EXPONENTIAL DECAY 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 n , n = 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 ∈ W 1 , ∞ (Ω) ∩ C ∞ (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 results given by Cavalcanti et. al in the reference [7].


Introduction.
We consider the following model of Klein-Gordon-Schrödinger equations with locally distributed damping where Ω is a bounded domain of R n , n = 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 ∈ W 1,∞ (Ω)∩C ∞ (Ω) nonnegative functions such that so that the nonlinearity ψ exists where the damping terms a(x)φ t and iαb(x)(−∆) are, in fact, effective and reciprocally. If the damping is effective in the 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 (7)) is a nonincreasing function of 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 [11,7]. While in [11] a full damping was in place in both equations, in contrast, in [7] a full damping has been considered in the Schröndinger equation but just a localized damping has been considered for the wave equation.
The main purpose of the present article is to generalize substantially both previous results just considering two localized dampings in both equations. A much more natural damping for the Schröndinger equation the should be ib(x)ψ instead of iαb(x)(−∆) 1 2 b(x)ψ; however, the second stronger one allows us to take advantage of a smoothing effect introduced by Aloui [1] for bounded domains, which will play an important role in the proof, as we will shown in section 4. As far as we are concerned this problem remains open for the weaker localized damping ib(x)ψ.

(3)
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 K-G-S equation has been widely studied, see for example the following references: [20,22,26,27,33,15] and references therein. The majority of works in the literature deal with linear dissipative terms acting in both equations, except for the works [21] and [11]. Very few is known regarding a localized damping acting in the wave equation for this system and, as far as we are concerned, there is no result in the literature dealing with a localized dissipation in both equations. A natural question arises from this context: it would be possible to consider a localized feedback i b(x)ψ acting in the Schrödinger equation (instead of the stronger present mechanism of damping given by the term iαb(x)(−∆) 1 2 b(x) ) in order to obtain some decay rate? This is a hard open problem to be solved yet since the so called smoothing effect, which is crucial in the proof is non longer valid. In fact, the smoothing effect is a natural property for the whole R n or non-trapping exterior domains as considered in the works of Constantin and Saut [14] or Burq, Gérard and Tzvetkov [10]. It has been considered previously in unbounded domains as powerful tool in order to achieve the exponential stability, see, for instance, the works [8] and [12]. On the other hand, for compact manifolds the smoothing effect has been introduced firstly by Aloui [1,2], by forcing it precisely by means of the stronger damping iαb(x)(−∆) 1 2 b(x) above mentioned where the the region ω such that b(x) > 0 satisfies the wellknown geometric control condition (GCG, in short). It is worth mentioning that the pioneers in using such property in order to stabilize the purely Schrödinger equation subject to the same kind of dissipation were Bortot and Corrêa [9]. In the present paper, we shall adopt similar ideas to our context and since we are considering a linear feedback acting in the Schrödinger equation and also a linear localized one acting in the wave equation, it is expected that the energy of the system decays to zero exponentially. To prove this fact is the main goal of this paper.
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 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, The following assumptions are made: Conjecture 1. We assume that a, b ∈ W 1,∞ (Ω)∩C ∞ (Ω) are nonnegative functions such that In addition, 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: The energy associated to problem (1) is defined by Now, we are in position to state our main results.
Setting H := {H 1 0 (Ω)∩H 2 (Ω)} 2 ×H 1 0 (Ω), 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 (8) will depend on L > 0. We shall denote d := c||b|| ∞ L, where c comes from the embedding H 1/2 (Ω) → L 4 (Ω). Now, we are in position to state our stabilization result. Theorem 2.2. Assume that assumptions (1) and (2) hold and moreover that α > or d is sufficiently small. Then, there exist C, γ positive constants such that 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.
The next proved theorem in [1] will be used in the proof of lemma 4.1.
Theorem 2.3. Let us consider b ∈ C ∞ such that ω = {b(x) = 0}} controls geometrically Ω and v is a solution of the following problem: Then, 3. 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 (12) 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 (12) and taking the real part, we obtain 1 2 where c is a positive constant which comes the embedding H and there exists C > 0, such that Now, considering v = φ m in the second equation of (12) and making use of Hölder and Young inequalities, we deduce 1 2 where C := C (||ψ 0 || 2 , ||φ 1 || 2 , ||∇φ 0 || 2 ). Applying Gronwall's lemma and considering (15) yields where C := C (||ψ 0 || 2 , ||φ 1 || 2 , ||∇φ 0 || 2 ), from which we deduce that The Second Estimate: Taking the derivative on time of the first equation in (12), considering u = ψ m and taking the real part of it, we have 1 2 (Ω) for n = 2 and using Hölder's generalized inequality and also considering the inequality ab ≤ 1 4ε a 2 + εb 2 and (14) to estimate the term on the right hand side of (21), we obtain 1 2 Taking the derivative in t of the second equation in (12), considering v = φ m and taking the real parte, we have 1 2 So, making use of the Hölder's generalized inequality combined with the inequality ab ≤ 1 4ε a 2 + εb 2 to estimate the term on the right hand side of (23), we get 1 2 Adding (22) and (24), integrating the result obtained and taking ε small enough, we infer where c and C are positive constants.
Making u = ψ m (0) in the first equation of (12) and v = φ m (0) in the second equation and observing the convergences in (12) Then, applying Gronwall's Lemma in (25), we obtain where and, furthermore, The Third Estimate: In (12) considering u = ψ m and taking the imaginary part, we obtain Noting that H 1 0 (Ω) → L 2 (Ω) and making use of the Hölder's inequality and taking inequality (5) into consideration, we deduce So, from (14), (19), (26), (32) and (33) we conclude Taking v = ∆φ m in the second equation of (12), we have We estimate the first term on the right hand side of (35) using the Hölder inequality and (5). Then, where 4. Uniform decay rates. In this section we work it regular solutions where L > 0. So, multiplying the first equation of (1) by ψ, the second equation by φ , integrating over Ω and making use of Green formula, we deduce that and Taking the real part in (38) and adding the obtained result with (39) we obtain Next, we will analyze the last term on the RHS of (40). We have, from Assumption 1, (5), (34), observing that for n = 2, D[(−∆) Combining (40) and (41) and considering α large enough such that β := α − where k = min{ 1 2 , β}. Remark 1. From (42) 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. We observe that in order to transform the energy E(t) in a non increasing function we could have considered d small enough instead of taking α sufficiently large, which would imply to take the initial data sufficiently small. Well, in any case, some kind of tribute must be paid in order to obtain uniform decay rates of the energy to the present system.
In order to prove Theorem (2.2) we proceed in several steps.
Step 1. Multiplying the first equation of problem (1) by ψ and the second equation by (q · ∇φ), where q ∈ (W 1,∞ (Ω)) n , and following (verbatim) the integration by parts of Lemma 3.7, Chap. I, of Lions [30] we deduce the following identity: In (43), for simplicity, we have omitted the variables of the functions under the integral signs and, in addition, we have used the convention of summation of repeated indexes.
Employing (43) with q(x) = m(x) = x − x 0 , for some x 0 ∈ R n , and taking (6) into account, we arrive at Now, multiplying the second equation of problem (1) by ξφ, with ξ ∈ W 1,∞ (Ω) and integrating by parts we obtain the following identity: Taking ξ = δ ∈ R in (45) and combining the obtained result with (44) we have Denoting and choosing δ = n−1 2 (having in mind that n = 2 in the present case) we deduce Next, we are going to estimate some terms in (48).
Now, from (81), (82) and (83) we deduce that On the other hand, from Assumption 1, namely, b(x) ≥ b 0 > 0 in ω, taking (87) into account and considering D[(−∆) From now on let us focus our attention on the coupled wave equation Let us divide our proof in two cases (in what concerns the limit φ above): (a) φ = 0.
Passing to the limit when k → +∞ in (89) taking into account the above convergence, we deduce that and for φ = v, we obtain, in the distributional sense that From Holmgren's uniqueness theorem for the wave equation we conclude that v ≡ 0, that is, φ ≡ 0. Returning to (90) we obtain the following elliptic equation for a. e. t ∈ (0, T ) : Multiplying (92) by φ we deduce that Ω |∇φ| 2 dx = 0, which implies that φ ≡ 0, which is a contradiction. Now, we consider the other case when we obtain Besides,Ê On the other hand, integrating (40) over (0, T ), we deduce From the fact that E k (t) ≥ E k (T ) for all t ∈ [0, T ] and taking (97) into account, we obtain Combining (79), (98) and making use of Cauchy-Schwarz inequality and considering the Assumption 1, we infer The last inequality yields for a large T , Having in mind that E k (t) ≤ E k (0) for all t ∈ [0, T ], applying inequality (99) and dividing both sides by Since in view of (81) we have then, from (100) there exists M > 0 such that , for all t ∈ [0, T ] and for all k ∈ N.
So, taking T 0 large enough for T > T 0 we obtain Thus, where, α < 1.
Proceeding in a similar way to that done previously from T to 2T and we deduce as before, E(2T ) ≤ αE(T ), for all T > T 0 , and, consequently, E(2T ) ≤ α 2 E(0), for all T > T 0 .
Let us consider, now, t > T 0 , then t = nT 0 + r for 0 ≤ r < T 0 . Thus, which implies the exponential stability.