two-dimensional sphere and {b > 0} ∩ N ε = ∅, where N ε is a neighbourhood of an equator of S 2 .
This result is generalised in [LR97] **for** a **wave** **equation** **damped** on a (small) part of **the** boundary. In this paper, **the** authors also make **the** following comment about **the** result they obtain:
“Notons toutefois qu’une ´etude plus approfondie de la localisation spectrale et des taux de d´ecroissance de l’´energie pour des donn´ees r´eguli`eres doit faire intervenir la dynamique globale du flot g´eod´esique g´en´eralis´e sur M . Les th´eor`emes [LR97, Th´eor`eme 1] et [LR97, Th´eor`eme 2] ne four- nissent donc que les bornes a priori qu’on peut obtenir sans aucune hypoth`ese sur la dynamique, en n’utilisant que les in´egalit´es de Carleman qui traduisent “l’effet tunnel”.”

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If Ω is a bounded manifold, **the** uniform positivity of hγi T (x, ξ) in Σ **for** some T > 0 implies that
**the** exponential **decay** (1.5) holds, as shown in **the** celebrated articles [30], [3] and [4] of Bardos, Lebeau, Rauch and Taylor. **The** assumption that there exists T > 0 such that hγi T (x, ξ) > 0 in
Σ is called **the** geometric control condition. **The** article [20] underlines in addition **the** importance of **the** value of min (x,ξ)∈Σ hγi T (x, ξ) in order to control **the** rate of **decay** of **the** high frequencies. In **the** case of an unbounded manifold, two situations have been investigated. First, some authors have considered **the** free **wave** **equation** (1.1) in an exterior domain (with γ ≡ 0 or γ > 0 only on a compact subset **the** exterior domain). They have shown that **the** local energy decays to zero in **the** sense that, under suitable assumptions, **the** energy of any solution escapes away from any compact set, see [18], [26] and [2] and **the** references therein. Secondly, several works have studied **the** **damped** **wave** **equation** in an unbounded manifold and with a non-linearity, but assuming that **the** damping satisfies γ(x) ≥ α > 0 outside a compact set, see [33], [10], [9] and [16].

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PRESSURE CONDITION
EMMANUEL SCHENCK
Abstract. We establish **the** presence of a spectral gap near **the** real axis **for** **the** **damped** **wave** **equation** on a manifold with negative curvature. This results holds under a dy- namical condition expressed by **the** negativity of a topological pressure with respect to **the** geodesic flow. As an application, we show an exponential **decay** of **the** energy **for** all initial data sufficiently regular. This **decay** is governed by **the** imaginary part of a finite number of eigenvalues close to **the** real axis.

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implies **the** exponential stabilization, which holds **for** **the** **wave** **equation** with localized viscous damping a(x)∂ t u (see [ LR05 , Zh18 ] **for** results using **the** multiplier methods)
1.2. **The** main result. To state our main result, we first make some geometric assumptions. Let Ω ⊂ R d with d ≥ 2. We consider **the** piecewise smooth damping a ∈ C ∞ (Ω 1 ), a| Ω\Ω 1 = 0, such that there exists α 0 > 0,

then it is extended to **the** full X 2 by a density argument, cf. [1]. As regards UGAS,
it is proved that under a linear cone condition in addition to **the** damping one (i.e., there exist positive a, b such that as 2 ≤ sσ(s) ≤ bs 2 **for** every s ∈ R), and exponential
stability is achieved, i.e., one can choose β(s, t) = Cse −µt **for** some positive constants C, µ, cf. [27] **for** instance. If **the** linear cone condition only holds in a neighborhood of zero then exponential stability cannot hold in general, as shown in [27] where σ is chosen as a saturation function, i.e., such as σ(s) = arctan(s). Besides **the** linear cone condition, several results establishing UGAS have been obtained, cf. [18, 20, 21, 28] where σ verifies a linear cone condition **for** large s and is either of polynomial type or weaker than any polynomial in a neighborhood of **the** origin, see also [27] **for** a extensive list of references. It has to be noticed that many of these studies deal with **wave** equations in dimension not necessarily equal to one and, **for** all of them, **the** estimates are obtained by refined arguments based on **the** multiplier method or highly nontrivial Lyapunov functionals.

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An **equation** related to ( 1 ), **the** 1d Goldstein-Taylor type model, has been studied in [ 7 ] where **the** authors do get explicit **rates** via comparing this **equation** to a **damped** **wave** **equation** **for** which explicit **rates** were obtained by Lebeau in [ 26 ].
**The** case where V is unbounded is treated in [ 22 ] by Han-Kwan and L´eautaud, where **the** authors study linear Boltzmann type equations **for** a general class of collision operators and external confining potential terms on a closed, smooth, connected and compact Riemannian manifold M (and in particular **the** **torus**). In this context, **the** authors indentify geometric control conditions in **the** natural phase space T ∗ M (similar to Definition 2 in **the** case M = T d ) allowing to completely

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where Ω is a bounded domain in R n , n ∈ N ∗ , with smooth boundary ∂Ω := Γ.
While there is a great number of papers regarding **the** Kirchhoff **equation** subject to a frictional damping, in contrast, there is just a few number of papers concerned with **the** Kirchhoff **equation** subject to a dissipation given by a memory term. We are aware solely **the** paper [ 22 ], where stronger conditions were considered on **the** kernel of **the** memory term. **The** assumption given in ( 1.7 ), firstly introduced in [ 20 ], is much more general and allows us to consider a wide class of kernels, and consequently, get new and optimal **decay** rate estimates then those ones considered previously in **the** literature **for** **the** linear viscoelastic **wave** **equation**. In **the** present paper, we combine techniques given in [ 20 ] with new ingredients inherent to **the** nonlinear character of **the** Kirchhoff **equation** ( 1.2 ).

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where Ω is a bounded domain in R n , n ∈ N ∗ , with smooth boundary ∂Ω := Γ.
While there is a great number of papers regarding **the** Kirchhoff **equation** subject to a frictional damping, in contrast, there is just a few number of papers concerned with **the** Kirchhoff **equation** subject to a dissipation given by a memory term. We are aware solely **the** paper [ 22 ], where stronger conditions were considered on **the** kernel of **the** memory term. **The** assumption given in ( 1.7 ), firstly introduced in [ 20 ], is much more general and allows us to consider a wide class of kernels, and consequently, get new and optimal **decay** rate estimates then those ones considered previously in **the** literature **for** **the** linear viscoelastic **wave** **equation**. In **the** present paper, we combine techniques given in [ 20 ] with new ingredients inherent to **the** nonlinear character of **the** Kirchhoff **equation** ( 1.2 ).

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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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An **equation** related to ( 1 ), **the** 1d Goldstein-Taylor type model, has been studied in [7] where **the** authors do get explicit **rates** via comparing this **equation** to a **damped** **wave** **equation** **for** which explicit **rates** were obtained by Lebeau in [26].
**The** case where V is unbounded is treated in [22] by Han-Kwan and L´eautaud, where **the** authors study linear Boltzmann type equations **for** a general class of collision operators and external confining potential terms on a closed, smooth, connected and compact Riemannian manifold M (and in particular **the** **torus**). In this context, **the** authors indentify geometric control conditions in **the** natural phase space T ∗ M (similar to Definition 2 in **the** case M = T d ) allowing to completely

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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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tially understood in [4, 8]. In particular, **the** existence of a spectral gap was established **for** all α 6= α ∗ = (2 − d)/2, but
its value was not stated **for** all values of α.
J. Denzler and R.J. McCann in [17, 18] formally linearized **the** fast diffusion flow (considered as a gradient flow of **the** en- tropy with respect to **the** Wasserstein distance) in **the** frame- work of mass transportation, in order to guess **the** asymptotic behaviour of **the** solutions of [ 1 ]. This leads to a different functional setting, with a different linearized operator, H α,d .

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Let us now mention two possible extensions of our result. In **the** case d = 4 one may expect to get uniqueness by combining **the** analysis of [8] with **the** critical H 1 theory **for**
(1.1). One may also expect to include **the** case s = 0 by elaborating on **the** arguments developed in [8] to treat this case. It is not clear to us what happens **for** s < 0 (and in [8] as well). In particular we do not know whether s = 0 is **the** optimal regularity one may achieve by our approach. Invariant Gibbs measures **for** dispersive equations were extensively studied (see e.g. [20, 3, 2, 19, 18, 16, 17, 6] ). In these papers **the** Gibbs measure is combined with a suitable local in time result (which can sometimes be quite involved) to get global existence and uniqueness on **the** support of **the** measure. By an extension of **the** method (using in particular Skorohod and Prokhorov theorems) we use in this paper one may construct a dynamics (without any uniqueness) on **the** support of a Gibbs measure and prove its invariance. We plan to give several relevant examples of this observation in [7]. We however do not see how to make work such an approach in **the** context of (1.1). Indeed, **the** present methods of renormalization of Gibbs measures are restricted to dimensions ≤ 2 (see [3]). Let us also recall that as mentioned above a global existence based on Gibbs measures only works **for** a very specific choice of **the** initial distribution. On **the** other hand, it has of course **the** advantage to give a quite remarkable dynamical property of **the** flow.

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In **the** static case, this kind of problem has been studied **for** a long time and a large number of results have been obtained us- ing **the** so-called homogenization theory applied to media showing rapid and periodical variations of their physical properties. Since **the** pioneering work of Auriault & Sanchez-Palencia (1977), nu- merous studies have been devoted either to **the** mathematical foun- dations of **the** homogenization theory in **the** static context (e.g. Bensoussan et al. 1978; Murat & Tartar 1985; Allaire 1992), to applications to **the** effective static behaviour of composite materials (e.g. Dumontet 1986; Francfort & Murat 1986; Abdelmoula & Marigo 2000; Haboussi et al. 2001a,b), to **the** application to **the** heat diffusion (e.g. Marchenko & Khruslov 2005), to porous me- dia (e.g. Hornung 1996), etc. In contrast, fewer studies have been devoted to **the** theory and its applications in **the** general dynamical context or to **the** non-periodic cases. However, one can **for** example refer to Sanchez-Palencia (1980), Willis (1981), Auriault & Bonnet (1985), Moskow & Vogelius (1997), Allaire & Conca (1998), Fish et al. (2002), Fish & Chen (2004), Parnell & Abrahams (2006), Milton & Willis (2007), Lurie (2009) or Allaire et al. (2009) **for** **the** dynamical context, to Briane (1994), Nguetseng (2003) or to Marchenko & Khruslov (2005) **for** **the** non-periodic case. Moczo et al. (2002) have also used a kind of local homogenization to take into account interfaces with **the** finite differences method. **The** specific case of a long **wave** propagating in finely layered media has been studied by Backus (1962) and **the** same results can be extracted from **the** 0th-order term of **the** asymptotic expansion im- plied in homogenization theory. Higher order homogenization in **the** non-periodic case has been studied by Capdeville & Marigo (2007) and Capdeville & Marigo (2008) **for** **wave** propagation in strati- fied media, but **the** extension to media characterized by 3-D rapid variation is not obvious from these works. Indeed, **the** non-periodic homogenization strategy suggested in Capdeville & Marigo (2007)

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2. Level structure of Pr IV
**The** ground state of Pr IV is [Xe]4f 2 3 H
4 . All of **the** term energies experimentally deduced so far are listed in **the** NIST compilation (Martin et al 1978 ) which contains 88 levels identified as belonging to **the** 4f 2 , 4f5f, 4f6p, 5d 2 even configurations and to **the** 4f5d, 4f6d, 4f6s, 4f7s, 5d6p odd configurations. This critical compilation was essentially based on **the** works of Sugar ( 1965 , 1971a ) and Crosswhite et al ( 1965 ) who observed **the** Pr IV spectrum emitted by sliding-spark discharges and reported line lists covering **the** wavelength region from 69.1 to 302.1 nm. **The** level compositions in **the** 4f 2 configuration were taken from **the** theoretical analysis of Goldschmidt ( 1968 ) later refined to include additional magnetic interactions by Goldschmidt et al ( 1968 ) and Pasternak ( 1970 ). Theoretical percentages **for** levels of **the** 4f6s, 4f5d, 4f6d, 4f6p and 4f5f configurations were taken from Sugar ( 1965 , 1971a ) supplemented by some of his unpublished results (Sugar 1971b ).

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is **the** trivial one u ≡ 0. Such a property is very important in Control Theory, as it is equivalent to **the** approximate controllability **for** linear PDE, and it is involved in **the** classical unique- ness/compactness approach in **the** proof of **the** stability **for** a PDE with a localized damping. **The** UCP is usually proved with **the** aid of some Carleman estimate (see e.g. [ 46 ]). **The** UCP **for** KdV was established in [ 48 ] by **the** inverse scattering approach, in [ 12 , 40 , 46 ] by means of Carleman estimates, and in [ 5 ] by a perturbative approach and Fourier analysis. **For** BBM, **the** study of **the** UCP is only at its early age. **The** main reason is that both x = const and t = const are characteristic lines **for** (1.1). Thus, **the** Cauchy problem in **the** UCP (assuming e.g. that u = 0 **for** x ≤ 0, and solving BBM **for** x ≥ 0) is characteristic, which prevents from applying Holmgren’s theorem, even **for** **the** linearized **equation**. **The** Carleman approach **for** **the** UCP of BBM was developed in [ 9 ] and in [ 47 ]. Unfortunately, Theorems 3.1-3.4 in [ 9 ] are not correct without further assumptions, as noticed in [ 49 ]. On **the** other hand, **the** UCP in [ 47 ] **for** **the** BBM-like **equation**

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