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

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.


Introduction
In this paper, we are concerned with global existence and uniform decay for the energy of solutions of Klein-Gordon-Schrödinger equations: where Ω ⊂ R n is a bounded domain, 1 ≤ n ≤ 3, with boundary Γ Γ 0 ∪ Γ 1 of class C 2 , where Γ 0 and Γ 1 are two disjoint pieces of Γ each having nonempty interior and f, g, h : Γ 1 → R are 2 Boundary Value Problems given functions. We will denote by ν the unit outward normal vector to Γ. Δ stands for the Laplacian with respect to the spatial variables; ' denotes the derivative with respect to time t.
Here z x, t is the normal displacement to the boundary at time t with the boundary point x.
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, u is complex scalar nucleon field while v and z 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 z , which physically means that the material of the surface is much more lighter than a liquid flowing along it. As far as h ≡ 1 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 h is nonnegative. When general acoustic boundary conditions, which had the presence of z in 1.1 5 , are prescribed on the whole boundary,  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 n 3. 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 1 ≤ n ≤ 3. 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 n ≥ 1.
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.

Notations and Main Results
We begin this section introducing some notations and our main results. Throughout this paper we define the Hilbert space Moreover, L p Ω -norm and L p Γ -norm are denoted by · p and · p,Γ , respectively. Denoting by ρ 0 : H 1 Ω → H 1/2 Γ and ρ 1 : H → H −1/2 Γ the trace map of order zero and the Neumann trace map on H, respectively, we have and the generalized Green formula

H 1 Hypotheses on Ω
Let Ω be a bounded domain in R n , 1 ≤ n ≤ 3 with boundary Γ of class C 2 . Here Γ 0 and Γ 1 are two disjoint pieces of Γ, each having nonempty interior and satisfying the following conditions: where ν represents the unit outward normal vector to Γ.
In physical situation, α and β are parameters representing the gratitude of diffusion and dissipation effects. Also, γ is a fluid density and μ 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 h.
We define the energy of system 1.1 by Now, we are in a position to state our main result.
where C 0 is a positive constant. Assume that H 1 and H 2 hold. Then problem 1.1 has a unique strong solution verifying

2.8
Moreover, if μ satisfies then one has the following energy decay: where C 1 and ω are positive constants.

Existence of Solutions
In this section, we prove the existence and uniqueness of solutions to problem 1.
For each η ∈ 0, 1 and m ∈ N, we consider satisfying the approximate perturbed equations where z 0m ∈ L 2 Γ 1 and for all w, y ∈ W m , ξ ∈ V m , 0 < T m ≤ T . The local existence of regular functions a jm , b jm , and c jm is standard, because 3.2 is a normal system of ordinary differential equation. A solution u, v, z to the problem 1.1 on some interval 0, T m will be obtained as the limit of u ηm , z ηm as m → ∞ and η → 0. Then, this solution can be extended to the whole interval 0, T , for all T > 0, as a consequence of the a priori estimates that will be proved in the next step.

The First Estimate
Replacing w, y, and ξ by u ηm , v ηm , and z ηm in 3.2 , respectively, we obtain Taking the real part in 3.3 , we get

3.6
On the other hand, by Young's inequality we have Substituting the above inequality in 3.6 , and then integrating 3.6 over 0, t with t ∈ 0, T m , we get 3.8 Using the fact that 0 < min x∈Γ 1 f x : f 0 , 0 < min x∈Γ 1 g x : g 0 , 2.7 and Gronwall's lemma, we obtain where C 2 is a positive constant which is independent of m, η, and t.

The Second Estimate
First of all, we are going to estimate z ηm 0 . By taking t 0 in 3.2 3 the third equation of 3.2 , we get Boundary Value Problems 7 By considering ξ z ηm 0 and hypotheses on the initial data, for all m ∈ N and η ∈ 0, 1 , we obtain Now, by replacing w and y by −Δu ηm and −Δv ηm in 3.2 , respectively, also differentiating 3.2 3 with respect to t, and then substituting ξ z ηm , we have 3.20 Replacing the above calculations in 3.12 and then taking the real part, we obtain On the other hand, we can easily check that ∇u ηm u ηm u ηm ∇u ηm ∇u ηm u ηm u ηm ∇u ηm 2 Re u ηm ∇u ηm .

3.23
Replacing 3.23 in 3.13 and using the imbedding V → L 2 Γ , we have

3.24
where c is an imbedding constant.

3.26
where C 3 is a positive constant. Using the hypotheses on α and β, 2.7 , 3.11 , and Gronwall's lemma, we obtain where C 4 is a positive constant which is independent of m, η, and t.

3.28
By considering w u ηm 0 and y v ηm 0 and hypotheses on the initial data, for all m ∈ N and η ∈ 0, 1 , we obtain where C 5 and C 6 are positive constants. Now by differentiating 3.2 with respect to t and substituting w u ηm , y v ηm , and ξ z ηm , we have

3.33
Taking the real part in 3.31 , we infer

3.38
On the other hand, we can easily check that u ηm u ηm u ηm u ηm u ηm u ηm u ηm u ηm 2 Re u ηm u ηm .

3.40
Replacing 3.40 in 3.32 , we have

3.42
Integrating 3.42 from 0 to t, we have

3.44
where C 7 is a positive constant which is independent of m, η, and t. According to 3.9 , 3.27 , and 3.44 , we obtain that We can see that 3.9 , 3.27 , and 3.44 are also independent of η. Therefore, by the same argument as 3.45 -3.61 used to obtain u η , v η , and z η from u ηm , v ηm , and z ηm , respectively, we can pass to the limit when η → 0 in u η , v η , and z η , obtaining functions u, v, and z 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 .

Uniqueness
Let u 1 , v 1 , z 1 and u 2 , v 2 , z 2 be two-solution pair to problem 1.1 . Then we put for all w, y ∈ V ∩ H 2 Ω and ξ ∈ L 2 Γ . By replacing w u, y v , and ξ z in 3.64 , it holds that Taking the real part in 3.65 , we get 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 By using the fact that Re z 1 z 2 Re z 1 z 2 for all z 1 , z 2 ∈ C, we obtain

3.70
Also, Boundary Value Problems

3.72
where C 8 is a positive constant. Replacing 3.70 and 3.72 in 3.68 , we have On the other hand, we can easily check that 3.74 where C 9 is a positive constant. Therefore, we can rewrite 3.66 as Adding 3.67 , 3.73 , and 3.75 , we get

3.76
Applying Gronwall's lemma, we conclude that u v z 0. This completes the proof of existence and uniqueness of solutions for problem 1.1 .

Uniform Decay
Multiplying the first equation of 1.1 by u and integrating over Ω, we get Taking the real part in the above equality, it follows that Now, multiplying the second equation of 1.1 by v and integrating over Ω, we have Taking into account 1.1 5 and 1.1 6 the fifth and sixth equations of 1.1 , we can see that Therefore 4.3 can be rewritten as

4.6
By choosing 1/2β and the hypotheses on β, we get So we conclude that E t is a nonincreasing function. Now we consider a perturbation of E t . For each > 0, we define

4.9
By definition of the function ρ t , Poincare's inequality, and imbedding theorem, we have

4.10
where c is a Poincare constant. Hence, from 4.8 and 4.10 , there exists a positive constant C 10 such that for all > 0 and t ≥ 0. This means that there exist positive constants C 11 and C 12 such that On the other hand, differentiating E t , we have E t E t ρ t , 4.13