ENERGY DECAY RATES FOR SOLUTIONS OF THE WAVE EQUATION WITH LINEAR DAMPING IN EXTERIOR DOMAIN

. In this paper we study the behavior of the energy and the L 2 norm of solutions of the wave equation with localized linear damping in exterior domain. Let u be a solution of the wave system with initial data ( u 0 ,u 1 ). We assume that the damper is positive at inﬁnity then under the Geometric Control Condition of Bardos et al [5] (1992), we prove that:


(Communicated by Irena Lasiecka)
Abstract. In this paper we study the behavior of the energy and the L 2 norm of solutions of the wave equation with localized linear damping in exterior domain. Let u be a solution of the wave system with initial data (u 0 , u 1 ). We assume that the damper is positive at infinity then under the Geometric Control Condition of Bardos et al [5] (1992), we prove that: 1. If (u 0 , u 1 ) belong to H 1 0 (Ω) × L 2 (Ω) , then the total energy Eu (t) ≤ C 0 (1 + t) −1 I 0 and u (t) 2 2. If the initial data (u 0 , u 1 ) belong to H 1 0 (Ω) × L 2 (Ω) and verifies d (·) (u 1 + au 0 ) L 2 < +∞, then the total energy Eu (t) ≤ C 2 (1 + t) −2 I 1 and u (t) 2 L 2 ≤ C 2 (1 + t) − Here ∆ denotes the Laplace operator in the space variables. a (x) is a nonnegative function in L ∞ (Ω). Let and H = H D (Ω) × L 2 (Ω) , the completion of (C ∞ c (Ω)) 2 with respect to the norm A is a unbounded operator on H with domain Let n ∈ N and (u 0 , u 1 ) ∈ D (A n ). Linear semigroup theory applied to (1) (see for example [2]), provides existence of a unique solution u in the class (u, ∂ t u) ∈ C k R + , D A n−k , with k ≤ n.
Moreover, if (u 0 , u 1 ) is in H 1 0 (Ω) × L 2 (Ω), then the system (1), admits a unique solution u in the class u ∈ C 0 R + , H 1 0 (Ω) ∩ C 1 R + , L 2 (Ω) . Let us consider the energy at instant t defined by The energy functional satisfies the following identity for every T ≥ 0. Zuazua [31], Nakao [26], Dehman et al [12], Aloui et al [1] and Joly et al [19] have considered the problem for the Klein-Gordon type wave equations with localized dissipations. For the Klein-Gordon equations the energy functional itself contains the L 2 norm and boundedness of L 2 norm of the solution is trivial. Thus under some assumptions on the support of the damper a we can show that the energy decays exponentially while for the system (1) the energy decay rate is weaker and more delicate.
In the case of damping localized in a compact set of Ω Nakao in [24] proved that the local energy decays exponentially if d is odd and polynomially if d is even under the Lions's geometric condition. Combining the definition of a non-trapping obstacle and the geometric control condition of Bardos et al [5], Aloui et al [2] introduced the exterior geometric control condition which is a sufficient condition for the stabilization of the local energy, more precisely they proved that the local energy decays exponentially if the space dimension d is odd. In [21] Khenissi showed that the estimate (3) holds if d is even. Recently in [9], using a nonlinear internal localized damping, Daoulatli obtained various decay rates, depending on the behavior of the damping term. We also mention the result of Daoulatli et al [10] on the behaviors of the local energy for solutions of the Lamé system in exterior domain. Finally, we quote the result of Bchatnia and Daoulatli [4] on the decay rates of the local energy for solutions of the wave equation with localized time dependent damping in exterior domain. On the other hand, Dan-Shibata [8] studied the local energy decay estimates for the compactly supported weak solutions of (1) with a (x) = 1 where If we assume that a (x) ≥ 0 > 0 in all of Ω, then we know that for weak solution u to the system (1) with initial data in H 1 0 (Ω) × L 2 (Ω). Nakao in [25] obtained the same estimates in (4) for a damper a which is positive near some part of the boundary (Lions's condition) and near infinity.
Furthermore Ikehata and Matsuyama in [18] obtained a more precise decay estimate for the total energy of solutions of the problem (1) with a (x) = 1 and for weighted initial data Especially this estimate seems sharp for d = 2 as compared with that of [8]. Ikehata in [17] derived a fast decay rate like (5) for solutions of the system (1) with weighted initial data and assuming that a (x) ≥ 0 > 0 at infinity and O = R d \Ω is star shaped with respect to the origin.
For another type of total energy decay property we refer the reader to [16,20,28,3,27] and references therein.
Before introducing our results we shall state several assumptions: Definition 1.1. (ω, T 0 ) geometrically controls Ω, i.e. every generalized geodesic travelling with speed 1 and issued at t = 0, meets the set ω in a time t < T 0 .
This condition is called Geometric Control Condition (see e.g. [5] ). We shall relate the open subset ω with the damper a by We note that according to [5] and [7] the Geometric Control Condition of Bardos et al is a sharp sufficient condition for the stabilization of the wave equation in bounded domain.
The goal of this paper is to prove that for a damper a positive near infinity and under the geometric control condition of Bardos et al [5], the estimates in (4) hold for all solutions of the system (1) with initial data in H 1 0 (Ω) × L 2 (Ω) and to show that the estimates in (5) hold for all solutions of the system (1) with weighted initial data. Moreover we show that for every p ∈ N * there exists a initial data in and for some C > 0 depending on the norm of the initial data.
Then there exists C 0 > 0 such that the following estimates for all t ≥ 0, hold for every solution u of (1) with initial data (u 0 , u 1 ) in H 1 0 (Ω) × L 2 (Ω), where As a corollary of theorem 1.2 we have: Proposition 1. Let n ∈ N * . We assume that Hyp A holds and (ω,T 0 ) geometrically controls Ω. Let (u 0 , u 1 ) in D (A n ), such that u 0 ∈ L 2 (Ω). Then the solution u of (1) satisfies E ∂ n t u (t) ≤ C n (1 + t) −n−1 I 0,n for all t ≥ 0, where C n is a positive constant independent of the initial data and Next we give the result on the decay rates for weighted initial data. We define Theorem 1.3. We assume that Hyp A holds and (ω,T 0 ) geometrically controls Ω.
Then there exists C 2 > 0 such that the following estimates We note that we obtain the same result of [17] when O is star shaped which is special case of non trapping obstacle. Our geometric condition allows us to consider also trapped obstacles in which the damper control the trapped rays.
Let ψ ∈ C ∞ c R d such that 0 ≤ ψ ≤ 1 and The key idea of the proof of theorem 1.2 and theorem 1.3 is to use the following auxiliary functional Then to show some observability estimates for the local energy using observability estimates for the wave equation in bounded domain. In bounded domain to prove such estimates we can use the theorem of propagation of singularities of Melrose-Sjöstrand or the notion of microlocal defect measure (see [14] and [13] for the definition and properties of microlocal defect measure) and exploiting their property. Especially the fact that the measure associated to a bounded sequence of solutions of the conservative wave equation propagates along the generalized bicharacteristic flow of Melrose-Sjöstrand.
In the sequel we suppose that the obstacle O is contained in B R0 for some positive R 0 .
The rest of the paper is organized as follows. The section 2 is devoted to the proof of theorem 1.2 and proposition 1 and in section 3 we give the proof of theorem 1.3.

Preliminary results.
We prove an observability estimate for the local energy of solutions of the system 1, the proof is based on an observalility estimate for the non homogeneous wave equation in bounded domain.
Proposition 2. We assume that Hyp A holds and (ω,T 0 ) geometrically controls Ω. Let δ > 0 and R > R 0 . There exist T > T 0 and C T,δ,R > 0, such that the following inequality t+T t holds for every t ≥ 0 and for all u solution of (1) with initial data (u 0 , u 1 ) in H.

Proof.
To prove this result we argue by contradiction: If (6) was false, there would exist a sequence of positive numbers (t n ) and a sequence of solutions (u n ) such that tn+T tn Setting It is clear that (7) , Let Z n be the solution of the following system Therefore E vn−Zn (0) = 0 and So we obtain the following energy identity for t ≥ 0. Using Holder's inequality, we deduce the following energy inequality Now using (8) and the fact that a (x) is a function in L ∞ (Ω) , we infer that On the other hand, (8) and (11) combined with the result above, gives The system (9) is conservative, therefore Therefore, along a subsequence,(Z n ) is convergent to a function with respect to the weak topology. Since Z satisfies Therefore we have We remind that x ∈ R d , |x| ≥ L ⊂ ω. By a classical result of unique continuation (see [29]), we see that there exists . Setting w n = θZ n . Using Poincare's inequality and (13) , we infer that w n is bounded in , therefore w n converges weakly to zero in L 2 ((0, T ) × (Ω ∩ B R2+1 )) . Now using Rellich's theorem we deduce that We take R 1 and R 2 such that, We multiply the Eq (9) by ϕψ 2 Z n and integrate over (0, T ) × Ω, we obtain Using Young's inequality and the fact that ϕ is in C ∞ c (0, T ), we infer that there exists a positive constant c such that Since the support of ψ is contained in {|x| ≥ L} and a (x) = 1 on {|x| ≥ L} , we get Combining the estimate above with (12) and (14) T − we note that in the inequality above we have used the fact that Let χ ∈ C ∞ c R d such that χ = 1 on {|x| ≤ 2L} and the support of χ is contained in {|x| ≤ R 1 } . Setting W n = χZ n , then W n is a solution of the following system 0)) .

In addition we have
. We take T > max (T 0 + 1, T 1 ) , therefore (ω ∩ B R1 , T − 2 ) geometrically controls Ω ∩ B R1 . Then using the observability estimate for the non homogenuous wave equation in bounded domain (see for example [11]), we obtain We note that to prove the estimate above we can use theorem of propagation of singularities of Melrose-Sjöstrand or the notion of microlocal defect measure (see [14] and [13] for the definition and properties of microlocal defect measure). Since gives So using (14) , we infer that Combining the estimate above with (12) , we obtain On the other hand, the energy estimate for the non homogeneous wave equation for s ≥ . Integrating the estimate above between and T − , we infer that there exists a positive constant C = C (T, ) Combining the result above with (14) and using the fact that χ = 1 on {|x| ≤ 2L} , we infer that This result combined with (15), gives with R 1 ≥ R + T.
Using the finite speed propagation property,we deduce that where t ∈ [0, T ] and −t ≤ s ≤ T − t. Let η be a nonnegative function in C ∞ c (0, T ) such that the support of η is contained in [ , T − ] and T 0 η (s) ds ≥ T.
Multiplying both sides of the estimate (19) by η (s + t) and integrating between −t and T − t, we obtain for all t ∈ [0, T ] . Using the fact that R 1 ≥ R + T and making some arrangement, we obtain Now passing to the limit and using (18), we deduce that By taking into account of the result above and (11) we deduce that The fact that the energy of v n is bounded, gives By the dominated convergence theorem, we infer that On the other hand, Poincare's inequality combined with (11) and (14) , gives

MOEZ DAOULATLI
So we conclude that In order to prove theorem 1.2 we need the following result on the auxiliary function X.
Setting v = (1 − ψ) u where u is a solution of (1) with initial data in H 1 0 (Ω)×L 2 (Ω). Let where k is a positive constant. We have Proof. Noting that for each (u 0 , u 1 ) in H 1 0 (Ω) × L 2 (Ω) the solution u of (1) is given as a limit of smooth solution u n with initial data (u n,0 , u n,1 ) smooth such that (u n,0 , u n,1 ) −→ n→+∞ (u 0 , u 1 ) in H 1 0 (Ω) × L 2 (Ω). Note that uniformly on the each closed interval [0, T ] for any T > 0. Therefore we may assume that u is smooth.
Using the fact that v is a solution of (21) and that Thus we obtain d dt where w = ψu. By taking into account that the support of (1 − ψ) is contained in the set {x ∈ Ω, a (x) > 0 }, we infer that We remind that w = ψu, therefore there exists positive constantC > 0 such that This gives Integrating the estimate above between t and t + T , we get (20).

Proof of Theorem 1.2.
In the sequel C, C T and C T,δ denote a generic positive constants and any changes from one derivation to the next will not be explicitly outlined.

MOEZ DAOULATLI
According to lemma 2.1, The estimate (6), gives t+T t Ω∩B 2L From the following estimate we deduce that t+T t Ω∩B 2L (20) and the estimate above gives It is clear that We choose δ and k such that Therefore using (27) we obtain iT Ω a (x) |∂tu| 2 dxds ≤ 0, this gives Using (20) and (26) , we conclude that there exists a positive constant C such that with Poincare's inequality and the fact that the energy of u is decreasing gives Combining the last two estimates, we get The energy decay estimate follows from (30) and the fact that This finishes the proof of theorem 1.2, now we give the proof of proposition 1.

2.3.
Proof of Proposition 1. Let n ∈ N and u solution of (1) with initial data (u 0 , u 1 ) in D (A n ) such that u 0 ∈ L 2 (Ω). We set u n = ∂ n t u. First we prove that for all n ∈ N there exists C n > 0 such that where We will prove the result above using induction. For n = 0, let u be a solution of (1) with initial data (u 0 , u 1 ) in D A 0 such that u 0 ∈ L 2 (Ω). From (30) we infer that there exists C 0 > 0, such that We assume that for p ∈ N there exists C p > 0 such that the following estimate holds, for all solution u of (1) with initial data (u 0 , u 1 ) in D (A p ) such that u 0 ∈ L 2 (Ω). Let u be a solution of (1) with initial data (u 0 , u 1 ) in D A p+1 such that u 0 ∈ L 2 (Ω).

MOEZ DAOULATLI
∂ t u (0) = u 1 ∈ L 2 (Ω). According to (33), we have Let ψ ∈ C ∞ c R d such that 0 ≤ ψ ≤ 1 and where k is a positive constant. u p+1 satisfies , Then we know from (29) that Multiplying the estimate above by (1 + t + T ) p+1 , we obtain Therefore using (33) , (34) and the fact that we deduce that for any q ∈ N * 1 2 We deduce that We remind that we have proved that +∞ 0 (1 + s) n E un (s) ds ≤ C n I 0,n . Now the energy decay estimate follows from the fact that for all t ≥ 0. By taking into account of the estimate above, we infer that, u is a solution of the following system therefore ∂ n−1 t u, ∂ n t u ∈ C 1 (R + , H) . Using Eq (36) , we infer that 3. Proof of Theorem 1.3. This section is devoted to the proof of theorem 1.3. We begin by giving some preliminary results.
3.1. Preliminary results. The following result is a observability estimate for the weighted local energy of solutions of the system (1) .
Proposition 3. We assume that Hyp A holds and (ω,T 0 ) geometrically controls Ω. Let δ > 0 and R > R 0 . There exist T > T 0 and C T,δ,R > 0, such that the following inequality holds for every t ≥ 0 and for all u solution of (1) with initial data (u 0 , u 1 ) in H 1 0 × L 2 . Proof. To prove this result we argue by contradiction: If (37) was false, there would exist a sequence of positive numbers (t n ) and a sequence of solutions (u n ) such that tn+T tn

MOEZ DAOULATLI
We may assume that t n −→ n→+∞ +∞ (if the sequence t n is bounded we can argue as in the proof of proposition 2). Setting and v n = (1 + t n + ·) 1/2 u n (t n + ·) λ n .
Therefore from (38) we infer that It is clear that v n is a solution of the following system   It is easy to see that (39), gives We conclude that tn Ω |u n (s)| 2 dxds We multiply the equation satisfied by u n by (1 + t) ∂ t u n and integrating between t n and t n + t, we obtain thus exploiting (39) , we infer that On the other hand, we have (1 + t n + t) E un (t n + t) + (1 + t n + t) −1 Ω |u n (t n + t)| 2 dx dt Let Z n be the solution of the following system    ∂ 2 t Z n − ∆Z n = 0 R + × Ω, Z n = 0 R + × Γ, (Z n (0) , ∂ t Z n (0)) = (v n (0) , ∂ t v n (0)) .  To complete the proof we argue as in the proof of the proposition 2, by taking into account of (42) and (43).
We integrate the inequality above between t and t + T , we obtain (44). Finally it is clear that, Therefore from (50) , we deduce (1 + s) E u (s) ds ≤ CI 1 .
The energy decay estimate follows from the fact that This finishes the proof of theorem 1.3.