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

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:

(1.1)

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.

In three dimension, [1–5] studied the global existence for the Cauchy problem to

(1.2)

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

We begin this section introducing some notations and our main results. Throughout this paper we define the Hilbert space with the norm , where is an -norm and ; without loss of generality we denote . Moreover, -norm and -norm are denoted by and , respectively. Denoting by and the trace map of order zero and the Neumann trace map on , respectively, we have

(2.1)

and the generalized Green formula

(2.2)

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

Let be a bounded domain in , with boundary of class . Here and are two disjoint pieces of , each having nonempty interior and satisfying the following conditions:

(2.3)

where represents the unit outward normal vector to .

Hypotheses on , , and

Assume that are continuous functions such that

(2.4)

where

(2.5)

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 .

We define the energy of system (1.1) by

(2.6)

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

Theorem 2.1.

Let satisfy the inequality

(2.7)

where is a positive constant. Assume that and hold. Then problem (1.1) has a unique strong solution verifying

(2.8)

Moreover, if satisfies

(2.9)

then one has the following energy decay:

(2.10)

where and are positive constants.

Note

By the hypothesis in , we have for all , 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 and be orthonormal bases of and , respectively, and define and . Let , , and be sequences of such that , strongly in , and strongly in . For each and , we consider

(3.1)

satisfying the approximate perturbed equations

(3.2)

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

Replacing , , and by , , and in (3.2), respectively, we obtain

(3.3)

(3.4)

(3.5)

Taking the real part in (3.3), we get

(3.6)

On the other hand, by Young's inequality we have

(3.7)

Substituting the above inequality in (3.6), and then integrating (3.6) over with , we get

(3.8)

Using the fact that , , (2.7) and Gronwall's lemma, we obtain

(3.9)

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

3.2. The Second Estimate

First of all, we are going to estimate . By taking in (3.2)₃ (the third equation of (3.2)), we get

(3.10)

By considering and hypotheses on the initial data, for all and , we obtain

(3.11)

Now, by replacing and by and in (3.2), respectively, also differentiating (3.2)₃ with respect to , and then substituting , we have

(3.12)

(3.13)

(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

(3.15)

Considering the equality

(3.16)

for all , we have

(3.17)

Hence,

(3.18)

Also,

(3.19)

Hence,

(3.20)

Replacing the above calculations in (3.12) and then taking the real part, we obtain

(3.21)

On the other hand, we can easily check that

(3.22)

Therefore,

(3.23)

Replacing (3.23) in (3.13) and using the imbedding , we have

(3.24)

where is an imbedding constant.

Adding (3.14), (3.21), and (3.24), we get

(3.25)

By choosing and integrating (3.25) from to , we have

(3.26)

where is a positive constant. Using the hypotheses on and , (2.7), (3.11), and Gronwall's lemma, we obtain

(3.27)

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

3.3. The Third Estimate

First of all, we are going to estimate and . By taking in (3.2), we get

(3.28)

By considering and and hypotheses on the initial data, for all and , we obtain

(3.29)

(3.30)

where and are positive constants.

Now by differentiating (3.2) with respect to and substituting , , and , we have

(3.31)

(3.32)

(3.33)

Taking the real part in (3.31), we infer

(3.34)

Considering the equality

(3.35)

for all , we have

(3.36)

Also,

(3.37)

From (3.34)–(3.37), we conclude

(3.38)

On the other hand, we can easily check that

(3.39)

Therefore,

(3.40)

Replacing (3.40) in (3.32), we have

(3.41)

Combining (3.33), (3.38), and (3.41), we obtain

(3.42)

Integrating (3.42) from to , we have

(3.43)

Therefore, using the hypotheses on and , (2.7), (3.11), (3.29), (3.30), and Gronwall's lemma, we get

(3.44)

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

According to (3.9), (3.27), and (3.44), we obtain that

(3.45)

(3.46)

(3.47)

(3.48)

(3.49)

(3.50)

(3.51)

(3.52)

(3.53)

From (3.45)–(3.52), there exist subsequences , , and , which we still denote by , and , respectively, such that

(3.54)

(3.55)

(3.56)

(3.57)

(3.58)

(3.59)

(3.60)

(3.61)

We can see that (3.9), (3.27), and (3.44) are also independent of . Therefore, by the same argument as (3.45)–(3.61) used to obtain , , and from , , and , respectively, we can pass to the limit when in , , and , obtaining functions , , and such that

(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 and be two-solution pair to problem (1.1). Then we put

(3.63)

From (3.2), we have

(3.64)

for all and . By replacing , , and in (3.64), it holds that

(3.65)

(3.66)

(3.67)

Taking the real part in (3.65), we get

(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

(3.69)

By using the fact that for all , we obtain

(3.70)

Also,

(3.71)

Hence by Hölder's inequality, (3.45), and (3.47), we deduce

(3.72)

where is a positive constant. Replacing (3.70) and (3.72) in (3.68), we have

(3.73)

On the other hand, we can easily check that

(3.74)

where is a positive constant. Therefore, we can rewrite (3.66) as

(3.75)

Adding (3.67), (3.73), and (3.75), we get

(3.76)

Applying Gronwall's lemma, we conclude that . 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 and integrating over , we get

(4.1)

Taking the real part in the above equality, it follows that

(4.2)

Now, multiplying the second equation of (1.1) by and integrating over , we have

(4.3)

Taking into account (1.1)₅ and (1.1)₆ (the fifth and sixth equations of (1.1)), we can see that

(4.4)

Therefore (4.3) can be rewritten as

(4.5)

Adding (4.2) and (4.5), we obtain

(4.6)

By choosing and the hypotheses on , we get

(4.7)

So we conclude that is a nonincreasing function.

Now we consider a perturbation of . For each , we define

(4.8)

where

(4.9)

By definition of the function , Poincare's inequality, and imbedding theorem, we have

(4.10)

where is a Poincare constant. Hence, from (4.8) and (4.10), there exists a positive constant such that

(4.11)

for all and . This means that there exist positive constants and such that

(4.12)

On the other hand, differentiating , we have

(4.13)

where

(4.14)

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

Estimates for

Using the first equation of (1.1), we can easily check that

(4.15)

Estimates for

Applying Green's formula, we deduce

(4.16)

Estimates for

Using the second equation of (1.1) and Young's inequality, we have

(4.17)

Estimates for

Similar to estimates for we have

(4.18)

By replacing (4.15)–(4.18) in (4.14) and choosing and , we conclude that

(4.19)

From (2.9) and (4.19), we obtain

(4.20)

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

Estimates for

From the fifth equation of (1.1), we have

(4.21)

Estimates for

By Young's inequality, we have

(4.22)

where is an arbitrary positive constant. By replacing (4.21) and (4.22) in (4.20), we get

(4.23)

We note that

(4.24)

where . By the above inequality and choosing such that

(4.25)

we conclude that

(4.26)

where is a positive constant. Now choosing sufficiently small, we obtain

(4.27)

where is a positive constant. Therefore,

(4.28)

From (4.12), we have

(4.29)

This implies the proof of Theorem 2.1 is completed.