The two and three-dimensional cases

In dimension higher than one, our hypotheses are less easy to verify. First, Hypotheses (ED) and (Grad) do not always hold. The hypothesis (ED) is equivalent to geometrical conditions on the support of γn, which are now well-understood. The case of Hypothesis (UED) is much more difficult and its study in dimension two or higher is still mostly open.

6.2.1 Hypothesis (ED)

It is now well-known that the following geometric condition is equivalent to (ED), see [6].

For eachn∈ N∪ {+∞}, there is a lengthLnsuch that all geodesics on Ω associated to the operator ∂_tt² +B and of length greater than Ln meet the support ofγn. In dimension one, the condition is trivially satisfied. In the higher dimensional case, the condition is more restrictive, since, for some examples, there exist geodesics of infinite length, which do not meet the support of γn.

γn

(ED) not satisfied (ED) satisfied

support of

6.2.2 Hypothesis (Grad)

Letn∈N∪ {+∞}be given. Let U0 ∈X be such that for all t≥0, we have Φ(Sn(t)U0) = Φ(U0). If U(t) =Sn(t)U0 = (u(t), v(t)), we thus have

ut(t) =v(t) and utt+B(u+ Γnut) = f(x, u).

We know that

∀t≥0, ∂

∂tΦ(U(t)) =< AnU|U >=− Z

Ω

γn|v|² = 0 . Hence, v =ut satisfies

∀t ≥0, v_tt+Bv=f_u^′(x, u)v and







v = 0 on supp(γn) if n∈N or

v = 0 and ^∂v_∂ν = 0 on supp(γ_∞)

. (6.15) To prove that Hypothesis (Grad) holds, we must show that (6.15) implies that ut=v = 0 on Ω×R₊. This unique continuation argument holds under geometrical conditions.

• If the support of γn contains a neighborhood of the boundary ∂Ω for n∈N and if the support of γ_∞ is equal to∂Ω, then (Grad) is satisfied (see [42]).

• Assume that supp(γ_∞) =ω_N, that the support of γ_n contains a neighborhood ofω_N and that there exists a point x0 ∈R^d such that

{x∈∂Ω/ (x−x0).ν >0} ⊂ωN , then (Grad) holds (see [28]).

x0 Dirichlet B.C.

• Let Ω be a domain with a boundary of class C¹. We assume that the support of γn

includes a neighborhood of supp(γ_∞) and that the boundary conditions on the whole boundary∂Ω are of Neumann type, that is thatωD =∅. In this case, [33] gives many

sufficient conditions for (Grad) to hold. In particular, if Ω is a disk, the fact that the support ofγ_∞ covers slightly more than a half circle is sufficient. Other examples are given.

less than a quarter turn sine graph

straight line

Neumann B.C.

Remark : For all the examples that we give here, one notices that (ED) is satisfied.

However, there is no reason that (Grad) implies (ED) in general.

6.2.3 Hypothesis (UED)

The methods, which were used in the one-dimensional case, cannot be generalized to dimensions two or three. For these dimensions, using an energy method, we obtain here a criterium equivalent to the property (UED). However, except for the particular cases where γn is uniformly bounded away from 0, it is very difficult to exhibit examples satisfying this criterium.

The following equivalence is very classical. The property (UED) is satisfied if and only if there exist two positive constants T and C such that, for all U0 ∈X and n ∈N, if we set U(t) = (u, ut)(t) =e^AntU0, then we have

Z T 0

Z

Ω

γn|ut|² ≥CkU0k²X . (6.16) We can weaken this criterium as follows.

Proposition 6.7. The uniform exponential decay property (UED) is satisfied if and only if there exist two positive constants T and C, independent of n, such that, for all U0 ∈X and n∈N, if we set U(t) = (u, u_t)(t) =e^AntU₀, then we have

Z T 0

Z

Ω

γ_n|u_t|² ≥C Z T

0

Z

Ω|u_t|² . (6.17)

Proof : The “only if” part is a direct consequence of the classical criterium (6.16). Indeed, the property (UED) implies that there exist two positive constants T and C such that, for allU0 ∈X and n ∈N,

Z T 0

Z

Ω

γ_n|u_t|² ≥CkU₀k²X = C T

Z T

0 kU₀k²Xdt≥ C T

Z T

0 kU(t)k²Xdt≥C Z T

0

Z

Ω|u_t|² .

In order to prove the “if” part of the equivalence, we introduce the following functional.

Let α be a positive number to be chosen. For allU = (u, v)∈X, we set F(U) = 1

2 Z

Ω

(|u|²+ 2uv+α|v|²+α|B^1/2u|²)dx . (6.18) For α large enough, the functional F is clearly equivalent to the energy in the sense that there exists a positive constant µsuch that

∀U ∈X, 1

µkUk²X ≤F(U)≤µkUk²X . (6.19) Let U0 ∈ D(An) and n∈N, we set U(t) = (u, ut)(t) = e^AntU0. As U(t) ∈ C¹(R₊, X), we can write

∂tF(U(t)) = Z

Ω

(uut+uutt+|ut|²+αututt+α(B^1/2u)(B^1/2ut))dx

= Z

Ω

(uut−γnuut−(Bu)u+|ut|²−αγn|ut|²

−αu_t(Bu) +α(B^1/2u)(B^1/2u_t))dx . Thus, for all ε >0, we have that

∂tF(U(t))≤ Z

Ω

ε(1 +γn)|u|²+ 1

ε(1 +γn)|ut|² − |B^1/2u|²+|ut|²−αγn|ut|² .

As Γn converges to Γ_∞ in L(D(B^1/2)), we know that there exists a positive constant C, independent of n, such that, for all u∈D(B^1/2), R

Ωγ_n|u|² ≤Ckuk²_D(B^1/2₎. Therefore, forε small enough and α large enough, (6.17) implies the existence of a time T and a positive constant C such that

Z T 0

∂tF(U(t))≤ −C Z T

0

F(t) .

Thus, using the density of D(An) in X, we obtain that, for allU0 ∈X, F(U₀)−F(e^AnTU₀)≥C

Z T 0

F(e^AntU₀)dt .

The inequalities (6.19) and the fact that e^Ant is a contraction imply that, for all U0 ∈X and k ∈N,

kU0k²X ≥ C µ²

Z kT

0 ke^AntU0k²Xdt≥ C

µ²kTke^AnkTU0k²X .

Forklarge enough, we obtain a timeT^′, independent ofn, such thatke^AnT′U0k^X ≤ ¹₂kU0k²X. This is well-known to imply the uniform exponential decay (UED).

It is difficult to find examples satisfying the criterium (6.17). Indeed, opposite to the cri-terium (EFW) obtained in the one-dimensional case, (6.17) involves functions U(t), which are solutions of an equation which depends ofn. However, Proposition 6.7 gives some very particular examples where (UED) is satisfied in dimension higher than one. This corollary is stated in a way so that it can be applied to a general sequence of dissipations, which satisfies (6.20) only.

Corollary 6.8. Let (γ_n⁰)n∈N be a sequence of non-negative functions in L^∞(Ω). Assume that there exists a positive constant C independent of n such that

∀u ∈D(B^1/2), ∀n ∈N, Z

Ω

γ_n⁰(x)|u(x)|²dx≤Ckuk²D(B^1/2) . (6.20) Then, for all η > 0, the uniform exponential decay property (UED) is satisfied for the sequence of dissipationsγn=η+γ_n⁰.

Notice that this result in not a priori trivial, since, as the sequence (γ_n⁰) is not neces-sary bounded in L^∞(Ω), overdamping phenomenas may occur. The fact thatγn ≥η > 0 seems slightly artificial from a mathematical point of view. However, it is not from the physical point of view since γn never really vanishes in the concrete cases. For example, ηcan be seen as the resistance of air when (2.5) models the propagation of waves in a room.

7 Examples

In this section, we give some examples illustrating our results. For each example, we define Ω, B and γ_n and say if the convergence of the attractors holds in the space X or only in X^−s (s > 0). Saying that the convergence holds in X^−s does not mean that there is no convergence in X. It only means that we are not able to prove it for the moment. Here we do not give explicit non-linearities for which Hypothesis (Hyp) is satisfied.

We recall that we denote by ∆N the Laplacian with Neumann boundary conditions.

Example 1 :

0 1/n 1

αn Ω =]0,1[, B =Id−∆N, α >0

γ_n(x) =

αn, x∈]0,_n¹[

0, elsewhere , γ_∞=αδ_x=0 . Convergence in X.

Example 2 :

α 1/n

1/n1/2

n

0 1

Ω =]0,1[, B =Id−∆N, α >0 γn(x) =

αn, x∈]^√1_n,^√1_n+ ¹_n[

0, elsewhere , γ_∞=αδx=0 . Convergence in X^−s.

Example 3 :

Ω is the disk of R², B =Id−∆_N, ω is an open subset of ∂Ω which covers strictly more than half of the circle γ_n(x) =

n, dist(x, ω)< _n¹

0, elsewhere , γ_∞=δ_x∈ω . Convergence in X^−s.

Example 4 : Ω⊂R², η >0,ω is any subset of ∂Ω B =Id−∆N

γ_n(x) =

n, dist(x, ω)< ¹_n

η, elsewhere , γ_∞=η+δ_x∈ω . Convergence inX.

Example 5 : For sake of simplicity, the abstract frame of this paper has not been defined so that this example fits in it. However, all the results given here are valid for this case.

Notice that we need the additional dissipation g, since the singular internal dissipation δx=a is not sufficient to obtain exponential decay (see [25]), or the gradient structure. We denote by ∆D the Laplacian with Dirichlet boundary conditions.

g(x)