Partial Interior Stabilization of a Coupled Wave Equations on an Exterior Bounded Obstacle

Similar works, based on the use of microlocal defect measures in the spirit of the article, have been achieved [5]. In the large time behavior of solutions of the wave equation were studied. The microlocal defect measures have been used to provide estimates of energy was shown in particular how these demonstrate the results of exact controllability, observation and stabilization [8]. without any assumption on the dynamics, the logarithmic decay of the local energy with respect to any Sobolev norm larger than the initial energy is proved [6]. In the two and three-dimensional system of linear thermoelasticity in a bounded smooth domain with Dirichlet boundary conditions were studied. In two space dimensions they proved a sufficient (and almost necessary) condition for the uniform decay under an assumption on the boundary of the domain and in three space dimensions sufficient conditions for the uniform decay are given [9].


Introduction
In this paper we study the stabilization of a coupled wave equations. More precisely, we consider the following initial and boundary value problem : The study of systems like (1)-(6) (and more generally coupled PDEs systems) is motivated by several physical considerations. In fact, There are many applied problems that can be modeled using coupled partial differential equations, for instance in heating processes, magnetohydrodynamics, quantum mechanics, optics, fluid dynamics....
Among the nowadays many contributions, using different methods and techniques, are given, and relevant reference therein [1,2].
One of the earliest tools in the stabilization analysis of partial differential equation is the micro-local default action of Gérard [3], Tartar [4].
Such techniques have been used firstly to study and to explicit the value of the best decay rate of damped waves equation [5], reduce the boundary and the regularity of the initial data or to show that the geometric condition for control by the board is required [6,7]. Similar works, based on the use of microlocal defect measures in the spirit of the article, have been achieved [5]. In the large time behavior of solutions of the wave equation were studied. The microlocal defect measures have been used to provide estimates of energy was shown in particular how these demonstrate the results of exact controllability, observation and stabilization [8]. without any assumption on the dynamics, the logarithmic decay of the local energy with respect to any Sobolev norm larger than the initial energy is proved [6]. In the two and three-dimensional system of linear thermoelasticity in a bounded smooth domain with Dirichlet boundary conditions were studied. In two space dimensions they proved a sufficient (and almost necessary) condition for the uniform decay under an assumption on the boundary of the domain and in three space dimensions sufficient conditions for the uniform decay are given [9].
Also, These techniques are used to study the stabilization of the wave equation in a domain with exterior Dirichlet condition [1], for the equation of damped waves equation in an outside field and under an "Exterior Geometric Control" condition inspired from the so-called microlocal condition of Bardos et al. [10] then for the stabilization of electromagnetic waves on an exterior bounded obstacle in 2D and 3D is treated, and under an exterior geometric control condition the behavior of the solution for large time is studied [11][12][13].
Later, in three dimension space and under a microlocal geometric condition, the rate of decay of the local energy for solutions of the Lamé system on exterior domain, with localized nonlinear damping was given in ref. [14].
Recently these techniques are also used to study the stabilization of different coupled equations and different results have been established in this domain, some results are given, by Duyckaerts [15] the exponential and the polynomial stabilization of a coupled hyperbolic-parabolic system of thermoelasticity are addressed with microlocal techniques, explained by Atallah-Baraket and Kammerer [16] the energy decay of thermoelasticity system with a degenerated second order operator in the Heat equation was studied, a stabilization problem for a coupled wave equations on a compact Riemanian manifold under a geometrical control condition was examined and a logarithmic decay result of the energy is given [13]. And finally in the exact controllability problem on a compact manifold for two coupled wave equations, with a control function acting on one of them only was treated [17].
For u=(u 1 ,u 2 ) solution of (7), we denote ( )( ) R E u t the local energy of at instant t>0 define by ( ) Now, according to the research of Moulahi [19], we recall that at the boundary point being the exterior normal to ∂Ω at x. with the assumption α≠1 we can consider (τ,η) as an element of T ∂Ω , and to look for its inverse image is the both characteristic sets means to look for λ∈R such that 1 ( , ; , ( )) = 0, ( , ; , and we write Hence, for the existence of such real λ, one of the two relations

 
According, we recall the following definition [12,14] Our aim in this work is to establish the energy decay and to give the best rate of convergence of a coupled damped wave equation On an exterior bounded obstacle. We prove this result in a geometric hypothesis and by using the arguments of the analysis microlocal.
The organization of this paper is as follows. In section 2, we give the main result and recalled some preliminary results. In section 3, we will study the poles of the resolvent, in the first, by means of conventional techniques is given a location on the low frequencies and by the defect measures theory we study the high frequencies. In section 4, the main results concerning the stability of systems are established. startsection section1@-3.5ex plus -1ex minus -.2ex2.3ex plus .2ex Preliminaries and Main result Let u=(u 1 ,u 2 ) then the system of equations (1)-(6) is equivalent to the following system Due to the nonlinear semi-group theory, it is well known that the problem (7) has an unique solution, obtained by using the Lummer-Philips theorem for an unbounded operator [18].
We consider the Hilbert space ( ) We can write the problem (7) as the following form where ( ) The problem (7) and (10) are equivalents if and only if that a A α has a domain ( ) The problem (7) has an unique solution, obtained by using the Hille-Yosida theorem for an unbounded operator. Now, we are going to make a description of a generalized bicharacteristic path and refer to the research of Lebeau G [5] for more details. The generalized bicharacteristic flow lives in  * Char T α ⊂ Ω  and for ρ∈Char α ,we denote by G(s,ρ) the generalized bicharacteristic path starting from ρ. Since char α is the disjoint union of char αΩ  T and  T if α>1or a Char α Ω  , L and  L if α<1. We shall consider separately the case where ρ belongs to each one of these sets. Moreover all the description below holds for |s| small, in the following we assume α>1.
Where (x(s),ξ) is the characteristic starting from the point (x,ξ) of [12] and we have one of the two relation be the outgoing (resp. incoming) bicharacterestic of  α . The generalized bicharacteristic path is such that Four possibilities may occur In particular, if 0<r one has x(s)∈Ω for small |s|≠0 We can see that the nature of the generalized bicharacteristic path changes when hitting the boundary, since it moves from char to char in 2 ii-and conversly from char to char in 2 iii-. Following ref. [14], we have: We will call generalized bicharacteristic path any curve which consists of generalized bicharacteristics of  α with possibility of moving from a characteristic manifold to another, at each of  ∂Ω , in the way indicated above.
In order to state the main results of this paper, we give the definition of outside geometric control condition ( OGCC) introduced [1] inspired of [5].
, T R >0 andω={a>0} We shall say that(ω,T R ) satisfy the outgoing geometric control condition (OGCC) above B R if every generalized geodesic path 1 γ derived, at time t=0 a point in ( ) T R + × Ω satisfies the following conditions •γ leave ×B R before the time T R .
•γ meet the region R + ×ω between the times 0 and T R .
Let t>0, we set that is a additive function and we set ( ) = ( ) lim Theorem 0.4 Assume α≠1 and under the hypothesis of (OGCC) above the B R , for any ( ) ∞ there exists c>0 such that for all g∈H supported in B R we have the following estimate of the energy with support in B R and for all λ≠0 and Imλ≤ 0 we have ( ) is the unique solution satisfies the outgoing radiation condition (OGRC) of the following problem: Firstly, we recall that u=(u 1 , u 2 ) satisfy the outgoing radiation condition if the following identity satisfy Now, let ψ the difference between two solution of (17). Then, ψ satisfy the homogenous problem with Dirichelet boundary. By integration on Ω R for R large enough, we have In particular, given that the real part of (19) is zero, gives ( ) In the following, we study the outgoing resolvent ( ) a α λ  on the real axis. We show firstly that it has no real pole and secondly it is bounded in the neighborhood of 0 in any angular sector does not meet the imaginary axis iR.

Boundedness of the Resolvent Near Zero
Before beginning the study of holomorphic of the resolvent ( ) a R α λ , Let us note that we can see (17) as a perturbation of the following problem in a free space The solution of the eqn. (23) is given by  is the free outgoing resolvent given by for r large [6].
equal to 1 on a neighborhood of ∂Ω with support in B R .The parameter ξ being chosen and subsequently fixed the following discussion. And w is completely determined by g and v is completely determined by w. The problem then is to determine the function g for which the function u verifies (22).
By (24) and the oscillatory integral theory we can see that Where C λ is bounded uniformly on any compact Riemanian Logarithmic surface [20]. Now we set = v w φ − Θ satisfy the following problem by the ellipticity argument we deduce that we obtain by eqn. (25) Moreover  λ contains only derivations of order less than or equal to 1 of φ, by the Rillich identity, we deduce that  λ is compact operator on ( ) 2 2 ( ) L Ω and this implies that ( ) a α λ  is meromorphic on C ( resp. Riemannian logarithmic surface ) if d odd (resp. d is even ).

Low Frequencies
First we prove that the resolvent    [6]. So in both cases we give a uniform bound of norm of the resolvent from 2 2 ( ( )) com L Ω into 1 0,loc H for λ close to zero and in the Λ γ . By choosing a finite number of real γ i it covers a neighborhood of upper half-plane (which is excluded 0)∪Λ γi which leads to the conclusion that the resolvent is bounded near zero and we have the assumption (1.1) in ref. [6].
which implies that one have to λ goes to zero and | ( ) / 2 | arg λ π π + ≤ , the following behavior: Proposition 0.8: α χ  does not allow the accumulation point, has no zero on the real axis and admits the following behavior where rank ( d )≤1 and  d is analytic at λ=0. We begin by the following lemma inspired from [2] which will be useful to the proof of our proposition.
which gives a good uniform bound on the norm of the resolvent from Then we obtain And u satisfy the (OGRC). It follows that

Studies of High Frequencies
This section is devoted to the proof of Theorem 0.11.
Theorem 0.11: There exists δ 0 >0and λ 0 >0 such that the truncated outgoing resolvent Firstly, we denote that the operator ( ) a R α λ defined by 2 2 ( ( )) L Ω in H is meromorphic on C (resp. the Riemann surface of the logarithm) if n is even (resp. odd), holomorphic on {Imλ<0}. Moreover, c, δ 0 and λ 0 don't depend of and we can check that . This allows us to limit our study to Re(λ)>0. The proof of (37) is based on a reductio ad absurdum argument. We assume that for any c ( in particular for n =c=n∈N, there exists f n ∈(L 2 ) 2 and such that Imλ n →0 and Reλ ≥ n ( we assume for example Reλ ≥ 0 )such that We note that n n n n a n n n n n n a n n n n equal to the id near the boundary and supported in B R . We set We can see that where  α is the outgoing free resolvent of the this inequality is deduced from the method of stationary phase for Reλ n → 0 ( [8]). We can see that is bounded n w in  L Ω in fact we have: such that µ(ω)≠0 On ω 1 ∪ω 2 where ω 1 and ω 1 are two defined by: We have µ(ω)=µ(ω 1 )+µ( ω 2 ) Or from Lemma 0.13µ( 1 )=0, in fact: if ρ∈ω 1 there exists s such that G(s)ρ∉B R then G(s)ρ is outgoing and by lemma 0.13 we obtain µ(ω 1 )=0, And it follows that µ(ω)=µ(ω 1 ) We note that if Ω is non-captive, ω=ω 1 On the other hand, And by using the fact that   And as µ(ω)≠0, it follows that for sufficiently large n we get that µ(G(s)ω 2 >1, which contradicts(41) startsection section1@-3.5ex plus -1ex minus -.2ex2.3ex plus .2ex Stabilization Using the Theorem 15 and the bound of resolvent in a neighborhood of zero we deduce the decreasing exponential (resp. polynomial) of energy in odd dimensional (resp. even dimensional). The Theorem 15 give a stabilization result by the boundary for the local energy for a coupled wave equation, on the exterior domain = \ d R O Ω . Some results of decreasing exponential has proved in ref. [1]. The proof is based on a method of the resolvent (Location of poles) in which we use a lemma recovery and a theorem of propagation for microlocal defect measures

Proof
We will proceed in similar way to that one in ref. [21]. Let consider the function ϕ∈ ∞ such that: Note that by a simple calculation, one can find that for Im<0