Energy Decay of Klein-Gordon-Schrödinger Type with Linear Memory Term ∗

This paper is concerned with the existence, uniqueness and uniform decay of the solutions of a Klein-Gordon-Schrodinger type system with linear memory term. The existence is proved by means of the Faedo-Galerkin method and the asymptotic behavior is obtained by making use of the multiplier technique combined with integral inequalities.


Introduction
This paper aims to prove the global existence and uniform decay for the following system iψ ′ + κ∆ψ + iαψ = φψ , x ∈ Ω ⊂ IR n , t > 0, (1.1) satisfying the following initial and boundary conditions ψ(x, 0) = ψ 0 (x), φ(x, 0) = φ 0 (x), φ ′ (x, 0) = φ 1 (x), x ∈ Ω, (1.3) ψ(x, t) = φ(x, t) = 0, x ∈ ∂Ω, t > 0, (1.4) where Ω is a bounded domain of R n , n ≤ 2 with κ, α, λ > 0. The variable ψ stands for the dimensionless low frequency electron field, whereas φ denotes the dimensionless low frequency density.This system in one dimension describes the nonlinear interaction between high frequency electron waves and low frequency ion plasma waves in a homogeneous magnetic field, adapted to model the UHH plasma heating scheme.The unusual form of the right side of equation (1.2), as compared to the corresponding Zakharov equation, is a consequence of the different low frequency coupling that was considered, i.e. the polarization drift instead of the ponderomotive force.
Systems of Klein-Gordon-Schrödinger type have been studied for many years.In [4] the authors proved the existence of a strong global attractor in H 2 (IR 3 ) × H 2 (IR 3 ) attracting bounded sets of H 3 (IR 3 ) × H 3 (IR 3 ) for a Klein-Gordon-Schrödinger system with Yukawa coupling.This was extended in [8] where the existence of a strong global attractor in H k (IR N ) × H k (IR N ), N = 1, 2, 3, attracting bounded sets of H k (IR N ) × H k (IR N ), k ≥ 1 was proved.For a dissipative system of Zakharov type I. Flahaut [3] proved the existence of a weak global attractor in H 1 0 ((0, L)) × H 1 0 ((0, L)) H 2 ((0, L)) × H 1 0 ((0, L)) H 3 ((0, L)) and obtained upper bounds for its Hausdorff and Fractal dimensions.In [6] the authors studied the one dimensional case of (1.1) -(1.2) and proved the global existence and uniqueness of the solutions and established the necessary conditions for the system to manifest energy decay.Later on the authors in [10] proved the existence of a global attractor in the space (H 1 0 (Ω)∩H 2 (Ω)) 2 ×H 1 0 (Ω) which attracts all bounded sets of (H 1 0 (Ω) ∩ H 2 (Ω)) 2 × H 1 0 (Ω) in the norm topology.
The rest of the paper is divided into four sections.In Section 2, the basic notation and assumptions made are stated along with the main results.In Section 3 the existence and uniqueness of the solutions of (1.1) - (1.4) in (H 1 0 (Ω) ∩ H 2 (Ω) 2 × H 1 0 (Ω) are established while in Section 4 the uniform decay of the solutions is proved.
Notation: Let us introduce some notations that will be used throughout this work.Denote by H s (Ω) both the standard real and complex Sobolev spaces on (Ω).For simplicity reasons sometimes we use H s , L s for H s (Ω), L s (Ω) and ||.||, (., .)for the norm and the inner product of L 2 (Ω) respectively as well as the symbol • denotes the inner product in IR n .Finally, C is a general symbol for any positive constant.

Assumptions and main result
Let us consider the Hilbert space L 2 (Ω) of complex valued functions on Ω endowed with the inner product and the corresponding norm We consider the Sobolev space H 1 (Ω) endowed with the scalar product We define the subspace of H 1 (Ω), denoted by H 1 0 (Ω), as the closure of C ∞ 0 (Ω) in the strong topology of H 1 (Ω).
Assumption 2.2 We assume that F (x) is a one dimensional vector function with We recall the following inequalities which will be used frequently later: ) and We define the energy of (1.1)-(1.2) as and therefore we have the following main result and Assumption 2.1 -2.2 hold .Then, there exists a unique solution for the system (1.1), (1.4)

Global Existence
Let us represent by w n a basis in H 1 0 (Ω) ∩ H 2 (Ω) formed by the eigenfunctions of −∆, also by V m the subspace of H 1 0 (Ω) ∩ H 2 (Ω) generated by the first m vectors and by with initial conditions In this section we derive a priori estimates for the solutions of the (3.1)-(3.
For the right hand side of the equation above we have 1 2 But from (3.1) we also obtain 2) and then integrating over Ω (3.2) becomes 1 2 Hence, by adding (3.4), (3.7) and (3.8) we have (3.9) EJQTDE, 2013 No. 11, p. 4 Evaluating the integrals of (3.9) by using Assumption 2.2, the compact embedding H 1 0 (Ω) ֒→ L 4 (Ω) and Young's inequality, we obtain Also, considering Cauchy-Schwarz Inequality, Young's Inequality and Assumption 2.1 we have the following estimate Combining the results above (3.9)can be rewritten as (3.10) Integrating the above expression over (0, t) and considering (3.3) it follows that Evaluating the following terms (3.12) Substituting the results above in (3.11) and applying Gronwall's Lemma we obtain the first estimate where L 1 is a positive constant independent of m ∈ N.

A Priori Estimate II
Let u = ∆ ψ′ m (t) + α∆ ψm (t) in (3.1), then by taking the real part and integrating over Ω we have 1 2 Therefore by integrating we have 1 2 Estimating the integrals on the right hand side of (3.18) using the Sobolev embedding theorem and Young's Inequality gives the following results

Re
Now evaluating the last term of (3.15) and taking into consideration Assumption 2.2 we evaluate the integrals on the right hand side also we obtain Substituting the expressions above into (3.18)gives the following result Integrating (3.19) over (0, t) and considering (3.3) it follows that Using Cauchy Schwarz inequality and Young's inequality imply Substituting the expression above into (3.20) and applying Gronwall's Lemma we obtain the second estimate where L 2 is a positive constant independent of m ∈ N.
But, by using (3.2) we have the following estimate where Analyzing the terms on the right hand side gives Similarly we have and Evaluating some of the integrals above by taking into consideration Young's inequality and the following embedding H 1 (Ω) ֒→ L q (Ω), with q ∈ [1, 6] and inequality (2.2) we obtain and into (3.30) and integrating over (0, t) we obtain (3.39) Therefore we can extract weekly * convergent subsequences denoted again as (ψ m , φ m ) such that ψ m → w * ψ, φ m → w * φ, The above convergences are sufficient to pass to the limit in (3.1) and (3.2) and it results thanks to the elliptic regularity that ψ ∈ L ∞ (0, ∞; H 1 0 (Ω) ∩ H 2 (Ω)).Following similar procedure as in Theorem 2.1 of [11] we prove the uniqueness of the solutions.Therefore the proof of Theorem 2.1 is completed.

Energy Decay
Due to the previous results the corresponding energy functional for the system (1.1) and (1.2) is The integral cannot affect the asymptotic value of the energy which remains positive as seeing below using (3.12) Define the modified energy as and taking into consideration that t 0 g(t − τ )(▽φ(τ ), ▽φ evaluating the integrals we have Following [5] , for ǫ > 0 we introduce the perturbed energy e pert (t) = e(t) + ǫp(t),  Now, let all the expressions within the brackets be simultaneously non positive or zero.To achieve this, we introduce the auxiliary constant ν > 0. Then choosing the constants to be and setting the third expression equal to zero, we determine the value of ν as ν = 2(µ − 1) 3µ .Next by requiring the first two expressions to be non positive we reach to the conclusion that for 16κλα > 6M 2 + 3C