From our point of view one of **the** most efficient approaches to **the** question of local en- ergy **decay** is **the** theory of resonances. Resonances correspond to **the** frequencies and rates of dumping of signals emitted by **the** black hole in **the** presence of perturbations (see [9, Chapter 4.35]). On **the** one hand these resonances are today an important hope of effectively detecting **the** presence of **a** black hole as we are theoretically able to measure **the** corresponding gravi- tational waves. On **the** other hand, **the** distance of **the** resonances to **the** real axis reflects **the** stability of **the** system **under** **the** perturbation: larger distances correspond to more stability. In particular **the** knowledge of **the** localization of resonances gives precise informations about **the** **decay** of **the** local **energy** and its rate. **The** aim of **the** present paper is to show how this method applies to **the** simplest model of **a** black hole: **the** De Sitter–Schwarzschild black hole. In **the** euclidean space, such results are already known, especially **for** non trapping geome- tries. **The** first result is due to Lax and Phillips (see their book [15, Theorem III.5.4]). They have proved that **the** cut-off propagator associated to **the** **wave** **equation** outside an obstacle in odd dimension ≥ 3 (more precisely **the** Lax–Phillips semi-group Z(t)) has an expansion in terms of resonances if Z(T ) is compact **for** **a** given T . In particular, there is **a** uniform exponential **decay** of **the** local **energy**. From Melrose–Sj¨ ostrand [18], this assumption is true

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Since **the** early works of Lax and Phillips [18, 21] on exterior Dirichlet problem **for** **the** **wave** operator ∂ t 2 − ∆, it is well known that geometry of **the** boundary plays **a** crucial role in **the** time **decay** properties
of **the** solution whether it is trapping or not. If **the** boundary is not trapping, it is known [22] that local **energy** decays at exponential rate. **For** **the** same problem, **under** very general trapping assumptions, N. Burq proved [3] that there exists an exponentially small neighbourhood of **the** **energy** real axis, free of resonances, which implies **a** logarithmic **decay** rate of local **energy**.

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We know from [ Har85 , Leb96 ] that **a** weak assumption on **the** absorption index **a** (**for** instance **the** dissipation is effective on any open subset of **the** domain) is enough to ensure that **the** **energy** goes to 0 **for** any initial datum.
Uniform exponential **decay** has been obtained in [ RT74 , BLR92 ] **under** **the** now usual geomet- ric control **condition**. Roughly speaking, **the** assumption is that any (generalized) bicharacteristic (or classical trajectory, or ray of geometric optics) meets **the** damping region (in **the** interior of **the** domain or at **the** boundary). **For** **the** free **wave** **equation** on **a** subset of R d , **the** spatial

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It is known that as soon as **the** damping b ≥ 0 is non trivial, **the** **energy** of every solution converge to 0 as t tends to infinity. On **the** other hand **the** rate of **decay** is uniform (and hence exponential) in **energy** space if and only if **the** geometric control **condition** [ 2 , 5 ] is satisfied. Here we want to explore **the** question when some trajectories are trapped and exhibit **decay** rates (assuming more regularity on **the** initial data). This latter question was previously studied in **a** general setting in [ 19 ] and on tori in [ 11 , 21 , 1 ] (see also [ 12 , 13 ]) and more recently by Leautaud-Lerner [ 18 ]. According to **the** works by Borichev-Tomilov [ 3 ], stabilization results **for** **the** **wave** **equation** are equivalent to resolvent estimates. On **the** other hand, Theorem 1.1 implies easily (see Section 2.2 ) **the** following resolvent estimate

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This limit proves that, along radial null geodesic, **a** particle goes to timelike infinity in finite Boyer-Lindquist time (recall that along these geodesic, t − r ∗ and t + r ∗ are constants).
This geometric property will lead to several differences compared to other spacetime. In **the** massless case, we would have to put boundary **condition**. In fact, even in **the** massive case, **a** confining potential appear near **the** Anti-de Sitter infinity. Combined with **the** usual bump created by **the** photon sphere, we expect that **the** **energy** will not **decay** really fast. In fact, **for** **the** **wave** **equation**, **the** local **energy** is decaying only logarithmically as was proven by G. Holzegel and J. Smulevici [24], [26]. In this work, we obtain **a** logarithmic lower bound as in [26].

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In this paper we provide **a** complete mathematical analysis of **the** optimal design problem (P γ,T ).
**The** article is structured as follows. In Section 2 (see in particular Theorem 2 ) we give **a** sufficient **condition** ensuring **the** existence and uniqueness of an optimal set. More precisely, we prove that, if **the** initial data **under** consideration belong to **a** suitable class of analytic functions, then there always exists **a** unique optimal domain ω, which has **a** finite number of connected components **for** any value of T > 0 but some isolated values. In Section 3 , we investigate **the** sharpness of **the** assumptions made in Theorem 2 . More precisely, in Theorem 3 we build initial data (y 0 , y 1 ) of

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tractor in **the** sense that there is an invariant compact set **A** ⊂ H 0 1 (Ω) ×L 2 (Ω), which consists of all **the** bounded trajectories and such that any regular set B bounded in (H 2 (Ω) ∩H 0 1 (Ω)) ×H 0 1 (Ω) is attracted by **A** in **the** topology of H 0 1 (Ω) ×L 2 (Ω). Notice that this concept of weak attractor is **the** one of Babin and Vishik in [3]. At this time, **the** asymptotic compactness property of **the** semilinear **damped** **wave** **equation** was not discovered and people thought that **a** strong attractor (attracting bounded sets of H 0 1 (Ω) × L 2 (Ω)) was impossible due to **the** lack of regularization property **for** **the** **damped** **wave** **equation**. Few years later, Hale [17] and Haraux [19] obtain this asymptotic compactness property and **the** existence of **a** strong attractor. Thus this notion of weak attractor has been forgotten. It is noteworthy that it appears again here. Notice that we cannot hope **a** better attraction property since even in **the** linear case, {0} is not an attractor in **the** strong sense.

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There is **a** natural choice of Lagrangian states which is associated to **the** vertical bundle of **the** **energy** layer. These particular states were used by Anantharaman and Nonnenmacher in **a** selfadjoint setting [1, 4] and also by Schenck in [28] in **the** context of **the** **damped** **wave** **equation**. In these references, these Lagrangian states remain **under** control up to large logarithmic times, due to **the** global structure of **the** geodesic flow (it was supposed to be Anosov). Indeed, **the** Anosov hypothesis implied that **the** associated Lagrangian submanifolds become uniformly close to **the** unstable foliation and that they do not develop caustics **under** **the** evolution (thanks to **the** absence of conjugate points) — see [28, §4] **for** details.

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information of **the** quasi-modes restricted to **the** interface by transmission conditions. Then in Section 4, we prove **the** propagation theorem **for** **the** hyperbolic problem in Ω 2 which will lead
to **a** contradiction. We need to analyze two semi-classical scales corresponding to **the** elliptic and hyperbolic region, connected by **the** transmission **condition** on **the** interface. Finally in Section 5, we construct **a** sequence of quasi-modes saturating **the** inequality ( 1.5 ) in **a** simple geometry. In particular this proves **the** optimality of **the** resolvent estimate. We collect various toolboxes in **the** final section of **the** appendix.

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We will now study **the** solutions of **the** quantization conditions (B.10) and (B.11), taking into account **the** relations (B.9) between **the** wavevectors k, k ′ and **the** **energy** E. To describe **the** full spectrum
(which we plan to present in **a** separate publication), we would need to consider several r´egimes, depending on **the** relative scales of E and h. However, since we are only interested here in proving Proposition B.1, we will focus on **the** r´egime leading to **the** smallest possible values of | Im e ζ | = | Re z|. What characterizes **the** corresponding eigenmodes v(x) ? From (B.3) we see that **the** mass of v(x) in **the** **damped** region, 2 R σ 1/2 |v(x)| 2 dx, should be small compared to its full mass. Intuitively, if such **a**

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— I R is **the** reflected **wave**: **the** phase has **a** singular, Airy type term (easy to deal with).
Since difficulties appear near rays issued from Q 0 which hit **the** boundary without being deviated, only
**the** diffracted **wave** part (containing I D ) will be dealt with here. Notice that this is **the** regime which
provides counter-examples in higher dimensions. We have I D (τ, Q, Q 0 ) =

mizing **the** tracking error by means of **a** control which is pointwise as small as possible. **The** appearance of **the** L ∞ –control costs leads to nondifferentiability.
**The** analytic and efficient numerical treatment of this nonsmooth problem by **a** semi-smooth Newton method stands in **the** focus of this work. We prove superlinear convergence of this iterative method and present numerical exam- ples.

focusing non-linear **wave** **equation**. To be published in Acta Math., 2006.
[KV08] Rowan Killip and Monica Vi¸san. **The** focusing **energy**-critical nonlinear schr¨ odinger **equation** in dimen- sions five and higher. Preprint, 2008.
[LS95] Hans Lindblad and Christopher D. Sogge. On existence and scattering with minimal regularity **for** semilinear **wave** equations. J. Funct. Anal., 130(2):357–426, 1995.

TIME-RESPONSE)
**A**. Spectral Rigidity ` **a** la Berry (diagonal approximation)
1. Form Factor
In **a** complex **wave** system, either **a** chaotic cavity or **a** disordered medium, it is not possible to give **a** detailed description of **the** frequency spectrum by providing **a** determined sequence of numbers. Hence, **the** frequency spectrum is too complicated to be explained level by level but may nevertheless be studied through **a** statistical approach, in **a** way quite analog to **the** statistical approach of **a** gas of interacting particles. In **the** study of spectral properties, **a** key role is played by **the** spectral correlations and their description in terms of adequate quantities will be our concern in this section. With these quantities we will be in **a** position to establish how certain universal features predicted by RMT can be recovered from **a** global knowledge of chaotic dynamics but also in what respect some non-universal behavior can be related to **the** shortest POs of **the** system.

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As show **the** results of Section 2 , if we suppose **a** suitable asymptotic behavior of **the** initial value (u(0) ∈ X if α 6 N 4 , u(0) ∈ H 1 (R N ) if α > 4
N ), then we have **a** sharp lower bound. In particular,
**under** **the** hypotheses of Section 2 , such results do not allow estimates of type ( 2.1 ), **for** any nontrivial solution of ( 1.1 ), **for** some r > 2 and δ > 0 (see Remark 2.6 ). In this section, we establish some lower bounds which eventually allow estimates on **the** above type, only if α is small enough (see Theorem 3.5 below). **The** loss of sharp estimate is compensated by **a** weaker assumption on u(0), that is u(0) ∈ L 2 (R N ) if α 6 4

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ing **the** whole space : this is **the** so called Sommerfeld radiation **condition** that represents **the** fact that, at infinity, **the** solution looks like an outgoing spheri- cal **wave** which decays as |x| − d−1 2 where d is **the** space dimension. In **the** case
of **a** homogeneous closed waveguide, that is when **the** domain of propagation is an infinite cylinder with bounded cross section, **the** behavior at infinity of **the** solution is quite different than in **the** homogeneous whole space : in particular, **the** solution does not **decay** any longer at infinity. As **a** consequence, **the** radi- ation **condition** is of quite different nature and relies on **the** decomposition of **the** solution as an infinite sum of evanescent modes and **a** finite sum of (appro- priately chosen) propagative modes : these are **the** so-called outgoing modes. **The** situation **for** **a** periodic closed waveguide that is considered in this paper is similar in nature to **the** case of **the** homogeneous waveguide. However, **the** notion of outgoing modes is much more delicate (we shall pay **a** lot of attention to **a** precise definition and description of such modes) and **the** analysis relies on quite different mathematical tools (**the** Floquet-Bloch transform, spectral theory of operators depending analytically on **a** parameter, complex contour integral techniques etc.).

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