://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
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq199_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq200_HTML.gif , we obtain
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ82_HTML.gif
      (3.70)
      Also,
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ84_HTML.gif
      (3.72)
      where http://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
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ86_HTML.gif
      (3.74)
      where http://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
      http://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
      http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq204_HTML.gif and integrating over http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq205_HTML.gif , we get
      http://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
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq206_HTML.gif and integrating over http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq207_HTML.gif , we have
      http://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
      http://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
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ94_HTML.gif
      (4.6)
      By choosing http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq208_HTML.gif and the hypotheses on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq209_HTML.gif , we get
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ95_HTML.gif
      (4.7)

      So we conclude that http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq211_HTML.gif . For each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq212_HTML.gif , we define
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ96_HTML.gif
      (4.8)
      where
      http://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 http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ98_HTML.gif
      (4.10)
      where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq215_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ99_HTML.gif
      (4.11)
      for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq216_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq218_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq219_HTML.gif such that
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq220_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ101_HTML.gif
      (4.13)
      where
      http://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 http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ103_HTML.gif
      (4.15)

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

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

      Estimates for http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ105_HTML.gif
      (4.17)

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

      Similar to estimates for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq225_HTML.gif we have
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq226_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_IEq227_HTML.gif , we conclude that
      http://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
      http://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 http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F132751/MediaObjects/13661_2010_Article_896_Equ109_HTML.gif
      (4.21)

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

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