Nicolas Burq
email: nicolas.burq@math.u-psud.fr

Iv Án Moyano
email: imoyano@unice.fr

A REMARK ON THE LOGARITHMIC DECAY OF THE DAMPED WAVE AND SCHR ÖDINGER EQUATIONS ON A COMPACT RIEMANNIAN MANIFOLD

In this paper we consider a compact Riemannian manifold (M, g) of class C 1 ∩ W 2,∞ and the damped wave or Schrödinger equations on M , under the action of a damping function a = a(x). We establish the following fact: if the measure of the set {x ∈ M ; a(x) = 0} is strictly positive, then the decay in time of the associated energy is at least logarithmic.

Introduction

Consider a compact Riemannian manifold (M, g) of class C 1 ∩ W 2,∞ , possibly with boundaries ∂M , endowed with a Lipschitz metric g. Denote d g x (or simply dx) the volume element in M associated to the metric g ∈ C 0 ∩ W 1,∞ and write vol g the associated volume on M . Let ∆ be the Laplace-Beltrami operator in (M, g). Recall that in local coordinates we may write

(1.1) ∆ = 1 √ det g ∂ i det gg ij ∂ j .
In this note we are interested in the evolution of respectively the wave equation and the Schrödinger equation under the influence of a damping term localised via a function 0 ≤ a(x), a ∈ L ∞ (M ) non trivial ( M a(x)dx > 0) which consequently may be supported in a small subset of M , namely E. We shall briefly recall these models.

The damped wave equation in M under the damping a∂ t u corresponds to the initial value problem

(1.2) ∂ 2 t u -∆u + a(x)∂ t u = 0, R + × M, (u, ∂ t u)| t=0 = (u 0 , u 1 ), M,
where (u 0 , u 1 ) is a given initial condition in the natural energy space H = H 1 (M ) × L 2 (M ). If ∂M = ∅, we impose the boundary conditions The energy associated to (1.2) is as usual

(1.4) E w (t, u 0 , u 1 ) = M |∂ t u(t)| 2 dx + M |∇ x u(t)| 2 dx, t ≥ 0, defined globally as u ∈ C 0 (R + ; H 1 (M )) ∩ C 1 (R + ; L 2 (M )).
The second model we are interested in, is the initial value problem for the Schrödinger equation under the action of the damping a = a(x), i.e., (1.5) i∂ t ψ + ∆ψ + ia(x)ψ = 0, R + × M, ψ| t=0 = ψ 0 , M, for a given ψ 0 ∈ L 2 (M ; C). Again if ∂M = ∅, we impose the boundary conditions (1.3). The energy associated to (1.5) is

(1.6) E S (t, ψ 0 ) = M |ψ(t)| 2 dx.
In this note we prove that if E ⊂ M is any measurable set with vol g (E) > 0, the energy functionals E w and E S decay at least logarithmically in time. This is the content of Theorems 1 and 2 below.

1.1. Main results. Since we assume a ≥ 0, M a(x)dx > 0, we deduce that there exists n > 0 such that the set

F n = {x ∈ M ; a(x) > 1 n }
has positive measure. As a consequence, with α = 1 n , β = a L ∞ , F = F n , We get that the damping function a = a(x) satisfies (1.7) α1 F (x) ≤ a(x) ≤ β, for almost all x ∈ M, with F ⊂ M of positive measure, not necessarily open. Our main result for the wave equation is the following.

Theorem 1. Let a ≥ 0, M a(x)dx > 0. Then, there exists a constant C = C(F ) > 0 such that for every

(1.8) (u 0 , u 1 ) ∈ H 2 (M ) ∩ H 1 0 (M ) × H 1 0 (M ), with Dirichlet boundary conditions, H 2 (M ) × H 1 (M ),
otherwise, the solution to the associated damped wave equation (1.2) satisfies

(1.9) E w (t, u 0 , u 1 ) ≤ C log(2 + t) 2 u 0 2 H 2 (M ) + u 1 2 H 1 (M ) , t ≥ 0,
where E w is the energy defined in (1.4).

See Section 1.2 for notation. In the case of the Schrödinger equation we obtain the next analogous result.

Theorem 2. Let a ≥ 0, M a(x)dx > 0. Then, there exists a constant C = C(F ) > 0 such that for every

(1.10) ψ 0 ∈ H 2 (M ) ∩ H 1 0 (M ), with Dirichlet boundary conditions, H 2 (M ),
otherwise, the solution to the associated Schrödinger equation (1.5) satisfies

(1.11) E S (t, ψ 0 ) ≤ C log(2 + t) 4 ψ 0 2 H 2 (M ) , t ≥ 0,
where E S is the energy defined in (1.6).

The strategy of proof combines the spectral inequalities obtained in [START_REF] Burq | Propagation of smallness and control for heat equations[END_REF] (see Theorem 3 below) with a sharp characterisation of the logarithmic decay of energy from [START_REF] Burq | Décroissance de l'énergie locale de l'équation des ondes pour le problème extérieur et absence de résonance au voisinage du réel[END_REF][START_REF] Batty | Non-uniform stability for bounded semi-groups on Banach spaces[END_REF] (see Theorem 4 in Section 2.1). We give some details concerning these results in Section 2.

1.2. Notation and setting. As mentioned above, the volume induced by g on M is defined as

vol g (A) = M 1 A (x) d g x,
for every Borel set A ⊂ M . In the case of the Euclidean flat space R d , me simply write |A| for the d-dimensional Lebesgue measure of a given Borel set A ⊂ R d . In both cases we denote B(x, r) the ball of radius r > 0 centred at a point x.

As M is compact, the Laplace-Beltrami operator on M defined in (1.1) has compact resolvent. Let (e k ) k∈N be the family of L 2 -normalised eigenfunctions of -∆, with eigenvalues λ 2 k → +∞ and satisfying

-∆e k = λ 2 k e k , e k | ∂M = 0 (Dirichlet condition) or ∂ ν e k | ∂M = 0 (Neumann condition).
Recall that (e k ) k∈N is a Hilbert basis of L 2 (M ) endowed with the usual inner product •, • . Moreover, the usual Sobolev norms on M can be defined using the spectral basis (e k ) k∈N as follows:

(1.12)

f 2 H s (M ) = k∈N (1 + λ 2 k )| f, e k | 2 , f ∈ H s (M ),
for every s ∈ R.

1.3. Previous work.

1.3.1.

Decay of damped waves. The study of the decay rates for (1.2) has been addressed by the seminal works [START_REF] Bardos | Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary[END_REF] and [START_REF] Lebeau | Equation des ondes amorties[END_REF]. These work establish an intimate relation between the rate decay of the energy and the support of the damping function. Under the geometric control condition of [START_REF] Bardos | Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary[END_REF] one can expect an exponential decay, as shown for instance in [START_REF] Bardos | Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary[END_REF][START_REF] Lebeau | Equation des ondes amorties[END_REF][START_REF] Burq | Exponential decay for the damped wave equation in unbounded domains[END_REF]. On the other hand, when the support of a does not satisfy a geometric control condition, the decay rate of the associated damped wave equation may be slower than exponential. We can find examples in the literature of polynomial decay [START_REF] Anantharaman | Sharp polynomial decay rates for the damped wave equation on the torus, with an appendix by S. Nonnenmacher[END_REF][START_REF] Phung | Polynomial decay rate for the dissipative wave equations[END_REF] or even logarithmic [START_REF] Lebeau | Equation des ondes amorties[END_REF][START_REF] Lebeau | Robbiano Stabilization of the wave equation by the boundary[END_REF][START_REF] Burq | Exponential decay for the damped wave equation in unbounded domains[END_REF][START_REF] Laurent | Logarithmic decay for damped hypoelliptic wave and Schrödinger equations[END_REF].

Under some hypothesis on the geometry of the manifold, such as the assumption that the manifold is compact and hyperbolic (negative curvature), it is possible to expect exponential decay in some (positive) Sobolev spaces as soon as a is smooth and non zero (cf. [START_REF] Jin | Damped wave equations on compact hyperbolic surfaces[END_REF]).

In this paper we establish the following fact: if | {x ∈ M ; a(x) = 0} | > 0, then the decay is at least logarithmic. We do not make any assumption on the curvature of the manifold. 1.3.2. Spectral inequalities. In the framework described in Section 1.2, Given a small subset E ⊂ M (of positive Lebesgue measure or at least not too small), we have studied in [START_REF] Burq | Propagation of smallness and control for heat equations[END_REF] how L p norms of the restrictions to E of arbitrary finite linear combinations of the form

φ = λ k ≤Λ u k e k (x)
can dominate Sobolev norms of φ on the whole M . Our result [START_REF] Burq | Propagation of smallness and control for heat equations[END_REF]Thm 2] is the following. Theorem 3. Let (M, g) be a Riemannian manifold of class C 1 ∩ W 2,∞ , possibly with boundaries ∂M . There exists δ ∈ (0, 1) such that for any m > 0, there exist C, D > 0 such that for any ω ⊂ M with vol g (ω) ≥ m and for any Λ > 0, we have

(1.13) φ = λ k ≤Λ u k e k (x) ⇒ φ L 2 (M ) ≤ Ce DΛ φ1 ω L 2 (M ) .
We shall use the spectral inequality (1.13) in Section 2.3.

1.4. Outline. In Section 2 we gather some facts about the tools used in the proof of our main result: Section 2.1 is devoted to a characterisation of logarithmic decay, Section 2.3 makes the link between the resolvent operators for waves and Schrödinger and the Helmholtz equation, and Section 2.3 is concerned with some estimates for solutions to the Helmholtz equation obtained thanks to the spectral inequalities mentioned before. The proof of Theorem 1 is carried out in Sections 3 and 4 which treat respectively the case of Dirichlet boundary conditions and the case of Neumann boundary conditions. Each of these two sections is also divided into high frequencies and low frequencies. Finally, Section 5 is concerned with the proof of Theorem 2, which is also divided into Section 5.1 (negative frequencies) and Section 5.2 (nonnegative frequencies).

Some tools

In this section we describe first (in Section 2.1) some abstract results relating the time decay of a semi-group with the growth of the resolvent operator at infinity. Next, in Section 2.2 we focus on the resolvent operators related to the wave equation (1.2) and the Schrödinger equation (1.5), which in both cases lead to a Helmholtz equation of the form ∆u + λu = S, for some parameter λ ∈ R and a source term S. Finally in Section 2.3 we use Theorem 3 to get some estimates for solutions to the Helmholtz equation that will be useful in the sequel.

2.1. Sufficient conditions for logarithmic decay. Consider a Hilbert space H and the functional equation

(2.1) dU dt = AU, t ≥ 0, with U (0) = U 0 ∈ H, for a possibly unbounded operator A with domain D(A) ⊂ H. As usual, z ∈ C belongs to the resolvent set ρ(A) whenever (A -z) -1 ∈ L(H). The spectrum of A is σ(A) = C \ ρ(A).
We focus next on the elements of ρ(A) lying on the imaginary axis. For every τ ∈ R we consider the resolvent mapping R(τ ) = (A -iτ ) -1 whenever iτ ∈ ρ(A).

2.1.1.

Growth of the resolvent and decay of the semi-group. Assume that A is the infinitesimal generator of a C 0 -continuous semi-group of operators in H that we denote (e tA ) t≥0 , so that the solution to (2.1) writes U (t) = e tA U 0 . Assume further that sup t≥0 e tA L(H) < +∞.

Batty and Duyckaerts have introduced in [START_REF] Batty | Non-uniform stability for bounded semi-groups on Banach spaces[END_REF] a quantitative approach to characterise the asymptotic behaviour of the semi-group, i.e., the fact that for some

k ∈ N * , (2.2) lim t→+∞ m k (t) = 0, with m k (t) = e tA (Id -A) -k L(H) , t ≥ 0,
in terms of the purely spectral condition

(2.3) σ(A) ∩ iR = ∅.
Observe that this condition ensures that the resolvent operators R(τ ) are well defined for any τ ∈ R. Moreover, it is possible to describe the decay rate of (2.2) in terms of the growth of the function

(2.4) M (µ) = sup |τ |≤µ (A -iτ ) -1 L(H) , µ ∈ [0, +∞).
The following result, obtained in [4, Thm. 3], guarantees logarithmic decay of all m k as long as M grows at most exponentially at infinity. Theorem 4 (Lebeau-Robbiano [START_REF] Lebeau | Robbiano Stabilization of the wave equation by the boundary[END_REF], Burq [START_REF] Burq | Décroissance de l'énergie locale de l'équation des ondes pour le problème extérieur et absence de résonance au voisinage du réel[END_REF], Batty-Duyckaerts, [3, Thm 1.5]). Assume that (2.3) holds and that

∃C, c > 0 such that M (µ) < Ce c|µ| , (resp. M (µ) < Ce c √ |µ| ) ∀µ ∈ R.
Then, for any k > 0 there exists C k such that

e tA (Id -A) -k L(H) ≤ C k log(2 + t) k , ∀t ≥ 0. (resp. e tA (Id -A) -k L(H) ≤ C k log(2 + t) 2k , ∀t ≥ 0.)
We shall use this result in Sections 3 and 5 to get the decay of solutions to (1.2) and (1.5).

2.2.

The resolvent operator and the Helmholtz equation. In this section we make explicit the choice of functional framework (compatible with Section 2.1) associated to the wave equation and the Schrödinger equations.

2.2.1.

The resolvent operator for waves. Following the notation of Section 2.1, let us set

(2.5) AU = 0 Id ∆ -a(x) , U = u v ,
in the Hilbert space H = H 1 (M ) × L 2 (M ) endowed with the natural inner product. As usual,

D(A) = H 2 (M )×H 1 (M ) if ∂M = ∅ or D(A) = (H 2 (M )∩H 1 0 (M ))×H 1 (M ) if
u v = (A -iτ ) -1 f g ⇔ v -iτ u = f, ∆u -av -iτ v = g. Using that v = iτ u + f , we find ∆u -a(iτ u + f ) + τ 2 u -iτ f = g, in M,
and hence u satisfies the Helmholtz equation

(2.7) ∆u + τ 2 u = g + (a + iτ )f + iaτ u, in M.

2.2.2.

The resolvent operator for Schrödinger. In this case we set H = L 2 (M ; C) and

(2.8)

A = i∆ -a(x), D(A) = H 2 (M ; C).
For ψ 0 ∈ H given, the solution of (1.5) is then given by the (2.9)

U (t)ψ 0 = e tA ψ 0 , t ≥ 0. Now, let τ ∈ R and consider the resolvent operator R(τ ) = (A -iτ ) -1 . If for some f ∈ H the function ψ ∈ D(A) is such that (A -iτ ) -1 ψ = f , then ψ satisfies the following Helmholtz equation: (2.10) ∆ψ -τ ψ + ia(x)ψ = -if, in M.

2.3.

Estimates for the Helmholtz equation. We state for convenience a unique continuation result for the Helmholtz equation that will be useful in Section 3.2. The unique continuation from small sets follows from the Remez inequalities obtained in [13, Section 1 Eq. ( 6) ].

Lemma 2.1. Let ω ⊂ M be a measurable set with vol g (ω) > 0. Let λ ∈ R be fixed and let u be the solution to the Helmholtz equation

(2.11) ∆u + λu = 0, in M.
Then, u satisfies the unique continuation principle on ω, i.e.,

(2.12)

u| ω = 0 ⇒ u| M = 0.
We state next an estimate for solutions to the Helmholtz equation with source that will be used in Section 3.1 to get a suitable exponential bound in the high frequency regime. The following result follows from the spectral inequality (1.13) in Theorem 3 above. Proposition 2.2. Let ω ⊂ M be a measurable set with vol g (ω) > 0. There exist constants C = C(ω) > 0 and D = D(ω) > 0 such that for every µ ∈ R and S ∈ L 2 (M ), the solution to the Helmholtz equation

(2.13) ∆u + µ 2 u = S, in M, u | ∂M = 0 (Dirichlet) or ∂ ν u | ∂M = 0 (Neumann) satisfies (2.14) u L 2 (M ) ≤ Ce D|µ| S L 2 (M ) + 1 ω u L 2 (M ) .
Proof. Let u be given by (2.13). If S ∈ L 2 (M ), using the orthonormal basis (e k ) as in Section 1.2, we can write

S = k∈N S k e k .
Then, we can split u into "hyperbolic" and "elliptic" frequencies as follows

u = 1 |∆+µ 2 |≤1 u + 1 |∆+µ 2 |>1 u,
where

1 |∆+µ 2 |>1 u = k∈N; µ 2 -k 2 >1 S k µ 2 -k 2 e k .
Thanks to this explicit expression, we have

(2.15) 1 |∆+µ 2 |>1 u L 2 (M ) ≤ S L 2 (M ) .
Now applying Theorem 3 on the "hyperbolic" frequencies, we get

1 |∆+µ 2 |≤1 u L 2 (M ) ≤ Ce D|µ| 1 |∆+µ 2 |≤1 u L 2 (ω) ≤ Ce D|µ| 1 |∆+µ 2 |≤1 u L 2 (ω) + 1 |∆+µ 2 |>1 u L 2 (ω) ≤ Ce D|µ| 1 |∆+µ 2 |≤1 u L 2 (ω) + S L 2 (M ) ≤ Ce D|µ| u L 2 (ω) + S L 2 (M ) ,
where we have used (2.15).

Proof of Theorem 1 for Dirichlet boundary conditions

Following the notation of Section 2.1, let τ ∈ R and consider the resolvent operator R(τ ) = (A -iτ ) -1 . Throughout this section we assume that the boundary conditions are of Dirichlet type only and use the notation of Section 2.2.1.

In Sections 3.1 we prove the resolvent estimate for the wave equations for τ ≥ τ * (with τ * a constant sufficiently large. Then in Section 3.2 we prove the estimate for τ ≤ τ * .

3.1. Proof of Theorem 1: High frequencies. Proposition 3.1. Let F ⊂ E ⊂ M and a damping a satisfying (1.7). Then there exist τ * ≥ 1 large enough and constants C h , c h > 0 independent of τ such that for every |τ | ≥ τ * we have

(3.1) U H 1 (M )×L 2 (M ) ≤ C h e c h |τ | (f, g) H 1 (M )×L 2 (M ) ,
for every h = (f, g) ∈ H and every

U = (u, v) = (A -iτ ) -1 h ∈ D(A).
Proof. First recall that for any

h = f g ∈ H, the element U = u v = (A -iτ ) -1 h

The first equation yields

v L 2 ≤ |τ | u L 2 + f L 2
and the second one, after multiplying by u and integrating, gives, for any |τ | ≥ 1,

∇u 2 L 2 ≤ M |((a + iτ )v + g) u| dx = M |((a + iτ )(f + iτ u) + g) u| dx (1 + β + τ 2 ) u L 2 ( f L 2 + g L 2 + u L 2 ) ,
where we have used that a ≤ β. We deduce

u H 1 ≤ C(1 + β + τ 2 ) ( f L 2 + g L 2 + u L 2 ) .
Hence, it is sufficient to estimate u L 2 to get an estimate on U H 1 (M )×L 2 (M ) . Let us focus on u L 2 . Recalling that u satisfies the Helmholtz equation (2.7), using Proposition 2.2 with

ω = F and S = g + (a + iτ )f + iaτ u, µ = τ, the estimate (2.14) yields u L 2 (M ) ≤ Ce D|τ | g + (a + iτ )f + iaτ u L 2 (M ) + 1 F u L 2 (M ) ≤ Ce D|τ | g L 2 (M ) + (|τ | + β) f L 2 (M ) + |τ | au L 2 (M ) + 1 F u L 2 (M ) ≤ (1 + |τ | + β)Ce D|τ | (f, g) H + Ce D|τ | |τ | au L 2 (M ) + 1 F u L 2 (M ) .
On the other hand, (1.7) implies

1 F u L 2 (M ) ≤ √ β α √ au L 2 (M ) ,
and

au L 2 (M ) ≤ β √ au L 2 (M ) .
As a result, we get

(3.2) u L 2 (M ) ≤ (1 + |τ | + β)Ce D|τ | (f, g) H + |τ | + 1 α βCe D|τ | √ au L 2 (M ) .
Next, we need to estimate √ au L 2 (M ) in terms of (f, g) H and τ . Using (2.7) we obtain

M (∆u + τ 2 u)u dx = M (g + (iτ + a)f + iaτ u) u dx and hence, - M |∇ x u| 2 dx + τ 2 M |u| 2 dx = M (g + (iτ + a)f ) u dx + iτ M a|u| 2 dx.
Taking the imaginary part, we find

|τ | √ au 2 L 2 (M ) = Im M (g + (iτ + a)f ) u dx ≤ 1 2 g + (iτ + a)f 2 L 2 (M ) + 2 u 2 L 2 (M ) ≤ 1 2 g 2 L 2 (M ) + 1 2 (|τ | + β 2 ) f 2 L 2 (M ) + 2 u 2 L 2 (M ) , for every > 0. Choosing = |τ |e -2D|τ | 2 |τ | + 1 α 2 βC 2 one finds √ au 2 L 2 (M ) ≤ 1 τ 2 |τ | + 1 α 2 βC 2 e 2D|τ | g 2 L 2 (M ) + (|τ | + β 2 ) f 2 L 2 (M ) + e -2D|τ | 4 |τ | + 1 α 2 βC 2 u 2 L 2 (M )
and thus,

√ au L 2 (M ) ≤ 1 + 1 α|τ | βC 1 + |τ | + β 2 e D|τ | (f, g) H + e -D|τ | 2 |τ | + 1 α √ βC u L 2 (M ) .
Now, we get from (3.2)

u L 2 (M ) ≤ (1 + |τ | + β)Ce D|τ | (f, g) H + |τ | + 1 α βCe D|τ | √ au L 2 (M ) ≤ (1 + |τ | + β)Ce D|τ | (f, g) H + 1 |τ | |τ | + 1 α 2 βC 2 e 2D|τ | 1 + |τ | + β 2 (f, g) H + 1 2 u L 2 (M ) .
Then, if τ * is large enough so that τ * ≥ 1, we have

1 2 u L 2 (M ) ≤ C (1 + |τ | + β) + e D|τ | Cβ(1 + |τ | + β 2 ) |τ | + 1 α 2 e D|τ | (f, g) H (3.3) ≤ C e 2D|τ | (f, g) H ,
for every |τ | ≥ τ * , where C is a positive constant depending only on α, β, C, D, τ * . Hence, estimate (3.1) follows for some constants C h , c h large enough, depending only on α, β, C, D, and τ * .

Proof of Theorem 1: Low frequencies (Dirichlet boundary conditions).

Proposition 3.2. Let F ⊂ E ⊂ M and a damping a satisfying (1.7). For any τ * > 0 (we shall choose τ * given by Proposition 3.1), there exist a constant C > 0 such that for any |τ | < τ * , every h = (f, g) ∈ H and every U such that U = (A -iτ ) -1 h.

(3.4) U H 1 (M )×L 2 (M ) ≤ C (f, g) H 1 (M )×L 2 (M ) .
Proof. In this case, we proceed by contradiction (we follow [START_REF] Burq | Exponential decay for the damped wave equation in unbounded domains[END_REF]Sect. 4]). Assume that the exponential growth is not true, i.e., that there exist sequences

(U n ) ⊂ H and (τ n ) ⊂ R with |τ n | ≤ τ * such that (3.5) U n H 1 (M )×L 2 (M ) = 1, ∀n ∈ N, and (A -iτ n )U n → 0 as n → +∞. Writing U n = u n v n , we have (3.6) v n -iτ n u n → 0, in H 1 (M ) and ∆u n -iτ n a(x)u n + τ 2 n u n → 0 in L 2 (
M ). Now multiplying the last limit by u n and integrating by parts we find

-∇u n 2 L 2 (M ) -iτ n M a(x)|u n | 2 dx + τ 2 n u n 2 L 2 (M ) → 0.
Taking real and imaginary parts yields

(3.7) -∇u n 2 L 2 (M ) + τ 2 n u n 2 L 2 (M ) → 0, and τ n M a(x)|u n | 2 dx → 0.
The sequence (τ n ) is bounded (in modulus) by τ * and consequently we can assume that it converges to some limit τ . We distinguish now two cases.

Case τ = 0. In this case, we would have

∇u n 2
L 2 (M ) → 0, thanks to (3.7). Hence, by Poincaré's inequality, we would also have

u n 2 L 2 (M ) → 0.
But then, the first part of (3.6) would also imply

v n 2 L 2 (M ) → 0. Henceforth, U n 2
H 1 (M ) → 0, which is a contradiction with (3.5).

Case τ = 0. In this case, using (3.7) we may write

lim n→∞ ∇u n 2 L 2 (M ) = lim n→∞ τ 2 u n 2 L 2 (M ) , and lim n→∞ M a(x)|u n | 2 (x) dx = 0.
Using (3.5) we also have lim

n→∞ v n 2 L 2 (M ) = lim n→∞ τ 2 u n 2 L 2 (M ) . Then, as (3.8) 1 = lim n→∞ U n 2 H 1 (M )×L 2 (M ) = lim n→∞ (1 + 2τ 2 ) u n 2 L 2 (M ) ,
which means that the sequence (u n ) is bounded in H 1 . Then, Rellich's compactness theorem implies that there exists u ∈ H 1 (M ) such that

u n → u in L 2 (M ) -strong, and
∇u n → ∇u in L 2 (M ) -weak,
and a fortiori, upon extracting a subsequence, we also have

u n → u a.e. M.
Now, thanks to Fatou's lemma and the second part of (3.7) we deduce

M a(x)|u| 2 dx ≤ lim inf n→∞ M a(x)|u n | 2 dx = 0
and hence, using (1.7), u = 0 a.e. in F.

This is enough to apply Lemma 2.1 which yields u = 0 a.e. in M.

But this is a contradiction with (3.8). This concludes the proof.

End of the proof of Theorem 1 (Dirichlet boundary conditions).

Once we have dealt with high and low frequencies in the previous sections, the proof of Theorem 1 is a consequence of Theorem 4.

Proof of Theorem 1. Combining Proposition 3.1 and Proposition 3.2, we find that the estimate

U H 1 ×L 2 ≤ Ce 2c|τ | h H 1 ×L 2 holds for every τ ∈ R, every h ∈ H 1 (M ) × L 2 (M ), U = R(τ )h and the constants C = C + C h , c = c .
As a consequence the function M defined by (2.4) satisfies in this case the growth

M (µ) ≤ Ce 2c|µ| ,
for every µ ∈ R. Hence, Theorem 4 yields that for any k ∈ N,

(3.9) U (t)(Id -A) -k L(H 1 ×L 2 ) ≤ C k log(2 + t) k , ∀t ≥ 0.
where we have used the notation of (2.5) and (2.6). Next, let k ∈ N be fixed and let

h 0 = u 0 u 1 ∈ H k+1 (M ) × H k (M ) Then, W 0 := (Id -A) k h 0 ∈ L 2 (M ) and inequality (3.9) implies E w (t, u 0 , u 1 ) = U (t)h 0 2 H 1 ×L 2 = U (t)(Id -A) -k h 0 2 H 1 ×L 2 ≤ C k log(2 + t) 2k W 0 2 H 1 ×L 2 ≤ C k log(2 + t) 2k (Id -A) k h 0 2 H 1 ×L 2 , for every t ≥ 0. The proof is completed by noticing that for k = 1 the domain of A is (H 2 (M ) ∩ H 1 0 (M )) × H 1 0 (M ) with Dirichlet boundary conditions (resp. H 2 (M ) × H 1 (M ) if ∂M = ∅), and
(Id -A)h 0 2 H 1 ×L 2 ≤ C h 0 2 H 2 ×H 1 .

Proof of Theorem 1: the case of Neumann boundary conditions

In the case τ = 0, we cannot use Poincaré's inequality (that we used when dealing with the low frequencies in Proposition 3.2) and we have to change slightly the functional framework. Here we follow the exposition in [START_REF] Burq | Stabilization of wave equations on the torus with rough dampings[END_REF]Appendix]. For the sake of completeness, we recall the argument (which is taken from [START_REF] Burq | Stabilization of wave equations on the torus with rough dampings[END_REF]Appendix]) below and focus on the low frequency regime |τ | ≤ τ * .

For s = 1, 2, we define Ḣs = H s (M )/R the quotient space of H s (M ) by the constant functions, endowed with the norm

u Ḣ1 = ∇u L 2 , u Ḣ2 = ∆u L 2 ,
(here u denotes the equivalence class of a function u in Ḣ). We define the operator

Ȧ = 0 Π ∆ -a on Ḣ1 × L 2 with domain Ḣ2 × H 1
, where Π is the canonical projection H 1 → Ḣ1 and ∆ is defined by ∆ u = (∆u) (independent of the choice of u ∈ u). The operator Ȧ is maximal dissipative and hence defines a semi-group of contractions on Ḣ = Ḣ1 × L 2 . Indeed for

U = u v , Re ȦU, U Ḣ = Re(∇u, ∇v) L 2 + ( ∆ u -av, v) L 2 = -(av, v) L 2 ,
and

(4.1) ( Ȧ -Id) u v = ḟ g ⇔ Πv -u = ḟ , and ∆ u -(a + 1)v = g ⇔ Πv -u = ḟ , and ∆v -(1 + a)v = g + ∆f ∈ H -1 (M )
and we can solve this equation by variational theory. Notice that this shows that the resolvent ( Ȧ -Id) -1 is well defined and continuous from Ḣ1 × L 2 to Ḣ2 × H 1 . We have further:

Lemma 4.1. The injection Ḣ2 × H 1 to Ḣ1 × L 2 is compact.
This follows from identifying Ḣs with the kernel of the linear form u → M u. We also have:

Corollary 4.2. The operator ( Ȧ -Id) -1 is compact on Ḣ.

On the other hand, it is very easy to show that for (u 0 , u 1 ) ∈ H 1 × L 2 , Π 0 0 Id e tA = e t Ȧ Π 0 0 Id , and hence, the logarithmic decay is equivalent to the logarithmic decay (in norm) of e t Ȧ (and consequently, according to Theorem 4 equivalent to resolvent estimates for Ȧ). The high frequency resolvent estimates in our new setting are handled with the exact same proof as for Dirichlet boundary conditions (we did not use Poincaré's inequality in this regime) and consequently we omit the proof. Let us focus on the low frequency regime and revisit our proof above in this new functional setting. We prove Proposition 4.3. Let F ⊂ E ⊂ M and a damping a satisfying (1.7). For any τ * > 0 (we shall choose τ * given by Proposition 3.1), there exist constants C > 0 such that for any |τ | < τ * . and every h = (f, g) ∈ H every U such that U = ( Ȧ -iτ ) -1 h.

(4.2) U Ḣ1 (M )×L 2 (M ) ≤ C (f, g) Ḣ1 (M )×L 2 (M ) ,
Proof. We argue by contradiction. Suppose there exist sequences

(τ n ), (U n ), (F n ) such that ( Ȧ -iτ n )U n = F n , U n Ḣ > n F n Ḣ.
Since U n = 0, we can assume U n Ḣ = 1. Extracting subsequences (still indexed by n for simplicity) we can also assume that τ n → τ ∈ R as n → ∞. We write

U n = un v n , F n = ḟn g n ,
and distinguish according to two cases.

5.1. Estimates when τ < 0. In this case, we may use Proposition 2.2 directly. We get the following result.

Proposition 5.2. Assume that (1.7) holds for some α, β and let f ∈ L 2 (M ; C) be given. Then, for any τ < 0, the resolvent (∆ -τ + ia) -1 satisfies

(∆ -τ + ia) -1 f L 2 (M ) ≤ 2(1 + C) 2 1 + β(1 + 1 α ) 2 e 2D √ |τ | f L 2 ,
where C, D are the constants in (2.14).

Proof. Recall that ψ = (∆ -τ + ia) -1 f satisfies the Helmholtz equation (2.10). Then, thanks to (2.14) there exist some constants C, D > 0 independent of τ such that

ψ L 2 (M ) ≤ Ce D √ |τ | i(f + aψ) L 2 (M ) + 1 ω ψ L 2 (M ) .
On the other hand, (1.7) implies

1 F ψ L 2 (M ) ≤ √ β α √ aψ L 2 (M ) and aψ L 2 (M ) ≤ β √ aψ L 2 (M ) . As a result, we get (5.1) ψ L 2 (M ) ≤ Ce D √ |τ | f L 2 + 1 + 1 α β √ aψ L 2 (M ) .
Next, using the Helmholtz equation (2.10) we obtain

M (∆ψ -τ ψ)ψ dx = M (-if -iaψ) ψ dx and hence, M |∇ x ψ| 2 dx + τ M |ψ| 2 dx = i M f ψ dx + i M a|ψ| 2 dx.
Now, taking the imaginary part and using Cauchy-Schwarz's and Young's inequalities we find

√ aψ 2 L 2 (M ) = Im M f ψ dx ≤ f L 2 (M ) ψ L 2 (M ) ≤ 1 4 f L 2 (M ) + ψ L 2 (M ) ,
for every > 0. Injecting this in (5.1) yields

ψ L 2 (M ) ≤ Ce D √ |τ | 1 + β(1 + 1 α ) 1 2 √ f L 2 + Ce D √ |τ | β (1 + 1 α ) ψ L 2 . Next, choosing = e -2D √ |τ | 4 1 + 1 α 2 βC 2 we get ψ L 2 (M ) ≤ 2Ce D √ |τ | 1 + β(1 + 1 α ) 2 Ce D √ |τ | f L 2 .
As a consequence, we get the exponential growth estimate

ψ L 2 (M ) ≤ 2(1 + C) 2 1 + β(1 + 1 α ) 2 e 2D √ |τ | f L 2 ,
for any τ > 0 given.

5.2.

Estimates when 0 ≤ τ . In this section we shall just rely on the following Poincaré-type inequality:

Proposition 5.3. Assume that a ≥ 0 and M a(x)dx > 0. Then there exists C P = C P (a) > 0 such that for all u ∈ H 1 (M ),

(5.2)

C P M (|∇ x u| 2 (x) + a(x)|u| 2 (x))dx ≥ u 2 H 1 (M ) .
Proof. We follow a standard proof and argue by contradiction. Otherwise, there would exist a sequence (u n ) ∈ H 1 (M ) (that we can assume of norm 1 in H 1 ) such that

M (|∇ x u n | 2 (x) + a(x)|u n | 2 (x))dx ≤ 1 n u n 2 H 1 (M ) .
Since (u n ) is bounded in H 1 (M ) and M is compact, by Rellich's compactness theorem there exists u ∈ H 1 such that we can extract a subsequence (still denoted by (u n )) such that u n -u L 2 → 0, as n → +∞.

Moreover, as ∇ x u n L 2 → 0, √ au n L 2 (M ) → 0, we deduce that ∇ x u = 0 and thus u must be constant in M . But

u n H 1 = 1, ∇ x u n L 2 → 0, u n -u L 2 → 0 ⇒ u L 2 = 1.
Hence, (5.2) follows for some positive constant C P .

Proposition 5.4. Assume that (1.7) holds for some α, β and let f ∈ L 2 (M ; C) be given. Then, there exists C > 0 such that for τ ≥ 0,and any f ∈ L 2 (M ), we have

(∆ -τ + ia(x)) -1 f L 2 (M ) ≤ 2C P f L 2 (M ) ,
where C P is the Poincaré's constant above.

Proof. For f ∈ L 2 (M ) given, let ψ = (A -τ + ia) -1 f . Recalling that ψ satisfies the Helmholtz equation (2.10), after multiplying by ψ and integrating by parts, we get where we have used the notation of (2.8) and (2.9). Next, let k ∈ N be fixed and let ψ 0 ∈ H 2k (M ) Then, Φ 0 := (Id -A) k ψ 0 ∈ L 2 (M ) and inequality (5.4) implies

ψ 2 H 1 ≤ 2C P f L 2 ψ L 2 ⇒ ψ H 1 ≤ 2C P f L 2 .
E S (t, ψ 0 ) = U (t)ψ 0 2 L 2 = U (t)(Id -A) -k Φ 0 2 L 2 ≤ C k log(2 + t) 4k Φ 0 2 L 2 ≤ C k log(2 + t) 4k (Id -A) k ψ 0 2

(1. 3 )

 3 u | ∂M = 0 (Dirichlet condition) or ∂ ν u | ∂M = 0 (Neumann condition).

  satisfies the Helmholtz equation (2.7) with the boundary conditions u | ∂M = 0 and the identities v -iτ u = f and ∆u -av -iτ v = g, in M.

|u| 2 M

 2 a(x)dx = M a(x)|u| 2 (x)dx = lim n→+∞ M a(x)|u n | 2 (x)dx,which implies that u = 0. This gives a contradiction with the fact that

2 M

 2 |ψ| 2 dx = -i M f ψ dx. The modulus of the left hand side in (5.3) is larger that 1 (|∇ψ| 2 (x) + a(x)|ψ| 2 (x))dx. Using Poincaré's inequality (5.2) in the left and Cauchy-Schwarz in the right we get

5. 3 . 1 α ) 2 ,

 312 Conclusion of the proof of Theorem 2.Proof of Theorem 2. Combining Proposition 5.4 and Proposition 5.2, we find that the estimateR(τ )f L 2 (M ) ≤ C 0 e 2D √ |τ | f L 2holds for every τ ∈ R, every f ∈ L 2 (M ) and the constantC 0 := max 2(1 + C) 2 1 + β(1 + 2C P .As a consequence the function M defined by (2.4) satisfies in this case the growthM (µ) ≤ C 1 e 2D √ |µ| ,for every µ ∈ R. Hence, Theorem 4 yields that for any k ∈ N,(5.4) U (t)(Id -A) -k L(L 2 ) ≤ C k log(2 + t) 2k , ∀t ≥ 0.

L 2 , 2 L 2 ≤ C ψ 0 2 H 2 .

 22222 for every t ≥ 0. The proof is completed by noticing that for k = 1 the domain of A is H 2 (M ) ∩ H 1 0 (M ) with Dirichlet boundary conditions (resp. H 2 (M ) with Neumann boundary conditions or if ∂M = ∅) and (Id -A)ψ 0

  we impose Dirichlet boundary conditions (the case of Neumann boundary conditions is slightly more involved and we deal with it in Section 4). The solution of (2.1) is given by

	(2.6)	U (t) = e tA u 0 u 1

and the solution of (1.2) is given by the first component of U (t). Let τ ∈ R and consider the resolvent operator R(τ ) = (A -iτ ) -1 . For any f g ∈ H, one has

Acknoledgments N. Burq is partially supported by the grant "ISDEEC" ANR-16-CE40-0013 and Institut Universitaire de France. I. Moyano is very grateful to AIMS Ghana, where part of this work was done during a visit in August 2021.

Zero frequency case: τ = 0. In this case, we have

We deduce that there exists (c n ) ⊂ C such that

As a consequence, we get v n = o(1) L 2 and ∆u n = o(1) L 2 , which implies that un = o(1) Ḣ1 . This contradicts U n Ḣ = 1. As a result (4.2) follows for τ = 0.

Low (nonzero) frequency case: τ ∈ R * . In this case, we have

We deduce

Extracting another subsequence, we can assume that

Taking the imaginary part of the scalar product with v in L 2 gives (since τ = 0) M a|v| 2 dx = 0, and consequently av = 0 which implies that v is an eigenfunction of the Laplace operator. But since the zero set of non trivial eigenfunctions has Lebesgue measure 0 in M , av = 0 implies that v = 0 (and consequently v n = o(1) L 2 ). Now, we have

but this contradicts U n Ḣ = 1 and (4.2) follows also in this case. This ends the proof.

Once we have established Proposition 4.3, the proof of Theorem 1 with Neumann boundary conditions follows the same lines of Section 3.3 without significant modifications. This ends the proof of Theorem 1.

Proof of Theorem 2: Schrödinger equation

In order to prove Theorem 2 it is enough to prove the following resolvent estimates.