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

- TaeGab Ha
^{1}Email author and - JongYeoul Park
^{2}

**2010**:132751

**DOI: **10.1155/2010/132751

© Tae Gab Ha and Jong Yeoul Park. 2010

**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

where is a bounded domain, , with boundary of class , where and are two disjoint pieces of each having nonempty interior and are given functions. We will denote by the unit outward normal vector to . stands for the Laplacian with respect to the spatial variables; denotes the derivative with respect to time . Here is the normal displacement to the boundary at time with the boundary point .

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, is complex scalar nucleon field while and are real scalar meson one.

Klein-Gordon-Schrödinger equations have been studied as many as ever by many authors (cf. [6–11], 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
, which physically means that the material of the surface is much more lighter than a liquid flowing along it. As far as
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
is nonnegative. When general acoustic boundary conditions, which had the presence of
in (1.1)_{5}, are prescribed on the whole boundary, Beale [12–14] 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. [15–21] 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
. 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
. 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
.

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

holds for all and . We denoted . By the Poincare's inequality, the norm is equivalent to the usual norm from . Now we give the hypotheses for the main results.

Hypotheses on

where represents the unit outward normal vector to .

Hypotheses on , , and

Moreover, , , , and is a real constant.

In physical situation,
and
are parameters representing the gratitude of diffusion and dissipation effects. Also,
is a fluid density and
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
.

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

Theorem 2.1.

where and are positive constants.

Note

By the hypothesis in , we have for all , so we can assume (2.9).

## 3. Existence of Solutions

where and for all , , . The local existence of regular functions , , and is standard, because (3.2) is a normal system of ordinary differential equation. A solution to the problem (1.1) on some interval will be obtained as the limit of as and . Then, this solution can be extended to the whole interval , for all , as a consequence of the a priori estimates that will be proved in the next step.

### 3.1. The First Estimate

where is a positive constant which is independent of , , and .

### 3.2. The Second Estimate

_{3}(the third equation of (3.2)), we get

_{3}with respect to , and then substituting , we have

where is an imbedding constant.

where is a positive constant which is independent of , , and .

### 3.3. The Third Estimate

where and are positive constants.

where is a positive constant which is independent of , , and .

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

### 3.4. Uniqueness

Applying Gronwall's lemma, we conclude that . This completes the proof of existence and uniqueness of solutions for problem (1.1).

## 4. Uniform Decay

_{5}and (1.1)

_{6}(the fifth and sixth equations of (1.1)), we can see that

So we conclude that is a nonincreasing function.

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

Estimates for

Estimates for

Estimates for

Estimates for

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

*Estimates for*

Estimates for

This implies the proof of Theorem 2.1 is completed.

## Authors’ Affiliations

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