EXPONENTIAL STABILITY FOR THE COUPLED KLEIN-GORDON-SCHR¨ODINGER EQUATIONS WITH LOCALLY DISTRIBUTED DAMPING

. The following coupled damped Klein-Gordon-Schr¨odinger equations are considered iψ where Ω is a bounded domain of R 2 , with smooth boundary Γ and ω is a neighbourhood of ∂ Ω satisfying the geometric control condition. Here χ ω represents the characteristic function of ω . Assuming that a,b ∈ L ∞ (Ω) are nonnegative functions such that a ( x ) ≥ a 0 > 0 in ω and b ( x ) ≥ b 0 > 0 in ω , the exponential decay rate is proved for every regular solution of the above system. Our result generalizes substantially the previous ones given by Cavalcanti et. al in the reference [9] and [1].


Introduction.
We consider the following model of Klein-Gordon-Schrödinger equations with locally distributed damping iψ t + ∆ψ + iαb(x)(|ψ| 2 + 1)ψ = φψχ ω in Ω × (0, ∞), φ tt − ∆φ + a(x)φ t = |ψ| 2 χ ω in Ω × (0, ∞), ψ = φ = 0 on Γ × (0, ∞), ψ(0) = ψ 0 ∈ H 1 0 (Ω) ∩ H 2 (Ω), where Ω is a bounded domain of R 2 , with smooth boundary Γ and ω is an open subset of Ω such that meas(ω) > 0 and satisfying the geometric control condition. In what follows, α is a positive constant and χ ω represents the characteristic function, that is, χ = 1 in ω and χ = 0 in Ω\ω. We consider a, b ∈ L ∞ (Ω) nonnegative functions such that so that the nonlinearity ψ exists where the damping terms a(x)φ t and iαb(x)(|ψ| 2 + 1)ψ (2) are, in fact, effective and reciprocally. If the damping is effective in whole domain, i. e., a(x) ≥ a 0 > 0 in Ω and b(x) ≥ b 0 > 0 in Ω we can consider χ ω ≡ 1 in Ω. This is required in order to turn the system dissipative. Indeed, the presence of the damping terms given in (2) is not necessary, by itself, to guarantee that the energy E(t) associated to problem (1) (see the definition of E(t) in (9)) is a nonincreasing function on the parameter t. This will clarified in section 4. Uniform decay rate estimates to problem (1) has been considered in the previous results due to Cavalcanti et. al [11], [9]. While in [11] a full damping was in place in both equations, in contrast, in [9] a full damping has been considered in the Schrödinger equation but just a localized damping has been considered for the wave equation.
It is worth mentioning that in the recent the article [1] the authors generalize both previous results just considering two localized dampings in both equations, namely: making use of the multipliers method combined with integral inequalities of energy and a regularizing effect due to Aloui [2], [3].
The main purpose of the present article is to generalize substantially all the previous results above mentioned by considering the weaker damped structure iαb(x)(|ψ| 2 + 1)ψ instead of iαb(x)(−∆) 1 2 b(x)ψ assumed in [1]. For this purpose, we make use of the observability inequality of both, the wave and the Schrödinger linear equations (see [7] and [16], [32] respectively), combined with other tools in order to prove the exponential decay rate as has been considered previously in [15] for the wave equation. This will be clarified during the proof. It is important to be mentioned that the use of the observability inequality associated to the linear problems of the wave and Schrödinger equations instead of the multiplier technique allows us to consider sharp regions ω satisfying the geometric control condition. Indeed, the inequalities given in (7) and (8) are proved by means of microlocal analysis and produce sharp regions when compared with the multiplier method. As a consequence, the present paper generalizes substantially our previous results not only with concerns to consider a weaker damping but also regarding the sharpness of the region ω (less damping as possible) where the damping acts.
Problem (1) (see [47]) has its origin in the canonical model of the Yukawa interaction of conserved complex nucleon field ψ with neutral real meson field φ given by Here, ψ is a complex scalar nucleon field while φ is a real scalar meson one and the positive constant µ represents the mass of a meson. Since we are considering a bounded domain, the term µ 2 φ does not affect our arguments in the proof of the asymptotic stability. So, for simplicity this term will be omitted.
It is important to note that problem (3) is not naturally dissipative. So, the introduction of the dissipative mechanisms given by the terms in (2) are necessary to force the energy to decay to zero when t goes to infinity. In fact, the dissipative KGS equation has been widely studied, see for example the following references: [22], [24], [28], [29], [21], [14] and references therein. The majority of works in the literature deal with linear dissipative terms acting in both equations, except for the works [23] and [11].
Our paper is organized as follows. In section 2 we give the precise assumptions and state our main result, in section 3 we give an idea of the proof of existence and in section 4 we give the proof of the main theorem.

Main result.
In what follows let us consider the Hilbert space L 2 (Ω) of complex valued functions on Ω endowed with the inner product and the corresponding norm ||u|| 2 2 = (u, u) . We also consider the Sobolev space H 1 (Ω) endowed with the scalar product (u, v) H 1 (Ω) = (u, v) + (∇u, ∇v) .
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). This space endowed with the norm induced by the scalar product (u, v) H 1 0 (Ω) = (∇u, ∇v) is, thanks to the Poincaré's inequality a Hilbert space. We set the norms In the particular case when n = 2 we have the Gagliardo-Nirenberg inequality, We denote by ω the intersection of Ω with a neighborhood of ∂Ω in R 2 and we will call it a neighborhood of the boundary of Ω.
The following assumptions are made: Conjecture 1. We assume that a, b ∈ L ∞ (Ω) are nonnegative functions such that In addition, If a(x) ≥ a 0 > 0 in Ω, then we consider χ ω ≡ 1 in Ω, in Ω, then we consider χ ω ≡ 1 in Ω. Conjecture 2. We assume that ω satisfies the geometric control condition (mentioned above). The standard example is when ω is a neighbourhood of Γ(x 0 ) where and ν(x) is the unit outward normal at x ∈ Γ.
As an example of a domain Ω satisfying the above assumption let us consider the figure 1 (See [35]), although there exist a wide assortment of much more interesting examples as those ones considered in Bardos, Lebeau and Rauch [7].
As a consequence of assumption (2) it follows that there exists a couple (ω, T 0 ), with T 0 > 0, such that the following observability inequalities occur: and for some positive constant C = C(ω, T 0 ) and for all T > T 0 . The proof of (7) can be found in [36] and [32] while the proof of (8) is established in [35] and [7].
The energy associated to problem (1) is defined by Now, we are in position to state our main results.
(Ω) and assuming that assumption 1 holds and that α ≥ 5 2a 0 b 0 , then, there exists a unique regular solution to problem (1) such that , in the next theorem, below, we provide a local uniform decay of the energy. Indeed, we shall consider the initial data taken in bounded sets of H, namely, ||{ψ 0 , φ 0 , φ 1 }|| H ≤ L, where L is a positive constant.This is strongly necessary due to the non linear character of system (1) and since the energy E(t) is not naturally a non increasing function of the parameter t. Thus, the constants, C and γ which appear in (10) will depend on L > 0. Now, we are in position to state our stabilization result. Theorem 2.3. Assume that the assumption of Theorem 2.2 are in place and the assumption (2) hold. Then, there exist C, γ positive constants such that the following decay rate holds for every regular solution of problem (1) in the class given in previous theorem, provided the initial data are taken in bounded sets of H.
On the other hand, take α sufficiently large is natural to guarantee the dissipativity of the system.

Existence and uniqueness.
In this section we derive a priori estimates for the solutions of the Klein-Gordon-Schrödinger system (1). In what follows, for simplicity, we will denote u t = u . Let us represent by {ω ν } a basis in H 1 0 (Ω)∩H 2 (Ω) formed by the eigenfunctions of −∆, by V m the subspace of H 1 0 (Ω)∩H 2 (Ω) generated by the first m vectors and by The approximate system (11) is a finite system of ordinary differential equations which has a solution in [0, t m [. The extension of the solution to the whole interval [0, T ], for all T > 0, is a consequence of the first estimate we are going to obtain below.

A priori estimates
The First Estimate: Considering u = ψ m in the first equation of (11) and taking the real part, we obtain 1 2 Multiplying (13) and there exist Now, considering v = φ m in the second equation of (11) and making use of Hölder and Young inequalities, we deduce 1 2 So, where C := C ( φ 1 2 , ∇φ 0 2 ).
4. Uniform decay rates. In this section we work it regular solutions {ψ(t), φ(t), φ t (t)} to problem (1), that is, those ones that lie in Multiplying (50) by ψ and integrating over Ω, we have Taking the real part in (51) we obtain Multiplying the second equation by φ , integrating over Ω and making use of Green formula, we deduce that 1 2 Adding (52) and (53) we obtain Next, we will analyze the last term on the RHS of (54). We have, from Assumption 1 and making use of the Cauchy-Schwarz inequality that Combining (54), (55) and since α ≥ 5 2a 0 b 0 follow that β := α − 1 2a0b0 > 0 and it holds that where k = min{ 1 2 , β}. Remark 2. From (56) we deduce two facts: (i) the map t ∈ (0, ∞) → E(t) is non increasing, and, in addition, (ii) we have the following inequality of the energy for 0 ≤ t 1 ≤ t 2 < +∞, which will be crucial in the proof.
Let T 0 > 0 considered sufficiently large for our purpose. We will prove the following lemma: Proof. We argue by contradiction. Let us suppose that (57) is not verified and let {ψ k (0), φ k (0), φ k (0)} be a sequence of initial data where the corresponding solutions Since E k (t) is non-increasing and E k (0) remains bounded then, we obtain a subsequence, still denoted by {ψ k , φ k } which verifies We also have, employing compactness results (see Theorem 5.1 in Lions [34]) that φ k → φ strongly in L 2 (0, T ; L 2 (Ω)).
Consider the following problems and Note thatψ k = v k + ω k is the solution of (84), where v k comes from (85) and ω k is the solution of (86). In addition,ψ k = v k + ω k = 0 in Γ × (0, ∞) and Let us define the following linear and continuous form Clearly T is linear. We will prove that T is continuous. In fact, z is the solution of (88), so z satisfies the integral equation Thus, taking into account that S(t) L(L 2 (Ω)) ≤ M, we arrive at the following estimate (Ω)×L 1 (0,T,L 2 (Ω)) ) which shows that T is continuous.

KLEIN-GORDON-SCHRÖDINGER 863
Take T 0 large enough. From the inequality of energy we deduce and from Lemma (4.1) we have of (98) follows that Combining (99) and (100) it holds that Thus, that is, where 0 < α < 1. So, taking T 0 large enough for T > T 0 we obtain E(T ) ≤ E(T 0 ) ≤ αE(0). Hence, where, α < 1. Proceeding in a similar way we have done previously from T to 2T and we deduce, as before, E(2T ) ≤ αE(T ), for all T > T 0 , and, consequently, In general, E(nT ) ≤ α n E(0), for all T > T 0 .