Romain Joly
email: romain.joly@ujf-grenoble.fr

Camille Laurent
email: laurent@ann.jussieu.fr

Stabilization for the semilinear wave equation with geometric control condition

Keywords: damped wave equation, stabilization, analyticity, unique continuation property, compact attractor AMS subject classification: 35B40, 35B60, 35B65, 35L71, 93D20, 35B41. Résumé

In this article, we prove the exponential stabilization of the semilinear wave equation with a damping effective in a zone satisfying the geometric control condition only. The nonlinearity is assumed to be subcritical, defocusing and analytic. The main novelty compared to previous results, is the proof of a unique continuation result in large time for some undamped equation. The idea is to use an asymptotic smoothing effect proved by Hale and Raugel in the context of dynamical systems. Then, once the analyticity in time is proved, we apply a unique continuation result with partial analyticity due to Robbiano, Zuily, Tataru and Hörmander. Some other consequences are also given for the controllability and the existence of a compact attractor.

Introduction

In this article, we consider the semilinear damped wave equation

   ✷u + γ(x)∂ t u + βu + f (u) = 0 (t, x) ∈ R + × Ω , u(t, x) = 0 (t, x) ∈ R + × ∂Ω (u, ∂ t u) = (u 0 , u 1 ) ∈ H 1 0 (Ω) × L 2 (Ω) (1.1)
where ✷ = ∂ 2 tt -∆ with ∆ being the Laplace-Beltrami operator with Dirichlet boundary conditions. The domain Ω is a connected C ∞ three-dimensional Riemannian manifold with boundaries, which is either: i) compact.

ii) a compact perturbation of R 3 , that is R 3 \ D where D is a bounded smooth domain, endowed with a smooth metric equal to the euclidean one outside of a ball.

iii) or a manifold with periodic geometry (cylinder, R 3 with periodic metric etc.). The nonlinearity f ∈ C 1 (R, R) is assumed to be defocusing, energy subcritical and such that 0 is an equilibrium point. More precisely, we assume that there exists C > 0 such that

f (0) = 0 , sf (s) ≥ 0 , |f (s)| ≤ C(1 + |s|) p and |f ′ (s)| ≤ C(1 + |s|) p-1
(1.2) with 1 ≤ p < 5.

We assume β ≥ 0 to be such that ∆β is a negative definite operator, that is that we have a Poincaré inequality Ω |∇u| 2 + β|u| 2 ≥ C Ω |u| 2 with C > 0. In particular, it may require β > 0 if ∂Ω = ∅ or if Ω is unbounded.

The damping γ ∈ L ∞ (Ω) is a non-negative function. We assume that there exist an open set ω ⊂ Ω, α ∈ R, x 0 ∈ Ω and R ≥ 0 such that ∀x ∈ ω , γ(x) ≥ α > 0 and Ω \ B(x 0 , R) ⊂ ω .

(1.3)

Moreover, we assume that ω satisfies the geometric control condition introduced in [START_REF] Rauch | Exponential decay of solutions to hyperbolic equations in bounded domains[END_REF] and [START_REF] Bardos | Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary[END_REF] (GCC) There exists L > 0 such that any generalized geodesic of Ω of length L meets the set ω where the damping is effective.

The associated energy E ∈ C 0 (X, R + ) is given by

E(u) := E(u, ∂ t u) = 1 2 Ω (|∂ t u| 2 + |∇u| 2 + β|u| 2 ) + Ω V (u) , (1.4)
where V (u) = u 0 f (s)ds. Due to Assumption (1.2) and the Sobolev embedding H 1 (Ω) ֒→ L 6 (Ω), this energy is well defined and moreover, if u solves (1.1), we have, at least formally,

∂ t E(u(t)) = - Ω γ(x)|∂ t u(x, t)| 2 dx ≤ 0 .
(1.5)

The system is therefore dissipative. We are interested in the exponential decay of the energy of the nonlinear damped wave equation (1.1), that is the following property:

(ED) For any E 0 ≥ 0, there exist K > 0 and λ > 0 such that, for all solutions u of (1.1) with E(u(0)) ≤ E 0 , ∀t ≥ 0 , E(u(t)) ≤ Ke -λt E(u(0))

Property (ED) means that the damping term γ∂ t u stabilizes any solution of (1.1) to zero, which is an important property from the dynamical and control points of view.

Our main theorem is as follows.

Theorem 1.1. Assume that the damping γ satisfies (1.3) and the geometric control condition (GCC). If f is real analytic and satisfies (1.2), then the exponential decay property (ED) holds.

Theorem 1.1 applies for nonlinearities f which are globally analytic. Of course, the nonlinearities f (u) = |u| p-1 u are not analytic if p ∈ {1, 3}, but we can replace these usual nonlinearities by similar ones as f (u) = (u/th(u)) p-1 u, which are analytic for all p ∈ [START_REF] Alinhac | Propagation de l'analyticité des solutions de systèmes hyperboliques non-linéaires[END_REF][START_REF] Babin | Attractors of Evolutionary Equations[END_REF]. Note that the estimates (1.2) are only required for s ∈ R, so that it does not imply that f is polynomial. Moreover, we enhance that (ED) holds in fact for almost all the nonlinearities f satisfying (1.2), including non-analytic ones.

More precisely, we set C 1 (R) = {f ∈ C 1 (R) such that there exist C > 0 and p ∈ [START_REF] Alinhac | Propagation de l'analyticité des solutions de systèmes hyperboliques non-linéaires[END_REF][START_REF] Babin | Attractors of Evolutionary Equations[END_REF] such that (1.2) holds } (1.6) endowed with Whitney topology (or any other reasonable topology). We recall that Whitney topology is the topology generated by the neighbourhoods

N f,δ = { g ∈ C 1 (R) | ∀u ∈ R , max(|f (u) -g(u)|, |f ′ (u) -g ′ (u)|) < δ(u) } (1.7)
where f is any function in C 1 (R) and δ is any positive continuous function. The set C 1 (R) is a Baire space, which means that any generic set, that is any set containing a countable intersection of open and dense sets, is dense in C 1 (R) (see Proposition 7.1). Baire property ensures that the genericity of a set in C 1 (R) is a good notion for "the set contains almost all non-linearity f ".

Theorem 1.2. Assume that the damping γ satisfies (1.3) and the geometric control condition (GCC). There exists a generic set G ⊂ C 1 (R) such that the exponential decay property (ED) holds for all f ∈ G.

The statements of both theorems lead to some remarks.

• Of course, our results and their proofs should easily extend to any space dimension d ≥ 3 if the exponent p of the nonlinearity satisfies p < (d + 2)/(d -2).

• Actually, it may be possible to get λ > 0 in (ED) uniform with respect to the size of the data. We can take for instance λ = λε where λ is the decay rate of the linear equation.

The idea is that once we know the existence of a decay rate, we know that the solution is close to zero for a large time. Then, for small solutions, the nonlinear term can be neglected to get almost the same decay rate as the linear equation. We refer for instance to [START_REF] Laurent | Control and stabilization of the Korteweg-de Vries equation on a periodic domain[END_REF] in the context of KdV equation. Notice that the possibility to get the same result with a constant K independent on E 0 is an open problem.

• The assumption on β is important to ensure some coercivity of the energy and to preclude the spatially constant functions to be undamped solutions for the linear equation. It has been proved in [START_REF] Dehman | Stabilization for the Nonlinear Klein Gordon Equation with critical Exponent[END_REF] for R 3 and in [START_REF] Laurent | On stabilization and control for the critical Klein Gordon equation on 3-D compact manifolds[END_REF] for a compact manifold that exponential decay can fail without this term β.

• The geometric control condition is known to be not only sufficient but also necessary for the exponential decay of the linear damped equation. The proof of the optimality uses some sequences of solutions which are asymptotically concentrated outside of the damping region. We can use the same idea in our nonlinear stabilization context. First, the observability for a certain time eventually large is known to be equivalent to the exponential decay of the energy. This was for instance noticed in [START_REF] Dehman | Stabilization for the Nonlinear Klein Gordon Equation with critical Exponent[END_REF] Proposition 2, in a similar context, see also Proposition 2.5 of this paper. Then, we take as initial data the same sequence that would give a counterexample for the linear observability. The linearizability property (see [START_REF] Gérard | Oscillations and Concentration Effects in Semilinear Dispersive Wave Equations[END_REF]) allows to obtain that the nonlinear solution is asymptotically close to the linear one. This contradicts the observability property for the nonlinear solution as it does for the linear case. Hence, the geometric control condition is also necessary for the exponential decay of the nonlinear equation.

• Our geometrical hypotheses on Ω may look strange, however they are only assumed for sake of simplicity. In fact, our results should apply more generally for any smooth manifold with bounded geometry, that is that Ω can be covered by a set of C ∞ -charts α i :

U i -→ α i (U i ) ⊂ R 3 such that α i (U i ) is equal either to B(0, 1) or to B + (0, 1) = {x ∈ B(0, 1), x 1 >
0} (in the case with boundaries) and such that, for any r ≥ 0 and s

∈ [1, ∞], the W r,s -norm of a function u in W r,s (Ω, R) is equivalent to the norm ( i∈N u • α -1 i s W r,s (α i (U i )) ) 1/s .
The stabilization property (ED) for Equation (1.1) has been studied in [START_REF] Haraux | Stabilization of trajectories for some weakly damped hyperbolic equations[END_REF], [START_REF] Zuazua | Exponential decay for semilinear wave equations with localized damping[END_REF], [START_REF] Zuazua | Exponential decay for the semilinear wave equation with localized damping in unbounded domains[END_REF] and [START_REF] Dehman | Stabilisation pour l'équation des ondes semi-linéaire[END_REF] for p < 3. For p ∈ [START_REF] Aloui | Stabilisation pour l'équation des ondes dans un domaine extérieur[END_REF][START_REF] Babin | Attractors of Evolutionary Equations[END_REF], our main reference is the work of Dehman, Lebeau and Zuazua [START_REF] Dehman | Stabilization and control for the subcritical semilinear wave equation[END_REF]. This work is mainly concerned with the stabilization problem previously described, on the Euclidean space R 3 with flat metric and stabilization active outside of a ball. The main purpose of this paper is to extend their result to a non flat geometry where multiplier methods cannot be used or do not give the optimal result with respect to the geometry. Other stabilization results for the nonlinear wave equation can be found in [START_REF] Aloui | Exponential energy decay for damped Klein-Gordon equation with nonlinearities of arbitrary growth[END_REF] and the references therein. Some works have been done in the difficult critical case p = 5, we refer to [START_REF] Dehman | Stabilization for the Nonlinear Klein Gordon Equation with critical Exponent[END_REF] and [START_REF] Laurent | On stabilization and control for the critical Klein Gordon equation on 3-D compact manifolds[END_REF].

The proofs of these articles use three main ingredients: (i) the exponential decay of the linear equation, which is equivalent to the geometric control condition (GCC),

(ii) a more or less involved compactness argument, (iii) a unique continuation result implying that u ≡ 0 is the unique solution of

✷u + βu + f (u) = 0 ∂ t u = 0 on [-T, T ] × ω . (1.8)
The results are mainly of the type: "geometric control condition" + "unique continuation" implies "exponential decay". This type of implication is even stated explicitly in some related works for the nonlinear Schrödinger equation [START_REF] Dehman | Stabilization and control for the nonlinear Schrödinger equation on a compact surface[END_REF] and [START_REF] Laurent | Global controllability and stabilization for the nonlinear Schrödinger equation on some compact manifolds of dimension 3[END_REF].

In the subcritical case p < 5, the less understood point is the unique continuation property (iii). In the previous works as [START_REF] Dehman | Stabilization and control for the subcritical semilinear wave equation[END_REF], the authors use unique continuation results based on Carleman estimates. The resulting geometric assumptions are not very natural and are stronger than (GCC). Indeed, the unique continuation was often proved with some Carleman estimates that required some strong geometric conditions. For instance for a flat metric, the usual geometric assumption that appear are often of "multiplier type" that is ω is a neighbourhood of {x ∈ ∂Ω |(xx 0 ) • n(x) > 0} which are known to be stronger than the geometric control condition (see [START_REF] Miller | Escape function conditions for the observation, control, and stabilization of the wave equation[END_REF] for a discussion about the links between these assumptions). Moreover, on curved spaces, this type of condition often needs to be checked by hand in each situation, which is mostly impossible.

Our main improvement in this paper is the proof of unique continuation in infinite time under the geometric control condition only. We show that, if the nonlinearity f is analytic (or generic), then one can use the result of Robbiano and Zuily [START_REF] Robbiano | Uniqueness in the Cauchy problem for operators with partially holomorphic coefficients[END_REF] to obtain a unique continuation property (iii) for infinite time T = +∞ with the geometric control condition (GCC) only.

The central argument of the proof of our main result, Theorem 1.1, is the unique continuation property of [START_REF] Robbiano | Uniqueness in the Cauchy problem for operators with partially holomorphic coefficients[END_REF] (see Section 3). This result applies for solutions u of (1.8) being smooth in space and analytic in time. If f is analytic, then the solutions of (1.1) are of course not necessarily analytic in time since the damped wave equations are not smoothing in finite time. However, the damped wave equations admit an asymptotic smoothing effect, i.e. are smoothing in infinite time. Hale and Raugel have shown in [START_REF] Hale | Regularity, determining modes and Galerkin methods[END_REF] that, for compact trajectories, this asymptotic smoothing effect also concerns the analyticity (see Section 5). In other words, combining [START_REF] Robbiano | Uniqueness in the Cauchy problem for operators with partially holomorphic coefficients[END_REF] and [START_REF] Hale | Regularity, determining modes and Galerkin methods[END_REF] shows that the unique solution of (1.8) is u ≡ 0 if f is analytic and if T = +∞. This combination has already been used by dynamicians in [26] and [START_REF] Joly | New examples of damped wave equations with gradient-like structure[END_REF] for p < 3.

One of the main interests of this paper is the use of arguments coming from both the dynamical study and the control theory of the damped wave equations. The reader familiar with the control theory could find interesting the use of the asymptotic smoothing effect to get unique continuation property with smooth solutions. The one familiar with the dynamical study of PDEs could be interested in the use of Strichartz estimates to deal with the case p ∈ [START_REF] Aloui | Stabilisation pour l'équation des ondes dans un domaine extérieur[END_REF][START_REF] Babin | Attractors of Evolutionary Equations[END_REF]. The main part of the proof of Theorem 1.1 is written with arguments coming from the dynamical study of PDEs. They are simpler than the corresponding ones of control theory, but far less accurate since they do not give any estimation for the time of observability. Anyway, such accuracy is not important here since we use the unique continuation property for (1.8) with T = +∞. We briefly recall in Section 8 how these propagation of compactness and regularity properties could have been proved with some arguments more usual in the control theory.

Moreover, we give two applications of our results in both contexts of control theory and dynamical systems. First, as it is usual in control theory, some results of stabilization can be coupled with local control theorems to provide global controllability in large time.

Theorem 1.3. Assume that f satisfies the conditions of Theorem 1.1 or belongs to the generic set G defined by Theorem 1.2. Let R 0 > 0 and ω satisfying the geometric control condition. Then, there exists T > 0 such that for any (u 0 , u 1 ) and (ũ 0 , ũ1 ) in H 1 0 (Ω)×L 2 (Ω), with

(u 0 , u 1 ) H 1 ×L 2 ≤ R 0 ; (ũ 0 , ũ1 ) H 1 ×L 2 ≤ R 0 there exists g ∈ L ∞ ([0, T ], L 2 (Ω)) supported in [0, T ]×ω such that the unique strong solution of ✷u + βu + f (u) = g on [0, T ] × Ω (u(0), ∂ t u(0)) = (u 0 , u 1 ). satisfies (u(T ), ∂ t u(T )) = (ũ 0 , ũ1 ).
The second application of our results concerns the existence of a compact global attractor. A compact global attractor is a compact set, which is invariant by the flow of the PDE and which attracts the bounded sets. The existence of such an attractor is an important dynamical property because it roughly says that the dynamics of the PDE may be reduced to dynamics on a compact set, which is often finite-dimensional. See for example [START_REF] Hale | Asymptotic behavior of dissipative systems[END_REF] and [START_REF] Raugel | Handbook of dynamical systems[END_REF] for a review on this concept. Theorems 1.1 and 1.2 show that {0} is a global attractor for the damped wave equation (1.1). Of course, it is possible to obtain a more complex attractor by considering an equation of the type

   ∂ 2 tt u + γ(x)∂ t u = ∆u -βu -f (x, u) (x, t) ∈ Ω × R + , u(x, t) = 0 (x, t) ∈ ∂Ω × R + (u, ∂ t u) = (u 0 , u 1 ) ∈ H 1 0 × L 2 (1.9) where f ∈ C ∞ (Ω × R, R
) is real analytic with respect to u and satisfies the following properties. There exist C > 0, p ∈ [1, 5) and R > 0 such that for all (x, u) .11) where x 0 denotes a fixed point of the manifold.

∈ Ω × R, |f (x, u)| ≤ C(1 + |u|) p , |f ′ x (x, u)| ≤ C(1 + |u|) p , |f ′ u (x, u)| ≤ C(1 + |u|) p-1 (1.10) (x ∈ B(x 0 , R) or |u| ≥ R) =⇒ f (x, u)u ≥ 0 . ( 1
Theorem 1.4. Assume that f is as above. Then, the dynamical system generated by (1.9) in H 1 0 (Ω) × L 2 (Ω) is gradient and admits a compact global attractor A. Of course, we would get the same result for f in a generic set similar to the one of Theorem 1.2.

We begin this paper by setting our main notations and recalling the basic properties of Equation (1.1) in Section 2. We recall the unique continuation property of Robbiano and Zuily in Section 3, whereas Sections 4 and 5 are concerned by the asymptotic compactness and the asymptotic smoothing effect of the damped wave equation. The proofs of our main results, Theorem 1.1 and 1.2, are given in Sections 6 and 7 respectively. An alternative proof, using more usual arguments from control theory, is sketched in Section 8. Finally, Theorems 1.3 and 1.4 are discussed in Section 9.

Acknowledgements: the second author was financed by the ERC grant GeCoMethods during part of the redaction of this article. Moreover, both authors benefited from the fruitful atmosphere of the conference Partial differential equations, optimal design and numerics in Benasque. We also would like to thank Mathieu Léautaud for his remarks about the optimality of Hypothesis (GCC) in the nonlinear context and Geneviève Raugel for her help for removing a non-natural hypothesis of Theorem 1.4.

Notations and basic properties of the damped wave equation

In this paper, we use the following notations:

U = (u, u t ) , F = (0, f ) , A = 0 Id ∆ -β -γ .
In this setting, (1.1) becomes

∂ t U(t) = AU(t) + F (U) .
We set X = H 1 0 (Ω) × L 2 (Ω) and for s ∈ [0, 1], X s denotes the space D((-

∆ + β) (s+1)/2 ) × D((-∆ + β) s/2 ) = (H 1+s (Ω) ∩ H 1 0 (Ω)) × H s 0 (Ω). Notice that X 0 = X and X 1 = D(A) (even if γ is only in L ∞ ).
We recall that E denotes the energy defined by (1.4). We also emphasize that (1.2) and the invertibility of ∆β implies that a set is bounded in X if and only if its energy E is bounded. Moreover, for all E 0 ≥ 0, there exists C > 0 such that

∀(u, v) ∈ X, E(u, v) ≤ E 0 =⇒ 1 C (u, v) 2 X ≤ E(u, v) ≤ C (u, v) 2 X (2.1)
To simplify some statements in the proofs, we assume without loss of generality that 3 < p < 5. It will avoid some meaningless statements with negative Lebesgue exponents since p = 3 is the exponent where Strichartz estimates are no more necessary and can be replaced by Sobolev embeddings. We recall that Ω is endowed with a metric g. We denote by d the distance on Ω defined by

d(x, y) = inf {l(c) |c ∈ C ∞ ([0, 1], Ω) with c(0) = x and c(1) = y }
where l(c) is the length of the path c according to the metric g. A ball B(x, R) in Ω is naturally defined by B(x, R) = {y ∈ Ω, d(x, y) < R} .

For instance, if Ω = R 3 \ B R 3 (0, 1), the distance between (0, 0, 1) and (0, 0, -1) is π (and not 2) and the ball B((0, 0, 1), π) has nothing to do with the classical ball B R 3 ((0, 0, 1), π) of R 3 .

Cauchy problem

The global existence and uniqueness of solutions of the subcritical wave equation (1.1) with γ ≡ 0 has been studied by Ginibre and Velo in [START_REF] Ginibre | The global Cauchy problem for the nonlinear Klein-Gordon equation[END_REF] and [START_REF] Ginibre | The global Cauchy problem for the nonlinear Klein-Gordon equation[END_REF]. Their method also applies for γ = 0 since this term is linear and well defined in the energy space X. Moreover, their argument to prove uniqueness also yields the continuity of the solutions with respect to the initial data. The central argument is the use of Strichartz estimates.

Theorem 2.1 (Strichartz estimates).

Let T > 0 and (q, r) satisfying

1 q + 3 r = 1 2 , q ∈ [7/2, +∞]. ( 2

.2)

There exists C = C(T, q) > 0 such that for every G ∈ L 1 ([0, T ], L 2 (Ω)) and every (u 0 , u 1 ) ∈ X, the solution u of

✷u + γ(x)∂ t u = G(t) (u, ∂ t u)(0) = (u 0 , u 1 )
satisfies the estimate

u L q ([0,T ],L r (Ω)) ≤ C u 0 H 1 (Ω) + u 1 L 2 (Ω) + G L 1 ([0,T ],L 2 (Ω)) .
The result was stated in the Euclidean space R 3 by Strichartz [START_REF] Strichartz | Restriction of Fourier transform to quadratic surfaces and decay of solutions to the wave equation[END_REF] and Ginibre and Velo with q ∈ (2, +∞]. Kapitanski extended the result to variable coefficients in [START_REF] Kapitanski | Some generalizations of the Strichartz-Brenner inequality[END_REF]. On a bounded domain, the first estimates were proved by Burq, Lebeau and Planchon [START_REF] Burq | Global existence for energy critical waves in 3-D domains[END_REF] for q ∈ [5, +∞] and extended to a larger range by Blair, Smith and Sogge in [START_REF] Blair | Strichartz estimates for the wave equation on manifolds with boundary[END_REF]. Note that, thanks to the counterexamples of Ivanovici [START_REF] Ivanovici | Counter-examples to the Strichartz estimates for the wave equation in general domains with boundary[END_REF], we know that we cannot expect some Strichartz estimates in the full range of exponents in the presence of boundaries.

From these results, we deduce the estimates for the damped wave equation by absorption for T small enough. We can iterate the operation in a uniform number of steps. Actually, for the purpose of the semilinear wave equation, it is sufficient to consider the Strichartz estimate

L 2p p-3 ([0, T ], L 2p (Ω)) which gives u p ∈ L 2 p-3 ([0, T ], L 2 (Ω)) ⊂ L 1 ([0, T ], L 2 (Ω)) because 1 < 2 p-3 < +∞. Theorem 2.

(Cauchy problem).

Let f satisfies (1.2). Then, for any (u 0 , u 1 ) ∈ X = H 1 0 (Ω) × L 2 (Ω) there exists a unique solution u(t) of the subcritical damped wave equation (1.1). Moreover, this solution is defined for all t ∈ R and its energy E(u(t)) is non-increasing in time.

For any E 0 ≥ 0, T ≥ 0 and (q, r) satisfying (2.2), there exists a constant C such that, if u is a solution of (1.1) with E(u(0)) ≤ E 0 , then

u L q ([0,T ],L r (Ω)) ≤ C u 0 H 1 (Ω) + u 1 L 2 (Ω) .
In addition, for any E 0 ≥ 0 and T ≥ 0, there exists a constant C such that, if u and ũ are two solutions of (1.1) with E(u(0

)) ≤ E 0 and E(ũ(0)) ≤ E 0 , then sup t∈[-T,T ] (u, ∂ t u)(t) -(ũ, ∂ t ũ)(t) X ≤ C (u, ∂ t u)(0) -(ũ, ∂ t ũ)(0) X .
Proof: The existence and uniqueness for small times is a consequence of Strichartz estimates and of the subcriticality of the nonlinearity, see [START_REF] Ginibre | The global Cauchy problem for the nonlinear Klein-Gordon equation[END_REF]. The solution can be globalized backward and forward in time thanks to the energy estimates (1.5) for smooth solutions. Indeed,

E(t) ≤ E(s) + C s t E(τ ) dτ
and thus, Gronwall inequality for t ≤ s and the decay of energy for t ≥ s show that the energy does not blow up in finite time. This allows to extend the solution for all times since the energy controls the norm of the space X by (2.1).

For the uniform continuity estimate, we notice that w = uũ is solution of

✷w + βw + γ(x)∂ t w = -wg(u, ũ) (w, ∂ t w)(0) = (u, ∂ t u)(0) -(ũ, ∂ t ũ)(0)
where g(s, s) =

1 0 f ′ (s + τ (s -s)) dτ fulfills |g(s, s)| ≤ C(1 + |s| p-1 + |s| p-1
). Let q = 2p p-3 , Strichartz and Hölder estimates give

(w, ∂ t w)(t) L ∞ ([0,T ],X)∩L q ([0,T ],L 2p ) ≤ C (w, ∂ t w)(0) X + C wg(u, ũ) L 1 ([0,T ],L 2 ) ≤ C (w, ∂ t w)(0) X + CT w L ∞ ([0,T ],L 2 ) +T θ w L q ([0,T ],L 2p ) u p-1 L q ([0,T ],L 2p ) + ũ p-1 L q ([0,T ],L 2p )
with θ = 5-p 2 > 0. We get the expected result for T small enough by absorption since we already know a uniform bound (depending on E 0 ) for the Strichartz norms of u and ũ. Then, we iterate the operation to get the result for large T .

Exponential decay of the linear semigroup

In this paper, we will strongly use the exponential decay for the linear semigroup in the case where γ may vanish but satisfies the geometric assumptions of this paper. In this case, (1.3) enables to control the decay of energy outside a large ball and the geometric control condition (GCC) enables to control the energy trapped in this ball.

Proposition 2.3. Assume that γ ∈ L ∞ (Ω) satisfies (1.
3) and (GCC). There exist two positive constants C and λ such that

∀s ∈ [0, 1] , ∀t ≥ 0 , |||e At ||| L(X s ) ≤ Ce -λt .
The exponential decay of the damped wave equation under the geometric control condition is well known since the works of Rauch and Taylor on a compact manifold [START_REF] Rauch | Exponential decay of solutions to hyperbolic equations in bounded domains[END_REF] and Bardos, Lebeau and Rauch [START_REF] Bardos | Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary[END_REF][START_REF] Bardos | Un exemple d'utilisation des notions de propagation pour le contrôle et la stabilisation de problèmes hyperboliques[END_REF] on a bounded domain. Yet, we did not find any reference for unbounded domain ( [START_REF] Aloui | Stabilisation pour l'équation des ondes dans un domaine extérieur[END_REF] and [START_REF] Khenissi | Equation des ondes amorties dans un domaine extérieur. (French. English and French summary) [Dissipative wave equation in an exterior domain[END_REF] concern unbounded domains but local energy only). It is noteworthy that the decay of the linear semigroup in unbounded domains seems not to have been extensively studied for the moment.

We give a proof of Proposition 2.3 using microlocal defect measure as done in Lebeau [START_REF] Lebeau | Equation des ondes amorties[END_REF] or Burq [START_REF] Burq | Contrôlabilité exacte des ondes dans des ouverts peu réguliers. (French) [Exact controllability of waves in nonsmooth domains[END_REF] (see also [START_REF] Burq | Condition nécéssaire et suffisante pour la contrôlabilite exacte des ondes[END_REF] for the proof of the necessity). The only difference with respect to these results is that the manifold that we consider may be unbounded. Since microlocal defect measure only reflects the local propagation, we thus have to use the property of equipartition of the energy to deal with the energy at infinity and to show a propagation of compactness (see [START_REF] Dehman | Stabilization and control for the subcritical semilinear wave equation[END_REF] for the flat case).

Lemma 2.4. Let T > L where L is given by (GCC). Assume that (U n,0 ) ⊂ X is a bounded sequence, which weakly converges to 0 and assume that

U n (t) = (u n (t), ∂ t u n (t)) = e At U n,0 satisfies T 0 Ω γ(x)|∂ t u n | 2 → 0 . (2.3)
Then, (U n,0 ) converges to 0 strongly in X.

Proof: Let µ be a microlocal defect measure associated to (u n ) (see [START_REF] Gérard | Microlocal Defect Measures[END_REF], [START_REF] Tartar | H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations[END_REF] or [START_REF] Burq | Mesures semi-classiques et mesures de défaut. (French) [Semiclassical measures and defect measures[END_REF] for the definition). Note that (2.3) implies that µ can also be associated to the solution of the wave equation without damping, so the weak regularity of γ is not problematic for the propagation and we get that µ is concentrated on {τ 2 -|ξ| 2 x = 0} where (τ, ξ) are the dual variables of (t, x). Moreover, (2.3) implies that γτ 2 µ = 0 and so µ ≡ 0 on S * (]0, T [×ω). Then, using the propagation of the measure along the generalized bicharacteristic flow of Melrose-Sjöstrand and the geometric control condition satisfied by ω, we obtain µ ≡ 0 everywhere. We do not give more details about propagation of microlocal defect measure and refer to the Appendix [START_REF] Lebeau | Equation des ondes amorties[END_REF] or Section 3 of [START_REF] Burq | Mesures semi-classiques et mesures de défaut. (French) [Semiclassical measures and defect measures[END_REF] (see also [START_REF] Gérard | Leichtnam Ergodic properties of eigenfunctions for the Dirichlet problem[END_REF] for some close propagation results in a different context). Since µ ≡ 0, we know that U n → 0 on

H 1 × L 2 (]0, T [×B(x 0 , R)) for every R > 0.
To finish the proof, we need the classical equipartition of the energy to get the convergence to 0 in the whole manifold Ω. Since γ is uniformly positive outside a ball B(x 0 , R), (2.3) and the previous arguments imply that

∂ t u n → 0 in L 2 ([0, T ]×Ω). Let ϕ ∈ C ∞ 0 (]0, T [) with ϕ ≥ 0 and ϕ(t) = 1 for t ∈ [ε, T -ε].
We multiply the equation by ϕ(t)u n and we obtain 0 = -

[0,T ]×Ω ϕ(t)|∂ t u n | 2 - [0,T ]×Ω ϕ ′ (t)∂ t u n u n + [0,T ]×Ω ϕ(t)|∇u n | 2 + [0,T ]×Ω ϕ(t)β|u n | 2 + [0,T ]×Ω ϕ(t)γ(x)∂ t u n u n . The L 2 -norm of u n (t) is bounded, while ∂ t u n → 0 in L 2 ([0, T ] × Ω)
, so the first, second and fifth terms converge to zero. Then, the above equation yields

[0,T ]×Ω ϕ(t) β|u n | 2 + |∇u n | 2 -→ 0.
Finally, notice that the energy identity U n,0

2 X = U n (t) 2 X + T 0 Ω γ(x)|∂ t u n | 2 shows that [0,T ]×Ω ϕ(t) β|u n | 2 + |∇u n | 2 ∼ U n,0 2 X T 0 ϕ(t)
and thus that U n,0 X goes to zero.

Proof of Proposition 2.3: Once Lemma 2.4 is established, the proof follows the arguments of the classical case, where Ω is bounded. We briefly recall them. We first treat the case s = 0. As in Proposition 2.5, the exponential decay of the energy is equivalent to the observability estimate, that is the existence of C > 0 and T > 0 such that, for any trajectory

U(t) = e At U 0 in X, T 0 Ω γ(x)|∂ t u| 2 ≥ C U(0) 2 X .
We argue by contradiction: assume that (2.4) does not hold for any positive T and C. Then, there exists a sequence of initial data U n (0) with U n (0) X = 1 and such that

n 0 Ω γ(x)|∂ t u n (t, x)| 2 dtdx -------→ n-→+∞ 0 , where (u n , ∂ t u n )(t) = U n (t) = e At U n (0). Let Ũn = U n (n/2 + •). We have n/2 -n/2 Ω γ(x)|∂ t ũn (t, x)| 2 dtdx ----→ n→∞ 0 ,
and, for any t ∈ [-n/2, n/2],

Ũn (t) 2 X = Ũn (-n/2) 2 X - t -n/2 Ω γ(x)|∂ t ũn (s, x)| 2 dsdx ----→ n→∞ 1 .
Therefore, we can assume that U n (0) converges to U ∞ (0) ∈ X, weakly in X. Moreover, for any T > 0, U n (t) and

∂ t U n (t) are bounded in L ∞ ([-T, T ], X) and L ∞ ([-T, T ], L 2 (Ω) × H -1 (Ω)
) respectively. Thus, using Ascoli's Theorem, we may also assume that U n (t) strongly converges to

U ∞ (t) in L ∞ ([-T, T ], L 2 (K) × H -1 (K)) where K is any compact of Ω. Hence, (u ∞ , ∂ t u ∞ )(t) = U ∞ (t) = e At U ∞ (0) is a solution of ✷u ∞ + βu ∞ = 0 on R × Ω ∂ t u ∞ = 0 on R × ω. (2.4) in L 2 × H -1 . Since U ∞ (0) ∈ X belongs to X, we deduce that, in fact, U ∞ (t) solves (2.4) in X.
To finish the proof of Proposition 2.3, we have to show that U ∞ ≡ 0. Indeed, applying Lemma 2.4, we would get that U n converges strongly to 0, which contradicts the hypothesis U n (0) X = 1. Note that U ∞ ≡ 0 is a direct consequence of a unique continuation property as Corollary 3.2. However, Corollary 3.2 requires Ω to be smooth, whereas Proposition 2.3 could be more general. Therefore, we recall another classical argument to show that U ∞ ≡ 0.

Denote N the set of function U ∞ (0) ∈ X satisfying (2.4), which is obviously a linear subspace of X. We will prove that N = {0}. Since γ(x)|∂ t u ∞ | 2 ≡ 0 for functions u ∞ in N and since N is a closed subspace, Lemma 2.4 shows that any weakly convergent subsequence of N is in fact strongly convergent. By Riesz Theorem, N is therefore finite dimensional. For any t ∈ R, e tA applies N into itself and thus A |N is a bounded linear operator. Assume that N = {0}, then A |N admits an eigenvalue λ with eigenvector Y = (y 0 , y 1 ) ∈ N. This means that y 1 = λy 0 and that (∆β)y 0 = λ 2 y 0 . Moreover, we know that y 1 = 0 on ω and so, if λ = 0, that y 0 = 0 on ω. This implies y 0 ≡ 0 by the unique continuation property of elliptic operators. Finally, if λ = 0, we have (∆β)y 0 = 0 and y 0 = 0, because, by assumption, ∆β is a negative definite operator.

So we have proved N = {0} and therefore U ∞ = 0, that is Ũn (0) converges to 0 weakly in X. We can then apply Lemma 2.4 on any interval [-n/2, -n/2 + T ] where L is the time in the geometric control condition (GCC) and obtain a contradiction to U n (0) X = 1.

Let us now consider the cases s ∈ (0, 1]. The basic semigroup properties (see [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]) shows that, if U ∈ X 1 = D(A), then e At U belongs to D(A) and

e At U X 1 = Ae At U X + e At U X = e At AU X + e At U X ≤ Ce -λt ( AU X + U X ) = Ce -λt U D(A) .
This shows Proposition 2.3 for s = 1. Notice that we do not have to require any regularity for γ to obtain this result. Then, Proposition 2.3 for s ∈ (0, 1) follows by interpolating between the cases s = 0 and s = 1 (see [START_REF] Tartar | An Introduction to Sobolev Spaces and Interpolation[END_REF]).

First nonlinear exponential decay properties

Theorem 2.2 shows that the energy E is non-increasing along the solutions of (1.1). The purpose of this paper is to obtain the exponential decay of this energy in the sense of property (ED) stated above. We first recall the well-known criterion for exponential decay.

Proposition 2.5. The exponential decay property (ED) holds if and only if, there exist T and C such that

E(u(0)) ≤ C(E(u(0)) -E(u(T ))) = C [0,T ]×Ω γ(x)|∂ t u(x, t)| 2 dtdx (2.5)
for all solutions u of (1.1) with E(u(0)) ≤ E 0 .

Proof: If (ED) holds then obviously (2.5) holds for T large enough since

E(u(0)) - E(u(T )) ≥ (1 -Ke -λT )E(u(0)). Conversely, if (2.5) holds, using E(u(T )) ≤ E(u(0)), we get E(u(T )) ≤ C/(C + 1)E(u(0)) and thus E(u(kT )) ≤ (C/(C + 1)) k E(u(0)).
Using again the decay of the energy to fill the gaps t ∈ (kT, (k + 1)T ), this shows that (ED) holds.

First, we prove exponential decay in the case of positive damping, which will be helpful to study what happens outside a large ball since (1.3) is assumed in the whole paper. Note that the fact that -∆ + β is positive is necessary to avoid for instance the constant undamped solutions.

Proposition 2.6. Assume that ω = Ω, that is that γ(x) ≥ α > 0 everywhere. Then (ED) holds.

Proof: We recall here the classical proof. We introduce a modified energy

Ẽ(u) = Ω 1 2 (|∂ t u| 2 + |∇u| 2 + β|u| 2 ) + V (u) + εu∂ t u with ε > 0. Since Ω |∇u| 2 + β|u| 2 controls u 2 L 2
, Ẽ is equivalent to E for ε small enough and it is sufficient to obtain the exponential decay of the auxiliary energy Ẽ. Using γ ≥ α > 0 and uf (u) ≥ 0, a direct computation shows for ε small enough that

Ẽ(u(T )) -Ẽ(u(0)) = T 0 Ω -γ(x)|∂ t u| 2 + ε|∂ t u| 2 + εγ(x)u∂ t u -ε(|∇u| 2 + β|u| 2 ) -εuf (u) ≤ -C T 0 (u, ∂ t u) 2 H 1 ×L 2 ≤ -C T 0 Ẽ(t)dt ≤ -CT Ẽ(T ) ,
where C is some positive constant, not necessarily the same from line to line. Thus, Ẽ(u(0)) -Ẽ(u(T )) ≥ CT Ẽ(u(T )) with CT > 0 and therefore Ẽ(u(0)) ≥ µ Ẽ(u(T )) with µ > 1. As in the proof of Proposition 2.3, this last property implies the exponential decay of Ẽ and thus the one of E.

A unique continuation result for equations with partially holomorphic coefficients

Comparatively to previous articles on the stabilization of the damped wave equations as [START_REF] Dehman | Stabilization and control for the subcritical semilinear wave equation[END_REF], one of the main novelties of this paper is the use of a unique continuation theorem requiring partially analyticity of the coefficients, but very weak geometrical assumptions as shown in Corollary 3.2. We use here the result of Robbiano and Zuily in [START_REF] Robbiano | Uniqueness in the Cauchy problem for operators with partially holomorphic coefficients[END_REF]. This result has also been proved independently by Hörmander in [START_REF] Hörmander | On the uniqueness of the Cauchy problem under partial analyticity assumptions, Geometrical optics and related topics[END_REF] and has been generalised by Tataru in [START_REF] Tataru | Unique continuation for operators with partially analytic coefficients[END_REF]. Note that the idea of using partial analyticity for unique continuation was introduced by Tataru [START_REF] Tataru | Unique continuation for solutions to PDE's; between Hörmander's theorem and Holmgren's theorem[END_REF] but it requires some global analyticity assumptions that are not fulfilled in our case. All these results use very accurate microlocal analysis and hold in a much more general framework than the one of the wave equation. However, for sake of simplicity, we restrict the statement to this case.

Theorem 3.1. Robbiano-Zuily, Hörmander (1998)

Let d ≥ 1, (x 0 , t 0 ) ∈ R d ×R and let U be a neighbourhood of (x 0 , t 0 ). Let (A i,j (x, t)) i,j=1,...,d , b(x, t), (c i (x, t)) i=1,...,d ) and d(x, t) be bounded coefficients in C ∞ (U, R). Let v be a strong solution of ∂ 2 tt v = div(A(x, t)∇v) + b(x, t)∂ t v + c(x, t).∇v + d(x, t)v (x, t) ∈ U ⊂ R d × R . (3.1)
Let ϕ ∈ C 2 (U, R) such that ϕ(x 0 , t 0 ) = 0 and (∇ϕ, ∂ t ϕ)(x, t) = 0 for all (x, t) ∈ U. Assume that:

(i) the coefficients A, b, c and d are analytic in time, (ii) A(x 0 , t 0 ) is a symmetric positive definite matrix, (iii) the hypersurface {(x, t) ∈ U , ϕ(x, t) = 0} is not characteristic at (x 0 , t 0 ) that is that we have

|∂ t ϕ(x 0 , t 0 )| 2 = ∇ϕ(x 0 , t 0 )|A(x 0 , t 0 )∇ϕ(x 0 , t 0 ) (iv) v ≡ 0 in {(x, t) ∈ U , ϕ(x, t) ≤ 0}.
Then, v ≡ 0 in a neighbourhood of (x 0 , t 0 ).

Proof: We only have to show that Theorem 3.1 is a direct translation of Theorem A of [START_REF] Robbiano | Uniqueness in the Cauchy problem for operators with partially holomorphic coefficients[END_REF] in the framework of the wave equation. To use the notations of [START_REF] Robbiano | Uniqueness in the Cauchy problem for operators with partially holomorphic coefficients[END_REF], we set x a to be the time variable and x b the space variable and we set (x 0 , t 0 ) = x 0 = (x 0 b , x 0 a ). Equation (3.1) corresponds to the differential operator

P = ξ 2 a -t ξ b A(x b , x a )ξ b -b(x b , x a )ξ a -c(x b , x a )ξ b -d(x b , x a ) with principal symbol p 2 = ξ 2 a -t ξ b A(x b , x a )ξ b .
All the statement of Theorem 3.1 is an obvious translation of Theorem A of [START_REF] Robbiano | Uniqueness in the Cauchy problem for operators with partially holomorphic coefficients[END_REF], except maybe for the fact that Hypothesis (iii) implies the hypothesis of pseudo-convexity of [START_REF] Robbiano | Uniqueness in the Cauchy problem for operators with partially holomorphic coefficients[END_REF]. We compute

{p 2 , ϕ} = 2ξ a ϕ ′ a -2 t ξ b A(x a , x b )ϕ ′ b . Let us set ζ = (x 0 a , x 0 b , iϕ ′ a (x 0 ), ξ b +iϕ ′ b (x 0 )), then {p 2 , ϕ} (ζ) = 0 if and only if i(ϕ ′ a (x 0 )) 2 -i t ϕ ′ b (x 0 ))A(x 0 )ϕ ′ b (x 0 ) -t ξ b A(x 0 )ϕ ′ b (x 0 ) = 0 . This is possible only if (ϕ ′ a (x 0 )) 2 = t ϕ ′ b (x 0 )A(x 0 )ϕ ′ b (x 0 )
, that is if the hypersurface ϕ = 0 is characteristic at (x 0 , t 0 ). Thus, if this hypersurface is not characteristic, then the pseudoconvexity hypothesis of Theorem A of [START_REF] Robbiano | Uniqueness in the Cauchy problem for operators with partially holomorphic coefficients[END_REF] holds.

The previous theorem allows to prove some unique continuation result with some optimal time and geometric assumption. This allows to prove unique continuation where the geometric condition is only, roughly speaking, that we do not contradict the finite speed of propagation.

Corollary 3.2. Let T > 0 (or T = +∞) and let b, (c i ) i=1,2,3 and d be smooth coefficients in C ∞ (Ω × [0, T ], R). Assume moreover that b, c and d are analytic in time and that v is a strong solution of

∂ 2 tt v = ∆v + b(x, t)∂ t v + c(x, t).∇v + d(x, t)v (x, t) ∈ Ω × (-T, T ) . (3.2)
Let O be a non-empty open subset of Ω and assume that v(x, t

) = 0 in O × (-T, T ). Then v(x, 0) ≡ 0 in O T = {x 0 ∈ Ω , d(x 0 , O) < T }. As consequences: a) if T = +∞, then v ≡ 0 everywhere, b) if v ≡ 0 in O × (-T, T
) and O T = Ω, then v ≡ 0 everywhere.

Proof: Since Ω is assumed to be connected, both consequences are obvious from the first statement. Let x 0 be given such that d(x 0 , O) < T . There is a point x * ∈ O linked to x 0 by a smooth curve of length l < T , which stays away from the boundary. We introduce a sequence of balls B(x 0 , r), . . . , B(x K , r) with r ∈ (0, T /K), x k-1 ∈ B(x k , r) and x K = x * , such that B(x k , r) stays away from the boundary and is small enough such that it is diffeomorphic to an open set of R 3 via the exponential map. Note that such a sequence of balls exists because the smooth curve linking x 0 to x K is compact and of length smaller than T . We also notice that it is sufficient to prove Corollary 3.2 in each ball B(x k , r). Indeed, this would enable us to apply Corollary 3.2 in B(x K , r) × (-T, T ) to obtain that v vanishes in a neighbourhood of x K-1 for t ∈ (-T +r, T -r) and then to apply it recursively in B(x K-1 , r) × (-T + r, Tr), . . . , B(x 1 , r) × (-T + (K -1)r, T -(K -1)r) to obtain that v(x 0 , 0) = 0.

From now on, we assume that x 0 ∈ B(x * , r) and that v vanishes in a neighbourhood O of x * for t ∈ (-r, r). Since d(x 0 , x * ) < r, we can introduce a non-negative function

h ∈ C ∞ ([-r, r], R) such that h(0) > d(x 0 , x * ), h(±r) = 0 and |h ′ (t)| < 1 for all t ∈ [-r, r].
We set U = B(x * , r) × (-r, r) and for any λ ∈ [0, 1], we define

ϕ λ (x, t) = d(x, x * ) 2 -λh(t) 2 .
Since r is assumed to be smaller than the radius of injectivity of the exponential map, ϕ λ is a smooth well-defined function. We prove Corollary 3.2 by contradiction. Assume that v(x 0 , 0) = 0. We denote by V λ the volume {(x, t) ∈ U, ϕ λ (x, t) ≤ 0}. We notice that r,r) where v vanishes, and that V 1 contains (x 0 , 0) where v does not vanish. Thus

V λ 1 ⊂ V λ 2 if λ 1 < λ 2 , that for small λ, V λ is included in O × (-
λ 0 = sup{λ ∈ [0, 1] , ∀(x, t) ∈ V λ , v(x, t) = 0}
is well defined and belongs to (0, 1). For t close to -r or r, h(t) is small and the section {x, (x, t) ∈ V λ 0 } of V λ 0 is contained in O where v vanishes. Therefore, by compactness, the hypersurface S λ 0 = ∂V λ 0 must touch the support of v at some point (x 1 , t 1 ) ∈ U (see Figure 1). In local coordinates, ∆ can be written div(A(x)∇.)+c(x)•∇. Moreover, ∇ϕ λ |A∇ϕ λ = |∇ g d(., x * )| 2 g = 1 where the index g means that the gradient and norm are taken according to the metric. Therefore, the hypersurface S λ 0 is non-characteristic at (x 1 , t 1 ) in the sense of Hypothesis (iii) of Theorem 3.1 since |∂ t ϕ λ (x, t)| = |λh ′ (t 1 )| < 1. Thus, we can apply Theorem 3.1 with ϕ = ϕ λ 0 at the point (x 1 , t 1 ), mapping everything in the 3d-euclidean frame via the exponential chart. We get that v must vanish in a neighbourhood of (x 1 , t 1 ). This is obviously a contradiction since (x 1 , t 1 ) has been taken in the support of v.

Asymptotic compactness

As soon as t is positive, a solution u(t) of a parabolic PDE becomes smooth and stays in a compact set. The smoothing effect in finite time of course fails for the damped wave equations. However, these PDEs admit in some sense a smoothing effect in infinite time. This effect is called asymptotic compactness if one is interested in extracting asymptotic subsequences as in Proposition 4.3, or asymptotic smoothness if one uses the regularity of globally bounded solutions as in Proposition 4.4. For the reader interested in these notions, we refer for example to [START_REF] Hale | Asymptotic behavior of dissipative systems[END_REF]. The proof of this asymptotic smoothing effect is based on the variation of constant formula U(t) = e At U 0 + t 0 e A(t-s) F (U(s))ds and two properties: -the exponential decay of the linear group (Proposition 2.3), which implies that the linear part e At U 0 asymptotically disappears, -the regularity of the nonlinearity F implying the compactness of the nonlinear term t 0 e A(t-s) F (U(s))ds (Corollary 4.2 below). Note that the subcriticality of f is the key point of this property and that our arguments cannot be extended as they stand to the critical case p = 5.

The purpose of this section is to prove some compactness and regularity results about undamped solutions as (1.8). Note that these results could also have been obtained with a more "control theoretic" proof (see section 8 for a sketch of the alternative proof) based on propagation results or observability estimates. Here, we have chosen to give a different one using asymptotic regularization, more usual in dynamical system. The spirit of the proof remains quite similar: prove that the nonlinearity is more regular than it seems a priori and use some properties of the damped linear equation.

Regularity of the nonlinearity

Since f is subcritical, it is shown in [START_REF] Dehman | Stabilization and control for the subcritical semilinear wave equation[END_REF] that the nonlinear term of 1.1 yields a gain of smoothness.

If v ∈ L ∞ ([0, T ], H 1+s (R 3 )) is a function with finite Strichartz norms v L q ([0,T ],L r (R 3 )) ≤ R, then χ(x)f (v) ∈ L 1 ([0, T ], H s+ε (R 3 )) and moreover χ(x)f (v) L 1 ([0,T ],H s+ε (R 3 )) ≤ C v L ∞ ([0,T ],H 1+s (R 3 )) .
The constant C depends only on χ, s, T , (q, r), R and the constant in Estimate (1.2). Theorem 4.1 is a copy of Theorem 8 of [START_REF] Dehman | Stabilization and control for the subcritical semilinear wave equation[END_REF], except for two points. First, we would like to apply the result to a solution v of the damped wave equation on a manifold possibly with boundaries, where not all Strichartz exponents are available. This leads to the constraint q ≥ 7/2 for the Strichartz exponents (q, r) of (2.2) (see Theorem 2.2). In the proof of Theorem 8 of [START_REF] Dehman | Stabilization and control for the subcritical semilinear wave equation[END_REF], the useful Strichartz estimate corresponds to r = 3(p-1)

1-ε and q = 2(p-1) p-3+2ε and it is required that q ≥ p -1, which yields ε ≤ (5p)/2. In this paper, we require also that q ≥ 7/2 which yields in addition ε ≤ (17 -3p)/14. Notice that p < 5 and thus both bounds are positive.

The second difference is that, in [START_REF] Dehman | Stabilization and control for the subcritical semilinear wave equation[END_REF], the function f is assumed to be of class C 3 and to satisfy

|f ′′ (u)| ≤ C(1 + |u|) p-2 and |f (3) (u)| ≤ C(1 + |u|) p-3 (4.1)
in addition of (1.2). Since Theorem 4.1 concerns the L 1 (H s ′ )-norm of χ(x)f (v) for s ′ = s+ε ≤ 1, we can omit Assumption (4.1). Actually, we make the assumption ε ≤ 1-s which is not present in [START_REF] Dehman | Stabilization and control for the subcritical semilinear wave equation[END_REF] and a careful study of their proof shows that (1.2) is not necessary under that assumption. Indeed, let f (u) = th 3 (u)|u| p . The function f is of class C 3 and satisfies (1.2) and (4.1). Hence, Theorem 8 of [START_REF] Dehman | Stabilization and control for the subcritical semilinear wave equation[END_REF] can be applied to f and we can bound the L 1 (H s ′ )-norm of f as in Theorem 4.1. On the other hand, we notice that

| f (u)| ∼ ±∞ |u| p , f ′ (u) ∼ ±∞ p|u| p-1 and f ′ (u) ≥ 0. Therefore, since f satisfies (1.2), there exists C > 0 such that |f (u)| ≤ C(1 + | f (u)|) and |f ′ (u)| ≤ C(1 + f ′ (u)). Thus, if say v > u for fixing the notations, |f (v) -f (u)| ≤ (v -u) 1 0 |f ′ (u + τ (v -u))|dτ ≤ C(v -u) + C(v -u) 1 0 f ′ (u + τ (v -u))dτ = C(v -u) + C( f (v) -f (u)) ≤ C|v -u| + C| f (v) -f (u)| .
For 0 < s < 1, using the above inequalities and the definition of the H s ′ -norm as

χf (u) 2 H s ′ = χf (u) 2 L 2 + R 6 |χ(x)f (u(x)) -χ(y)f (u(y))| 2 |x -y| 2s ′ dxdy , we obtain χf (u) L 1 (H s ′ ) ≤ C u L ∞ (H 1 ) + C χ f (u) L 1 (H s ′ )
where χ is another cut-off function with larger support. Hence, for 0 < s < 1, the conclusion of Theorem 4.1 holds not only for f but also for f . If s ′ = 1, we just apply the chain rule and the proof is easier. Note that the above arguments show that the constant C depends on f through Estimate 1.2 only. Notice in addition that, since f is only C 1 , we cannot expect χf (v) to be more regular than H 1 and that is why we also assume ε ≤ 1s.

In this paper, we use a generalisation of Theorem 8 of [START_REF] Dehman | Stabilization and control for the subcritical semilinear wave equation[END_REF] to non-compact manifolds with boundaries. Corollary 4.2. Let R > 0 and T > 0. Let s ∈ [0, 1) and let ε = min(1s, (5p)/2, (17 -3p)/14) > 0 with p as in (1.2). There exist (q, r) satisfying (2.2) and C > 0 such that the following property holds.

If v ∈ L ∞ ([0, T ], H 1+s (Ω) ∩ H 1 0 (Ω)) is a function with finite Strichartz norms v L q ([0,T ],L r (Ω)) ≤ R, then f (v) ∈ L 1 ([0, T ], H s+ε 0 (Ω)) and moreover f (v) L 1 ([0,T ],H s+ε 0 (Ω)) ≤ C v L ∞ ([0,T ],H s+1 (Ω)∩H 1 0 (Ω))
. The constant C depends only on Ω, (q, r), R and the constant in Estimate (1.2).

Proof: Since we assumed that Ω has a bounded geometry in the sense that Ω is compact or a compact perturbation of a manifold with periodic metric, Ω can be covered by a set of C ∞ -charts α i : U i -→ α i (U i ) ⊂ R 3 such that α i (U i ) is equal either to B(0, 1) or to B + (0, 1) = {x ∈ B(0, 1), x 1 > 0} and such that, for any s ≥ 0 the norm of a function u ∈ H s (Ω) is equivalent to the norm

( i∈N u • α -1 i 2 H s (α i (U i )) ) 1/2 .
Moreover, the Strichartz norm

L q ([0, T ], L r (α i (U i )) of v • α -1 i is uniformly controlled from above by the Strichartz norm L q ([0, T ], L r (U i )) of v which is bounded by R.
Therefore, it is sufficient to prove that Corollary 4.2 holds for Ω being either B(0, 1) or B + (0, 1). Say that Ω = B + (0, 1), the case Ω = B(0, 1) being simpler. To apply Theorem 4.1, we extend v in a neighbourhood of B + (0, 1) as follows. For x ∈ B + (0, 2), we use the radial coordinates x = (r, σ) and we set ṽ(x) = ṽ(r, σ) = 5v(1r, σ) -20v(1r/2, σ) + 16v(1r/4, σ) .

Then, for x = (x 1 , x 2 , x 3 ) ∈ B -(0, 2), we set ṽ(x) = 5v(-x 1 , x 2 , x 3 ) -20v(-x 1 /2, x 2 , x 3 ) + 16v(-x 1 /4, x 2 , x 3 ) .
Notice that ṽ is an extension of v in B(0, 2), which preserves the C 2 -regularity, and that the H s -norm for s ≤ 2 as well as the Strichartz norms of ṽ are controlled by the corresponding norms of v. Let χ ∈ C ∞ 0 (R 3 ) be a cut-off function such that χ ≡ 1 in B + (0, 1) and χ ≡ 0 outside B(0, 2). Applying Theorem 4.1 to χ(x)f (χ(x)ṽ) yields a control of f (v) L 1 ([0,T ],H s+ε (B + (0,1))) by v L ∞ ([0,T ],H s+1 (Ω)) . Finally, notice that f (0) = 0 and thus, the Dirichlet boundary condition on v naturally implies the one on f (v).

Asymptotic compactness and regularization effect

As explained in the beginning of this section, using Duhamel formula U(t) = e At U 0 + t 0 e A(t-s) F (U(s))ds and Corollary 4.2, we obtain two propositions related to the asymptotic smoothing effect of the damped wave equations.

Proposition 4.3. Let f ∈ C 1 (R) satisfying (1.2), let (u n 0 , u n 1 ) be a sequence of initial data which is bounded in X = H 1 0 (Ω) × L 2 (Ω)
and let (u n ) be the corresponding solutions of the damped wave equation (1.1). Let (t n ) ∈ R be a sequence of times such that t n → +∞ when n goes to +∞.

Then, there exist subsequences (u φ(n) ) and (t φ(n) ) and a global solution u ∞ of (1.1) such that

∀T > 0 , (u φ(n) , ∂ t u φ(n) )(t φ(n) + .) -→ (u ∞ , ∂ t u ∞ )(.) in C 0 ([-T, T ], X) .
Proof: We use the notations of Section 2. Due to the equivalence between the norm of X and the energy given by (2.1) and the fact that the energy is decreasing in time, we know that U n (t) is uniformly bounded in X with respect to n and t ≥ 0. So, up to taking a subsequence, it weakly converges to a limit U ∞ (0) which gives a global solution U ∞ . We notice that, due to the continuity of the Cauchy problem with respect to the initial data stated in Theorem 2.2, it is sufficient to show that U φ(n) (t φ(n) ) → U ∞ (0) for some subsequence φ(n). We have

U n (t n ) = e Atn U n (0) + tn 0 e sA F (U n (t n -s)) ds = e Atn U n (0) + ⌊tn⌋-1 k=0 e kA 1 0 e sA F (U n (t n -k -s)) ds + tn ⌊tn⌋ e sA F (U n (t n -s)) ds = e Atn U n (0) + ⌊tn⌋-1 k=0 e kA I k,n + I n (4.2) Theorem 2.2 shows that the Strichartz norms u n (t n -k -.) L q ([0,1],L r (Ω))
e kA I k,n + I n X ε ≤ ⌊tn⌋-1 k=0 Ce -λk M + M ≤ M 1 + C 1 -e -λ .
Moreover, Proposition 2.3 shows that e Atn U n (0) goes to zero in X when n goes to +∞. Therefore, by a diagonal extraction argument and Rellich Theorem, we can extract a subsequence

U φ(n) (t φ(n) ) that converges to U ∞ (0) in H 1 0 (B) × L 2 (B) for all bounded set B of Ω.
To finish the proof of Proposition 4.3, we have to show that this convergence holds in fact in X and not only locally. Let η > 0 be given. Let T > 0 and let Ũn be the solution of (1.1) with Ũn (0) = U n (t n -T ) and with γ being replaced by γ, where γ(x) ≡ γ(x) for large x and γ ≥ α > 0 everywhere. By Proposition 2.6, Ũn (T ) X ≤ η if T is chosen sufficiently large and if n is large enough so that t n -T > 0. Since the information propagates at finite speed in the wave equation, U n (t n ) ≡ Ũn (T ) outside a large enough bounded set and thus U φ(n) (t φ(n) ) has a X-norm smaller than η outside this bounded set. On the other hand, we can assume that the norm of U ∞ (0) is also smaller than η outside the bounded set. Then, choosing n large enough, U φ(n) (t φ(n) ) -U ∞ (0) X becomes smaller than 3η.

The trajectories U ∞ appearing in Proposition 4.3 are trajectories which are bounded in X for all times t ∈ R. The following result shows that these special trajectories are more regular than the usual trajectories of the damped wave equation.

Proposition 4.4. Let f ∈ C 1 (R) satisfying (1.
2) and let E 0 ≥ 0. There exists a constant M such that if u is a solution of (1.1), which exists for all times t ∈ R and satisfies

sup t∈R E(u(t)) ≤ E 0 , then t → U(t) = (u(t), ∂ t u(t)) is continuous from R into D(A) and sup t∈R (u(t), ∂ t u(t)) D(A) ≤ M .
In addition, M depends only on E 0 and the constants in (1.2).

Proof: We use a bootstrap argument. For any t ∈ R and n ∈ N,

U(t) = e nA U(t -n) + n-1 k=0 e kA 1 0 e sA F (U(t -k -s)) ds .
Using Proposition 2.3, when n goes to +∞, we get

U(t) = +∞ k=0 e kA 1 0 e sA F (U(t -k -s)) ds . ( 4
|u(x)| ≤ K u H 2 .
In particular, the solution u in the statement of Proposition 4.4 belongs to C 0 (Ω × R, R) and sup (x,t)∈Ω×R |u(x, t)| ≤ KM.

Proof: Proposition 4.5 follows directly from the fact that Ω has a bounded geometry and from the classical Sobolev embedding H 2 ֒→ C 0 in the ball B(0, 1) of R 3 .

Smoothness and uniqueness of non-dissipative complete solutions

In this section, we consider only a non-dissipative complete solution, that is a solution u * existing for all times t ∈ R for which the energy E is constant. In other words, u * (t) solves

   ∂ 2 tt u * = ∆u * -βu * -f (u * ) (x, t) ∈ Ω × R , u * (x, t) = 0 (x, t) ∈ ∂Ω × R ∂ t u * (x, t) = 0 (x, t) ∈ supp(γ) × R (5.1)
Since the energy E is not dissipated by u * (t), we can write E(u * ) instead of E(u * (t)). Yet, an interesting fact that will be used several times in the sequel is that such u * is, at the same time, solution of both damped and undamped equations. The purpose of this section is: -First show that u * is analytic in time and smooth in space. The central argument is to use a theorem of J.K. Hale and G. Raugel in [START_REF] Hale | Regularity, determining modes and Galerkin methods[END_REF].

-Then use the unique continuation result of L. Robbiano and C. Zuily stated in Corollary 3.2 to show that u * is necessarily an equilibrium point of (1.1).

-Finally show that the assumption sf (s) ≥ 0 imply that u * ≡ 0.

We enhance that the first two steps are valid and very helpful in a more general framework than the one of our paper.

Smoothness and partial analyticity of u *

First, we recall here the result of Section 2.2 of [START_REF] Hale | Regularity, determining modes and Galerkin methods[END_REF], adapting the statement to suit our notations.

Theorem 5.1. [START_REF] Hale | Regularity, determining modes and Galerkin methods[END_REF] Let Y be a Banach space. Let P n ∈ L(Y ) be a sequence of continuous linear maps and let Q n = Id -P n . Let A : D(A) → Y be the generator of a continuous semigroup e tA and let G ∈ C 1 (Y ). We assume that V is a complete mild solution in Y of

∂ t V (t) = AV (t) + G(V (t)) ∀t ∈ R .
We further assume that (i) {V (t), t ∈ R} is contained in a compact set K of Y . (ii) for any y ∈ Y , P n y converges to y when n goes +∞ and (P n ) and (Q n ) are sequences of L(Y ) bounded by K 0 . (iii) the operator A splits in A = A 1 + B 1 where B 1 is bounded and A 1 commutes with P n . (iv) there exist M and λ > 0 such that e At L(Y ) ≤ Me -λt and e (A 1 +QnB 1 )t L(QnY,Y ) ≤ Me -λt for all t ≥ 0.

(v) G is analytic in the ball B Y (0, r), where r is such that r ≥ 4K 0 sup t∈R V (t) Y . More precisely, there exists ρ > 0 such that G can be extended to an holomorphic function of

B Y (0, r) + iB Y (0, ρ). (vi) {DG(V (t))V 2 |t ∈ R, V 2 Y ≤ 1 } is a relatively compact set of Y . Then, the solution V (t) is analytic from t ∈ R into Y .
More precisely, Theorem 5.1 is Theorem 2.20 (which relates to Theorem 2.12) of [START_REF] Hale | Regularity, determining modes and Galerkin methods[END_REF] applied with Hypothesis (H3mod) and (H5).

Proposition 4.4 shows that u * is continuous in both space and time variables. We apply Theorem 5.1 to show that, because f is analytic, u * is also analytic with respect to the time.

Proposition 5.2. Let f ∈ C 1 (R) satisfying (1.
2) and let E 0 ≥ 0. Let K and M be the constants given by Propositions 4.4 and 4.5. Assume that f is analytic in [-4KM, 4KM]. Then for any non-dissipative complete solution u * (t) solving (5.1) and satisfying E(u * ) ≤ E 0 , t → u * (., t) is analytic from R into X α with α ∈ (1/2, 1). In particular, for all x ∈ Ω, u * (x, t) is analytic with respect to the time.

Proof: Theorem 5.1 uses strongly some compactness properties. Therefore, we need to truncate our solution to apply the theorem on a bounded domain (of course, this is not necessary and easier if Ω is already bounded).

Let χ ∈ C ∞ 0 (Ω) be such that ∂χ ∂ν = 0 on ∂Ω, χ ≡ 1 in {x ∈ Ω, γ(x) = 0} and supp(χ) is included in a smooth bounded subdomain O of Ω. Since Proposition 4.4 shows that u * ∈ C 0 (R, D(A)) and since u * is constant with respect to the time in supp(γ), (1χ)u * is obviously analytic from R into D(A). It remains to obtain the analyticity of χu * .

In this proof, the damping γ needs to be more regular than just L ∞ (Ω). We replace γ by a damping γ ∈ C ∞ (Ω), which has the same geometrical properties (GCC) and (1.3) and which vanishes where γ does. Notice that γ∂ t u * ≡ 0 ≡ γ∂ t u * , therefore replacing γ by γ has no consequences here.

Let v = χu * , we have

∂ 2 tt v + γ(x)∂ t v = ∆v -βv + g(x, v) (x, t) ∈ O × R + , v(x, t) = 0 (x, t) ∈ ∂O × R + (5.2) with g(x, v) = -χ(x)f (v + (1 -χ)u * (x)) -2(∇χ.∇u * )(x) -(u * ∆χ)(x)
. We apply Theorem 5.1 with the following setting.

Let Y = X α = H 1+α (O)∩H 1 0 (O)×H α 0 (O) with α ∈ (1/2, 1). Let V = (v, ∂ t v) and let G(v) = (0, g(., v)). We set A = A 1 + B 1 = 0 Id ∆ -β 0 + 0 0 0 -γ .
Let (λ k ) k≥1 be the negative eigenvalues of the Laplacian operator on O with Dirichlet boundary conditions and let (ϕ k ) be corresponding eigenfunctions. We set P n to be the canonical projections of X on the subspace generated by ((ϕ k , 0)) k=1...n and ((0, ϕ k )) k=1...n .

To finish the proof of Proposition 5.2, we only have to check that the hypotheses of Theorem 5.1 hold.

The trajectory V is compact since we know by Proposition 4.4 that it is bounded in X 1 , which gives (i).

Hypothesis (ii) and (iii) hold with K 0 = 1 by construction of P n and because

B 1 is bounded in Y since γ belongs to C ∞ (Ω).
The first part of Hypothesis (iv) follows from Proposition 2.3. The second estimate e (A 1 +QnB 1 )t L(QnY,Y ) ≤ Me -λt means that the restriction Q n AQ n of A to the high frequencies of the wave operator also generates a semigroup satisfying an exponential decay. By a result of Haraux in [START_REF] Haraux | Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps[END_REF], we know that this also holds, see Section 2.3.2 of [START_REF] Hale | Regularity, determining modes and Galerkin methods[END_REF] for the detailed arguments.

We recall that u * (x, .) is constant outside χ -1 (1) and belongs locally to H 1+α since u * ∈ D(A). Therefore, the terms (1χ)u * (x), ∇χ.∇u * and u * ∆χ appearing in the definition of g are in H 1 . Moreover, they satisfy Dirichlet boundary condition on ∂Ω since u * ≡ 0 and ∂ ν χ ≡ 0 there. Of course, they also satisfy Dirichlet boundary condition on the other parts of ∂O since χ ≡ 0 outside O. Notice that α > 1/2 and thus

H 1+α (O) ∩ H 1 0 (O) is an algebra included in C 0 . Therefore, (1.2) shows that G is of class C 1 in the bounded sets of Y . Since u ∈ [-4KM, 4KM] → f (u) ∈ R is analytic, it can be extended to a holomorphic function in [-4KM, 4KM] + i[-ρ, ρ]
for small ρ > 0. Using again the embedding H 1+α (O) ֒→ C 0 (O) and the definitions of K and M, we deduce that (v) holds.

Finally, for

V 2 = (v 2 , ∂ t v 2 ) with V 2 Y ≤ 1 DG(V (t))V 2 = (0, -χ(x)f ′ (v(t) + (1 - χ)u * (x))v 2 ) is relatively compact in Y since v(t) is bounded in H 2 ∩ H 1 0 due to Proposition 4.4 and therefore v 2 ∈ H 1+α → χ(x)f ′ (v(t) + (1 -χ)u * (x))v 2 ∈ H α is a compact map.

This yields (vi).

Once the time-regularity of u * is proved, the space-regularity follows directly. Proposition 5.3. Let f and u * be as in Proposition 5.2, then u * ∈ C ∞ (Ω × R).

Proof: Proposition 5.2 shows that u * and all its time-derivatives belongs to X α with α ∈ (1/2, 1). Due to the Sobolev embeddings, this implies that any time-derivative of u * is Hölder continuous. Writing

∆u * = ∂ 2 tt u * + βu * + f (u * ) (5.3)
and using the local elliptic regularity properties (see [START_REF] Miranda | Partial differential equations of elliptic type[END_REF] and the references therein for example), we get that u * is locally of class C 2,λ in space for some λ ∈ (0, 1). Thus, u * is of class C 2,λ in both time and space. Then, we can use a bootstrap argument in (5.3) to show that u * is of class C 2k,λ for all k ∈ N.

Identification of u *

The smoothness and the partial analyticity of u * shown in Propositions 5.2 and 5.3 enable us to use the unique continuation result of [START_REF] Robbiano | Uniqueness in the Cauchy problem for operators with partially holomorphic coefficients[END_REF].

Proposition 5.4. Let f and u * be as in Proposition 5.2, then u * is constant in time, i.e. u * is an equilibrium point of the damped wave equation (1.1).

Proof:

Setting v = ∂ t u * , we get ∂ 2 tt v = ∆v -βv -f ′ (u * )v.
Propositions 5.2 and 5.3 show that u * is smooth and analytic with respect to the time and moreover v ≡ 0 in supp(γ). Thus, the unique continuation result stated in Corollary 3.2 yields v ≡ 0 everywhere.

The sign assumption on f directly implies that 0 is the only possible equilibrium point of (1.1).

Corollary 5.5. Let f ∈ C 1 (R) satisfying (1.2) and let E 0 ≥ 0. Let K and M be the constants given by Propositions 4.4 and 4.5 and assume that f is analytic in [-4KM, 4KM]. Then, the unique solution u * of (5.1) with E(u * ) ≤ E 0 is u * ≡ 0.

Proof: Due to Proposition 5.4, u * is solution of ∆u *βu * = f (u * ). By multiplying by u * and integrating by part, we obtain

Ω |∇u * | 2 + β|u * | 2 dx = -Ω u * f (u * ) dx,
which is non-positive due to Assumption (1.2). Since β ≥ 0 is such that ∆β is negative definite, this shows that u * ≡ 0.

6 Proof of Theorem 1.1

Due to Proposition 2.5, Theorem 1.1 directly follows from the following result. Proposition 6.1. Let f ∈ C 1 (R) satisfying (1.2) and let E 0 ≥ 0. Let K and M be the constants given by Propositions 4.4 and 4.5. Assume that f is analytic in [-4KM, 4KM] and that γ is as in Theorem 1.1. Then, there exist T > 0 and C > 0 such that for any u solution of (1.1) with E(u)(0) ≤ E 0 satisfies

E(u)(0) ≤ C [0,T ]×Ω γ(x) |∂ t u| 2 dtdx.
Proof: We argue by contradiction: we assume that there exists a sequence (u n ) of solutions of (1.1) and a sequence of times (T n ) converging to +∞ such that

[0,Tn]×Ω γ(x) |∂ t u n | 2 dtdx ≤ 1 n E(u n )(0) ≤ 1 n E 0 . (6.1) Denote α n = (E(u n )(0)) 1/2 . Since α ∈ [0, √ E 0 ]
, we can assume that α n converges to a limit α when n goes to +∞. We distinguish two cases: α > 0 and α = 0.

• First case: α n -→ α > 0 Notice that, due to (2.1), (u n , ∂ t u n )(0) X is uniformly bounded from above and from below by positive numbers. We set u * n = u n (T n /2 + .). Due to the asymptotic compactness property stated in Proposition 4.3, we can assume that u * n converges to a solution u * of (1.1) in C 0 ([-T, T ], X) for all time T > 0. We notice that

E(u n (0)) ≥ E(u * n (0)) = E(u n (0)) - [0,Tn/2]×Ω γ(x) |∂ t u n | 2 ≥ (1 -1/n)E(u n (0))
and thus E(u * (0)) = α 2 > 0. Moreover, (6.1) shows that γ∂ t u * n converges to zero in L 2 ([-T, T ], L 2 (Ω)) for any T > 0 and thus ∂ t u * ≡ 0 in supp(γ). In other words, u * is a non-dissipative solution of (1.1), i.e. a solution of (5.1) with E(u * ) = α 2 ≤ E 0 . Corollary 5.5 shows that u * ≡ 0, which contradicts the positivity of E(u * (0)).

• Second case: α n -→ 0 The assumptions on f allow to write f (s) = f ′ (0)s + R(s) with

|R(s)| ≤ C(|s| 2 + |s| p ) and |R ′ (s)| ≤ C(|s| + |s| p-1 ) . (6.2)
Let us make the change of unknown w n = u n /α n . Then, w n solves

✷w n + γ(x)∂ t w n + (β + f ′ (0))w n + 1 α n R(α n w n ) = 0 (6.3) and [0,Tn]×Ω γ(x) |∂ t w n | 2 dtdx ≤ 1 n . (6.4)
Denote W n = (w n , ∂ t w n ). Due to the equivalence between norm and energy given by (2.1), the scaling w n = u n /α n implies that (w n (0), ∂ t w n (0)) X is uniformly bounded from above and from below by positive numbers. Moreover, (6.1) implies

W n (t) X = (U n (t)) X α n ≥ C E(u n (t)) 1/2 α n ≥ C (E(u n )(0) -α 2 n /n) 1/2 α n ≥ C 2 > 0 (6.5)
for any t ∈ [0, T n ] and n large enough. We set f n = 1/α n R(u n ) and F n = (0, f n ). The stability estimate of Proposition 2.2 implies that u n L q ([k,k+1],L r ) ≤ Cα n uniformly for n, k ∈ N. In particular, combined with (6.2), this gives

f n L 1 ([k,k+1],L 2 ) = 1 α n R(α n w n ) L 1 ([k,k+1],L 2 ) ≤ C(α n + α p-1 n )
We can argue as in Proposition 4. where à is the modified damped wave operator

à = 0 Id ∆ -β -f ′ (0) -γ .
Notice that à satisfies an exponential decay as A in Proposition 2.3 since (1.2) implies f ′ (0) ≥ 0. By summing up as in Proposition 4.3, we get

W n (T n ) X ≤ Ce -λTn + C(α n + α p-1 n )
which goes to zero, in a contradiction with (6.5).

As a direct consequence of Proposition 6.1, we obtain a unique continuation property for nonlinear wave equations. Notice that the time of observation T required for the unique continuation is not explicit. Thus, this result is not so convenient as a unique continuation property. But it may be useful for other nonlinear stabilization problems as ✷u + γ(x)g(∂ t u) + f (u) = 0. Corollary 6.2. Let f ∈ C 1 (R) satisfying (1.2) and let E 0 ≥ 0. Assume that f is analytic in R and that ω is an open subset of Ω satisfying (GCC). Then, there exist T > 0 such that the only solution u of

✷u + βu + f (u) = 0 on [-T, T ] × Ω ∂ t u ≡ 0 on [-T, T ] × ω . (6.7) with E(u)(0) ≤ E 0 is u ≡ 0.
Proof: Corollary 6.2 is a straightforward consequence of Proposition 6.1 since we can easily construct a smooth damping γ supported in ω and such that Supp(γ) satisfies (GCC). We only have to remark that u solution of (6.7) is also solution of (1.1).

7 Proof of Theorem 1.2

Before starting the proof of Theorem 1.2 itself, we prove that C 1 (R) is a Baire space, that is that any countable intersection of open dense sets is dense. This legitimizes the genericity in C 1 (R) as a good notion of large subsets of C 1 (R). We recall that C 1 (R) is defined by (1.6) and endowed by Whitney topology, the open sets of which are generated by the neighbourhoods N f,δ defined by (1.7).

Proposition 7.1. The space C 1 (R) endowed with Whitney topology is a Baire space.

Proof: The set C 1 (R) is not an open set of C 1 (R), and neither a submanifold. It is a closed subset of C 1 (R), but C 1 (R) endowed with Whitney topology is not a completely metrizable space, since it is not even metrizable (the neighbourhoods of a function f are not generated by a countable subset of them). Therefore, we have to come back to the basic proof of Baire property as in [START_REF] Golubitsky | Stable mapping and their singularities[END_REF].

Let U be an open set of C 1 (R) and let (O n ) n∈N be a sequence of open dense sets of C 1 (R). By density, there exists a function f 0 ∈ C 1 (R) in U ∩ O 0 and by openness, there exists a positive continuous function δ 0 such that the neighbourhood N f 0 ,δ 0 is contained in U ∩ O 0 . By choosing δ 0 small enough, one can also assume that N f 0 ,2δ 0 ⊂ U ∩ O 0 and that sup u∈R |δ 0 (u)| ≤ 1/2 0 . By recursion, one constructs similar balls N fn,δn ⊂ N

f n-1 ,δ n-1 ⊂ N f 0 ,δ 0 such that N fn,2δn ⊂ U ∩O n and that sup u∈R |δ n (u)| ≤ 1/2 n . Since C 1 ([-m,
m], R) endowed with the uniform convergence topology is a complete metric space, the sequence (f i ) converges to a function f ∈ C 1 (R, R) uniformly in any compact set of R. By construction, the limit f satisfies

∀n ∈ N, ∀u ∈ R , max(|f (u) -f n (u)|, |f ′ (u) -f ′ n (u)|) ≤ δ n (u) < 2δ n (u) , (7.1)
as well as f (0) = 0 and uf (u) ≥ 0 since any f n satisfies (1.2). Moreover, there exist C > 0 and p ∈ [1, 5) such that f 0 satisfies

|f 0 (u)| ≤ C(1 + |u|) p and |f ′ 0 (u)| ≤ C(1 + |u|) p-1 . (7.2) Since max(|f (u) -f 0 (u)|, |f ′ (u) -f ′ 0 (u)|) ≤ δ 0 (u) ≤ 1, f also satisfies (7.
2) with a constant C ′ = C + 1. Therefore, f satisfies (1.2) and thus belongs to C 1 (R). In addition, f satisfying (7.1) and N fn,2δn being contained in U ∩ O n , we get f ∈ U ∩ O n for all n. This shows that ∩ n∈N O n intersects any open set U and therefore is dense in C 1 (R).

Proof of Theorem 1.2:

We denote by G n the set of functions f ∈ C 1 (R) such that the exponential decay property (ED) holds for E 0 = n. Obviously, G = ∩ n∈N G n and hence it is sufficient to prove that G n is an open dense subset of C 1 (R). We sketch here the main arguments to prove this last property.

G n is a dense subset: let N be a neighbourhood of f 0 ∈ C 1 (R). Up to choosing N smaller, we can assume that the constant in (1.2) is independent of f ∈ N . Due to Propositions 4.4 and 4.5, there exist constants K and M such that, for all f ∈ N , all the global nondissipative trajectories u of (1.1) with E(u) ≤ n are such that u L ∞ (Ω×R) ≤ KM. We claim that we can choose f ∈ N as close to f 0 as wanted such that f is analytic on [-4KM, 4KM] and still satisfies (1.2). Then, Proposition 6.1 shows that f satisfies (ED) with E 0 = n i.e. that f ∈ G n .

To obtain this suitable function f , we proceed as follows. First, we set a = 4KM and notice that it is sufficient to explain how we construct f in [-a, a]. Indeed, one can easily extend a perturbation

f of f 0 in [-a, a] satisfying f (s)s ≥ 0 to a perturbation f of f 0 in R, equal to f 0 outside of [-a -1, a + 1] and such that f (s)s ≥ 0 in [-a -1, a + 1]. We construct f in [-a, a] as follows. Since f 0 (s)s ≥ 0, we have that f ′ 0 (0) ≥ 0. We perturb f 0 to f 1 such that f 1 (0) = 0, f ′ 1 (s) ≥ ε > 0 in a small interval [-η, η] and sf 1 (s) ≥ 2ε in [-a, -η] ∪ [η, a]
, where ε could be chosen as small as needed. Then we perturb f 1 to obtain a function f 2 which is analytic in [-a, a] and satisfies f

′ 2 (s) > 0 in [-η, η], sf 2 (s) ≥ ε in [-a, -η] ∪ [η, a] and |f 2 (0)| < ε/a. Finally, we set f (s) = f 2 (s) -f 2 (0)
and check that f is analytic and satisfies sf (s) ≥ 0 in [-a, a]. Moreover, up to choosing ε very small, f is as close to f 0 as wanted.

G n is an open subset: let f 0 ∈ G n . Proposition 2.3 shows the existence of a constant C and a time T such that, for all solution u of (1.1),

E(u(0)) ≤ E 0 =⇒ E(u(0)) ≤ C T 0 Ω γ(x)|∂ t u(x, t)| 2 dxdt . (7.
3)

The continuity of the trajectories in X with respect to f ∈ C 1 (R) is not difficult to obtain: using the strong control of f given by Whitney topology, the arguments are the same as the ones of the proof of the continuity with respect to the initial data, stated in Theorem 2.2. Thus, (7.3) holds also for any f in a neighbourhood N of f 0 , replacing the constant C by a larger one. Therefore, Proposition 2.3 shows that N ⊂ G n and hence that G n is open.

A proof of compactness and regularity with the usual arguments of control theory

In this section, we give an alternative proof of the compactness and regularity properties of Propositions 4.3 and 4.4. We only give its outline since it is redundant with previous results of the article. Moreover, it is quite similar to the arguments of [START_REF] Dehman | Stabilization and control for the subcritical semilinear wave equation[END_REF]. Yet, the arguments of this section are interesting because they do not require any asymptotic arguments and they show a regularization effect through an observability estimate with a finite time T , which can be explicit. However, for the moment, it seems impossible to obtain an analytic regularity similar to Proposition 5.2 with this kind of arguments. Instead of using a Duhamel formula with an infinite interval of time (-∞, t) as in (4.3), the main idea is to use as a black-box an observability estimate for T large enough, T being the time of geometric control condition,

U 0 2 X s ≤ C Be tA U 0 2 L 2 ([0,T ],X s ) (8.1)
where

A = 0 Id ∆ -β -γ and B = 0 0 0 -γ .
The first aim is to prove that a solution of (5.1), globally bounded in energy, is also globally bounded in X s for s ∈ [0, 1]. We proceed step by step. First, let show that it is bounded in X ε .

• we fix T > large enough to get the observability estimate (8.1). By the existence theory on each [t 0 , t 0 + T ], u |[t 0 ,t 0 +T ] is bounded in Strichartz norms, uniformly for t 0 ∈ R. Since the nonlinearity is subcritical, Corollary 4.2 gives that f (u) is globally bounded in L 1 ([t 0 , t 0 + T ], H 1+ε ).

• we decompose the solution into its linear and nonlinear part by the Duhamel formula U(t) = e A(t-t 0 ) U(t 0 ) + t t 0 e A(t 0 -τ ) f (U(τ ))dτ = U lin + U N lin .

Since f (u) is bounded in L 1 ([t 0 , t 0 +T ], H 1+ε ), U N lin is uniformly bounded in C([t 0 , t 0 + T ], X ε ).

• we will now use the linear observability estimate (8.1) with s = ε, applying it to U lin :

U(t 0 ) 2 X ε = U lin (t 0 ) 2 X ε ≤ C t 0 +T t 0 γ(x)∂ t u lin 2 H ε . (8.2)
Then, using triangular inequality, we get

t 0 +T t 0 γ(x)∂ t u lin 2 H ε ≤ 2 t 0 +T t 0 γ(x)∂ t u 2 H ε + 2 t 0 +T t 0 γ(x)∂ t u N lin 2 H ε ≤ 2 t 0 +T t 0 γ(x)∂ t u N lin 2 H ε ≤ C
where we have used ∂ t u ≡ 0 on ω and U N lin is bounded in C([t 0 , t 0 + T ], X ε ). Combining this with (8.2) for any t 0 ∈ R, we obtain that U is uniformly bounded in X ε on R.

Repeating the arguments, we show that u is bounded in X 2ε , X 3ε and so on. . . until X 1 . Similar ideas allows to prove some theorem of propagation of compactness in finite time, replacing the asymptotic compactness property of Proposition 4.3.

As said above, an advantage of this method, compared to the one used in Propositions 4.3 and 4.4, is that it allows to propagate the regularity or the compactness on some finite interval of fixed length. Yet, it seems that such propagation results are not available in the analytic setting. Indeed, it seems that, for nonlinear equations, the propagation of analytic regularity or of nullity in finite time is much harder to prove. We can for instance refer to the weaker (with respect to the geometry) result of Alinhac-Métivier [START_REF] Alinhac | Propagation de l'analyticité des solutions de systèmes hyperboliques non-linéaires[END_REF] or the negative result of Métivier [START_REF] Métivier | Counterexamples to Hölmgren's uniqueness for analytic non linear Cauchy problems[END_REF].

Applications

Control of the nonlinear wave equation

In this subsection, we give a short proof of Theorem 1.3, which states the global controllability of the nonlinear wave equation. The first step consists in a local control theorem. Theorem 9.1 (Local control). Let ω satisfying the geometric control condition for a time T . Then, there exists δ such that for any (u 0 , u 1 ) in H 1 0 (Ω) × L 2 (Ω), with (u 0 , u 1 ) H 1 0 ×L 2 ≤ δ there exists g ∈ L ∞ ([0, T ], L 2 ) supported in [0, T ] × ω such that the unique strong solution of ✷u + βu + f (u) = g on [0, T ] × Ω (u(0), ∂ t u(0)) = (u 0 , u 1 ).

satisfies (u(T ), ∂ t u(T )) = (0, 0).

Proof:

The proof is exactly the same as Theorem 3 of [START_REF] Dehman | Stabilization and control for the subcritical semilinear wave equation[END_REF] or Theorem 3.2 of [START_REF] Laurent | On stabilization and control for the critical Klein Gordon equation on 3-D compact manifolds[END_REF]. The main argument consists in seeing the problem as a perturbation of the linear controllability which is known to be true in our setting. Now, as it is very classical, we can combine the local controllability with our stabilization theorem to get global controllability.

Sketch of the proof of Theorem 1.3:

In a first step, we choose as a control g = -γ(x)∂ t u where u is solution of (1.1) with initial data (u 0 , u 1 ). By uniqueness of solutions, we have u = u. Therefore, thanks to Theorem 1.1, for a large time T 1 , only depending on i) The positive trajectories of bounded sets are bounded. Indeed, (1.11) implies that, for x ∈ B(x 0 , R), we have V (x, u) = u 0 f (x, ξ)dξ ≥ 0. Moreover, for x ∈ B(x 0 , R), V (x, .) is non-increasing on (-∞, -R) and non-decreasing on (R, ∞). Thus, V (x, u) is bounded from below for x ∈ B(x 0 , R) and ∀(u, v) ∈ X , E(u, v) ≥ 1 2 (u, v) 2 X + vol(B(x 0 , R)) inf V .

The Sobolev embeddings H 1 (Ω) ֒→ L p+1 (Ω) shows that the bounded sets of X have a bounded energy. Since the energy E is non increasing along the trajectories of (1.9), we get that the trajectory of a bounded set is bounded.

ii) The dynamical system is asymptotically smooth. The asymptotic compactness exactly corresponds to the statement of Proposition 4.3. Let us briefly explain why it can be extended to the case where f depends on x. The key point is the extension of Corollary 4.2. First notice that, since we have assumed that f (x, 0) = 0 on ∂Ω, f (x, u) satisfies Dirichlet boundary condition if u does. Then, it is not difficult to see that the discussion following Theorem 4.1 can be extended to the case f depending on x by using estimates (1.10). Corollary 4.2 follows then, except for a small change: since it is possible that f (x, 0) = 0 for some x ∈ Ω, the conclusion of Corollary 4. ) . Then the proof of Proposition 4.3 is based on Corollary 4.2, the boundedness of the positive trajectories of bounded sets (both could be extended to the case where f depends on x as noticed above) and an application of Proposition 2.6 outside of a large ball. We conclude by noticing that, for x large, f (x, u)u ≥ 0 and γ(x) ≥ α > 0 and thus Proposition 2.6 can still be applied exactly as in the proof of Proposition 4.3.

iii) The dynamical system generated by (1.9) is gradient, that is that the energy E is non-increasing in time and is constant on a trajectory u if and only if u is an equilibrium point of (1.9). This last property is shown in Proposition 5.4 for f independent of x but can be easily generalised for f = f (x, u). Notice that the proof of this property is the one, where the analyticity of f is required since the unique continuation property of Section 3 is used. Finally, we enhance that the gradient structure of (1.9) is interesting from the dynamical point of view since it implies that any trajectory u(t) converges when t goes to +∞ to the set of equilibrium points.

iv) The set of equilibrium points is bounded. The argument is similar to the one of Corollary 5.5: if e is an equilibrium point of (1.9) then (1.11) implies that where we have bounded f (x, u)u from below exactly as we have done for V (x, u) in i).

Figure 1 :

 1 Figure 1: The proof of Corollary 3.2

Theorem 4 . 1 .

 41 [START_REF] Dehman | Stabilization and control for the subcritical semilinear wave equation[END_REF] Let χ ∈ C ∞ 0 (R 3 , R), R > 0 and T > 0. Let s ∈ [0, 1) and let ε = min(1s, (5p)/2, (17 -3p)/14) > 0 with p and f as in (1.2). There exist (q, r) satisfying (2.2) and C > 0 such that the following property holds.

e

  3 and writeW n (T n ) = e ÃTn W n (0) + -Ãs F n (k + s) ds + e Ã(Tn-⌊Tn⌋)Tn-⌊Tn⌋ 0 e -Ãs F n (⌊T n ⌋ + s) ds .(6.6)

  2 should be replaced byf (x, v) L 1 ([0,T ],H s+ε 0 (Ω)) ≤ C(1 + v L ∞ ([0,T ],H s+1 (Ω)∩H 1 0 (Ω))

1 2 |∇e| 2 +

 2 β|e| 2 = -Ω f (x, e)edx ≤ -vol(B(x 0 , R)) inf (x,u)∈Ω×R f (x, u)u ,

  Using Proposition 2.3 again and summing up, we get that the last terms of (4.2) are bounded in H 1+ε (Ω)×H ε (Ω) uniformly in n by

	⌊tn⌋-1
	k=0

are uniformly bounded since the energy of U n is uniformly bounded. Therefore, Corollary 4.2 and Proposition 2.3 show that the terms I n,k = 1 0 e sA F (U n (t nks)) ds, as well as I n , are bounded by some constant M in H 1+ε (Ω) × H ε (Ω) uniformly in n and k.

  Proposition 4.5. The Sobolev embedding H 2 (Ω) ֒→ C 0 (Ω) holds and there exists a constant K such that ∀u ∈ H 2 (Ω) , sup

	x∈Ω

.3) Moreover, arguing exactly as in the proof of Proposition 4.3, we show that Proposition 2.3 and Corollary 4.2 imply that Equality (4.3) also holds in X ε . Hence, U(t) is uniformly bounded in X ε . Then, using again Proposition 2.3 and Corollary 4.2, (4.3) also holds in X 2ε etc. Repeating the arguments and noting that, until the last step, ε only depends on p, we obtain that U(t) is uniformly bounded in X 1 = D(A).

Since the constant C of Corollary 4.2 only depends on f through Estimate (1.2), the same holds for the bound M here.

Existence of a compact global attractor

In this subsection, we give the modification of the proofs of this paper necessary to get Theorem 1.4 about the existence of a global attractor.

The energy associated to (1.9) in X = H 1 0 (Ω) × L 2 (Ω) is given by

The existence of a compact global attractor for (1.9) is well known for the Sobolev subcritical case p < 3. The first proofs in this case go back to 1985 ( [START_REF] Hale | Asymptotic behavior of dissipative systems[END_REF] and [START_REF] Haraux | Two remarks on hyperbolic dissipative problems. Nonlinear partial differential equations and their applications[END_REF]), see [START_REF] Raugel | Handbook of dynamical systems[END_REF] for other references. The case p = 3 as been studied in [START_REF] Babin | Attractors of Evolutionary Equations[END_REF] and [START_REF] Arrieta | A damped hyperbolic equation with critical exponent[END_REF]. For p ∈ (3, 5), Kapitanski proved in [START_REF] Kapitanski | Minimal compact global attractor for a damped semilinear wave equation[END_REF] the existence of a compact global attractor for (1.9) if Ω is a compact manifold without boundary and if γ(x) = γ is a constant damping. Using the same arguments as in the proof of our main result, we can partially deal with the case p ∈ (3, 5) with a localized damping γ(x) and with unbounded manifold with boundaries.

Assume that f satisfies the assumptions of Theorem 1.4. We first claim that we can assume in addition that f (x, 0) = 0 on ∂Ω in order to guarantee the Dirichlet boundary condition for f (x, u) if u ∈ H 1 0 (Ω). Indeed, let ϕ be the solution of ∆ϕβϕ = f (x, 0) with Dirichlet boundary conditions, which is well defined and smooth, since by assumptions f (x, 0) is smooth and compactly supported. Let χ be a smooth compactly supported cutoff function such that ∂χ ∂ν = 0 on ∂Ω and χ ≡ 1 in the ball B(x 0 , R) of Hypothesis (1.11). If f (x, 0) = 0 on ∂Ω, we consider the new variable ũ = uχϕ and the new equation

where f (x, ũ) = f (x, ũ + χϕ(x))χf (x, 0) + 2∇ϕ.∇χ + ϕ∆χ. One directly checks that u solves (1.9) if and only if ũ solves (9.1). Moreover f is smooth, analytic in ũ and satisfies Hypotheses (1.10) and (1.11) and in addition f (x, 0) = 0 on the boundary. Clearly, since our change of variables is a simple translation, obtaining a compact global attractor for (9.1) is equivalent to obtaining a compact global attractor for (1.9). In what follows, we may thus assume that f (x, 0) = 0 in addition to the hypotheses of Theorem 1.4. The arguments of this paper show the following properties.

It is well known (see [START_REF] Hale | Asymptotic behavior of dissipative systems[END_REF] or Theorem 4.6 of [START_REF] Raugel | Handbook of dynamical systems[END_REF] for examples) that Properties i)-iv) yield the existence of a compact global attractor. Hence, we obtain the conclusion of Theorem 1.4.


Nicolas Maestripieri

Martin Paegelow
email: paegelow@univ-tlse2.fr

Gilles Selleron
email: selleron@univ-tlse2.fr

Monitoring, modeling and predicting timber plantations dynamics. The case of San Juan de la Costa (Chile)

Keywords:

Introduction

Intensive forestry is a notable manifestation of land use/cover changes (LUCC) [START_REF] Bull | Industrial forest plantation subsidies: Impacts and implications[END_REF] characterized by its well-known environmental and socio-economic impacts at multiple scales [START_REF] Arroyo | Plant Invasions in Chile: Present Patterns and Future Predictions[END_REF]. The Chilean industrial forestry is focused on the intensification of the pulp and cellulose production, in order to improve its place into the world pulp market. The DL 701 (1974 -and its repeated updating) covering up to 75% the plantations costs, was the keystone of the forest expansion of pine and eucalyptus species. The timber plantations are mainly concentrated from the 8th Region of Bío Bío to the 10th Region de Los Lagos (Lake Region). However, their distribution is not uniform and there is a clear interregional imbalance. Given that, the corporate strategy is to expand their forest heritage, the companies actually focus on new suitable areas for afforestation, as the municipality of San Juan de la Costa. With the weakest national HDI, this municipality is particularly exposed to develop a spatially explicit model of land cover change. The goal is to develop future and contrasted scenarios in order to implement sustainable policies.

Methods

Study area

The study area is located at the north-west of the Lake Region, between parallels 73°47' and 73°18" (western longitude) and 40°14' and 40°44' (southern latitude) (Figure 1). The native forest covering this area is part of one of the highest priority areas recognized by the World Wild Fund (WWF). According to the National Statistics Institute (INE, 2002), more than 89% of the population live in the countryside (7929 habitants). The indigenous people called "Mapuche-Huilliche" represent 62% of this rural territory. The municipality is divided by two topographical units, the Coastal range and the Piedmont. A supervised multi-temporal classification was performed to get land use / cover (LUC) at these three dates. Four LUC classes are distinguished: timber plantations, native forest, "other" category (including second growth forest, shrubland, grassland and bare land) and open water. The "other" category allows focusing on the timber plantations and the native forest and thus simplifies the modeling process. More than three hundred field points were taken with a GPS in order to estimate the global accuracy of each classification. Analysis of land use/cover changes (LUCC) occurs in three stages, with (i) quantifying losses and gains, the analysis (ii) of the spatial distribution of these changes, and (iii) trajectories of evolutions. Then we have refined this analysis by the identification, characterization and weighing of many factors that have induced this expansion. Spatial analysis, expert analysis, stakeholder interviews and bibliographic revision assess drivers of timber plantation dynamics at local, regional, national and global scales.

Modeling future dynamics

Taking into account the past dynamics, the aim is reaching a vision of the future land use/cover changes (2017, 2026 and 2035), more specifically about the possible influences of multinational forest companies in a municipality where indigenous land recognition is lacking. Two spatially explicit models are tested: CA-Markov [START_REF] Eastman | IDRISI Andes. Guide to GIS and Image Processing[END_REF] and Land Change Modeler [START_REF] Eastman | The Land Change Modeler for Ecological Sustainability[END_REF]. With CA-Markov the spatial allocation of future dynamics is supervised by Multi-Criteria Evaluation (MCE) which generates suitability maps in order to allocate Markov chain predicted quantities of changes (Paegelow and Camacho, 2008). Land Change Modeler allows an automatic calibration of the model using a multi-layer perceptron (MLP) which is an artificial neural network (ANN). The spatial allocation procedure is performed by a multi-objective land allocation (MOLA). As for CA-Markov, quantities of expected changes are predicted by Markov chain. Both model tools are used to predict LUC in 2008, based on training data from 1986-1999. This is the calibration step before prospective scenarios. 2008 predicted LUC is compared to observed LUC in 2008 by various statistical tests.

Results

Land use and cover change in San Juan de la Costa

The native forest is the predominant land cover with 110,775 ha in 1986 (71.7%), 104,322 ha in 1999 (67.5 %) and 96,857 ha in 2008 (62.7 %). They are mainly localized on the Coastal Range, and in a more fragmented way on the Piedmont (Figure 2). The "other" category shows an increase of 44% between 1986 and 1999. The timber plantations present the largest increase with 95%, covering almost 10,000 ha in 2008 and are located on the Piedmont. Finally the water resources remain stable with 408 ha (0.26%). The major change was a conversion of 12,700 ha from native forest to "other" (12% of deforestation) between 1986 and 1999 (Figure 3). At the same period, timber plantations regained 1,483 ha from native forest (1% of substitution) and 2,329 ha from "other" (5%). Between 1999 and 2008 the process of substitution of native forest and "other" became more intensive (respectively 1,883 ha and 3,803 ha). The loops represent persistence and the arrows changes.

LUCC driving forces

The LUCC are influenced by a wide range of environmental and social factors (direct and indirect). According to Eric Lambin and Helmut Geist (2007), these factors have an influence on land use decisions "across a wide range of spatial scales, from household level decisions that influence local land use practices, to policies and economic forces that can alter land use regionally and even globally". Succinctly the main drivers are the Decree Law 701, land ability, afforestation costs, property taxes, land property, cultural factors (household consumption), unemployment, poverty, migration, slope, altitude, road network (accessibility), distance from existing land cover, distance from ports, social conflicts, etc.

Scenarios and future land use/cover dynamics

The first is a business as usual scenario (SC-TEND), based on the assumption that previous tendencies will continue. It reaches to 2017 in order to fit the time-frame used during the calibration of the model (9 years). Land availability, DL 701 update (2012), bio-energy, and the approval of the native forest law are taken into account. Results show an increase of timber plantation (15,590 ha, + 58%), while native forest decline remarkably (90,184 ha, -7%) and the "other" category slightly increases (48,295 ha, + 2%)1 .

The next two prospective scenarios are developed for 2035 because they take more time to implement themselves, given the growth of native species.

The sustainable scenario (SC-SUST) is based on a sustainable vision of the resource management, the recognition of the indigenous lands with the setting up of development politics. SC-SUST integrates suggestions of the Agrupación de Ingenieros Forestales por el Bosque Nativo (AIFBN), the Mapu Lahual Territory Project, incentive to agroforestry, promotion and distribution of native forest resources (firewood). This scenario (i) stimulates the native forest expansion regaining new areas (100,600 ha, + 3%), (ii) decreases the exotic species expansion after an increase in 2017 and 2026 (10,600 ha, + 8%) and shows a reduction of the "other" category (42,700 ha, -8%).

The third scenario is a preservation scenario (SC-0P) with an eco-centric vision. Assumptions include relocation of forest companies which leave the municipality, public demand in environmental advantages, protected areas planning and ecotourism. This scenario shows a net increase of native forest (119,500 ha, + 23%) and the timber plantations collapse (5,400 ha, -45%).

At last the intensive scenario (SC-2E) called "Sembremos Futuro" (in radical opposition with the previous scenario) is based upon an increase of the incentive for afforestation, making the assumption that international pulp market and competition explodes, land abandonment by small landowners, establishment of a pulp/cellulose factory in the region and redefinition of the routing of Coastal Range Road linking Valdivia to Puerto Montt. It extends the trend of sustained manner until 2026. The plantation areas could reach more than 54,973 ha (+ 456 %), while the native forest could fall below of 71,930 ha (-26%).

The spatial rendering of each scenario is realized by CA-Markov. Even if both models give acceptable results as for spatial allocationat least visually -, CA-Markov turns out to be more efficient than LCM for the baseline dynamics. The statistical operations show this model presents better results. Furthermore, conceptually, it gives more leewayto build backcasting scenarioand to measure the relative importance of the criteria (assignation of weighting factors).

Conclusion

Prospective modeling, and more precisely the spatial dimension, proves to be an advantage to model spatial interactions between environmental and socio-economic dynamics, which was largely lacking in the municipality of San Juan de la Costa. These four scenarios give contrasting visions of future land use/cover dynamics, and represent a basis of reflection in order to suggest efficient environmental policies. "What we support is prospective thinking, that means to prepare oneself to anticipated changes (pre-activity) and to promote actions leading to the desirable objectives (pro-activity)" (Godet, 1990, p.731).

Figure 1 .

 1 Figure 1. Location of the study area at the north-west of the Lake Region, Chile

Figure 2 .

 2 Figure 2. Land use/cover in San Juan de la Costa in 1986, 1999 and 2008.

Figure 3 .

 3 Figure 3. LUCC for 1986-1999 (up) and 1999-2008 (bottom) (in percent).The loops represent persistence and the arrows changes.

Open water is not modelized because of its persistence between1986-1999 and 1999-2008.

Data and land use/cover change detection

In order to predict the future LUCC we applied a prospective scenario-based approach following [START_REF] Godet | Introduction à la Prospective. Seven key ideas and one scenario method[END_REF], [START_REF] Hatem | La Prospective. Pratiques et Méthodes[END_REF], De [START_REF] Jouvenel | La démarche prospective. Un bref guide méthodologique[END_REF] and [START_REF] Houet | Modélisation prospective spatialisée à l'échelle locale : approche méthodologique. Application à la gestion de l'eau en Bretagne[END_REF]. The elaboration of the scenarios is based on a retrospective LUCC analysis of the municipality. To do this, we used three Landsat TM images (path 233, row 88) taken in 1986, 1999 and 2008.