Compactness of trajectories to some nonlinear secondorder evolution equations and applications.

Faouzia ALOUIUPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions,

F-75005, Paris, France.aloui@ann.jussieu.fr

Imen BEN HASSENInstitut preparatoire aux etudes d’ingenieur de Bizerte

BP 64, 7021 Jarzouna, Bizerte, Tunisia.imen benhassen@yahoo.fr

Alain HARAUX (1, 2)1. UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions,

F-75005, Paris, France.2- CNRS, UMR 7598, Laboratoire Jacques-Louis Lions,

Boıte courrier 187, 75252 Paris Cedex 05, France.haraux@ann.jussieu.fr

Abstract

Under suitable growth and coercivity conditions on the nonlinear damping operatorg, we establish boundedness or compactness properties of trajectories to the equation

u(t) + g(u(t)) + Au(t) = h(t), t ∈ R+,

where A is a positive selfadjoint operator and g is a nonlinear damping operator. Thecompactness results are used to prove the existence of almost periodic solutions whenh is almost periodic, and to generalize some recent results of Chergui and Ben Hassen-Chergui concerning convergence to equilibrium when a nonlinear term depending on uis added and h dies off sufficiently fast for t large.

AMS classification numbers: 34A34, 34D20, 35B40, 35L10, 35L90

Keywords: Second order equation, compactness, almost periodic , Lojasiewicz inequality.

1

1 Introduction.

In this paper we investigate the asymptotic behavior of solutions to the equation

u(t) + g(u(t)) + Au(t) + f(u(t)) = h(t), t ∈ R+, (1.1)

where V is a real hilbert space, A ∈ L(V, V ′) is a symmetric, positive, coercive operator ,g ∈ C(V, V ′) is monotone, f is a gradient operator satisfying some appropriate conditionsand h is a forcing term. We are especially interested in the two following cases

1) f = 0 and h is almost periodic.

2) h tends to 0 sufficiently fast at infinity in t and f is the gradient of an analyticfunctional or more generally the gradient of a potential satisfying the Lojasiewicz gradientin equality in a sense which will be specified later.

Case 1) has been intensively studied in the Literature, covering the following topics:existence of almost periodic solutions, asymptotic behavior of the general solution, rate ofdecay to 0 of the difference of two solutions in the energy space in the best cases, cf.e.g.[1,5, 6, 12, 15, 14, 24, 19, 25]. Until now, although boundedness of trajectories for t ≥ 0is sufficient in the linear case and when h is time-periodic, the treatment of the generalcase has always required the existence of a precompact trajectory, and the problem is thatit is difficult to distinguish between different trajectories at this level: if we were able toexhibit a precompact orbit without knowing anything about the others, it would mean thatwe can (by going to the positive ω-limit set) localize an almost periodic orbit and thisis precisely what becomes impossible in the nonlinear case. Therefore we are condemnedto prove compactness of all trajectories or nothing (note that for g tangent to 0 at theorigine, extra regularity of the initial state or even of the forcing term will not help when tbecomes large). Another difficulty is the following: if the existence of bounded trajectoriescan be proved by combining a coerciveness property of g “at infinity” with some growthrestrictions, compactness requires a global, uniform kind of coerciveness. In the past moreand more general compactness results have been obtained, but only when g is a Nemytskiitype operator. When g is a non-local operator or involves differential operators in space, thetheory remained to be done: this is the main object of the present paper.

For case 2), compactness is the vital starting point. In the past several significativeadvances have been done, cf.e.g. [21, 16, 17, 18, 20, 11, 9, 10, 2, 3, 4], the other tool herebeing the Lojasiewicz gradient inequality [22, 23]. But here even the case h = 0 is non-trivialsince the set of equilibria needs not have any particular structure except for the restrictionsinduced by the existence of a Lojasiewicz inequality : we know for instance in advance thatthe potential energy is constant on continua inside the set of equilibria, a property which canfail for C∞ and even Gevrey potentials. The fact that precompactness of trajectories hadbeen proved only for Nemytskii type damping operators limited until now the convergenceresults with non-linear damping to those damping operators. Therefore the second innovationof this paper is to contain the first convergence results in case 2) in presence of a non-localdamping term.

2

The plan of the paper is as follows: In Section 2, we introduce the basic tools used inthe statements and proofs of the main results. Section 3 is devoted to the initial valueproblem for (1.1). Sections 4 and 5 contain the statement and proof of the boundednessresult and the compactness result, respectively. Sections 6 and 7 contain respectively thestatement and proof of the asymptotic almost periodicity and the semilinear convergenceresult, respectively. Finally Section 8 is devoted to the application to PDE models withnon-local damping terms.

2 Some useful tools.

In this section, we collect quite a few results of general interest which will reveal essentialfor the proofs of our main results. We also need to recall the definitions of some well knownmathematical objects as well as their basic properties in the exact functional framework thatshall be used in the main sections containing our new results.

2.1 Monotonicity theory

Let H be a real Hilbert space endowed with an inner product (., .)H. We recall that a mapA defined on a part D = D(A) with values in H is monotone if

∀(U, U) ∈ D ×D, (AU −AU , U − U)H ≥ 0.

In addition A is called maximal monotone if

∀F ∈ H, ∃U ∈ D(A) AU + U = F .

The following result is well-known (cf. H. Brezis [7]) .

Proposition 2.1. if A is maximal monotone, for each T > 0, each U0 ∈ D(A) and F =F (t) ∈ W 1,1(0, T ;H) there is a unique function U ∈ W 1,1(0, T ;H) with U(t) ∈ D(A)) foralmost all t ∈ (0, T ) , U(0) = U0 and such that for almost all t ∈ (0, T )

U ′(t) +AU(t) = F (t). (2.1)

In addition if for some U0 ∈ D(A) and H ∈ W 1,1(0, T ;H) we consider the solution U ∈W 1,1(0, T ;H) with U(t) ∈ D(A)) for almost all t ∈ (0, T ) , U(0) = U0 of

U ′(t) +AU(t) = F (t)

then the difference satisfies the inequality

∀t ∈ [0, T ], |U(t)− U(t)| ≤ |U0 − U0|+∫ t

0

|F (s)− F (s)|ds

This proposition allows to define by density, for any U0 ∈ D(A) and F = F (t) ∈L1(0, T ;H) a weak solution of (2.1) such that U(0) = U0, cf. H. Brezis [7].

3

2.2 A class of nonlinear operators

In the applications to non-local dissipations we shall use the following simple inequalities.Let X be any Hilbert space with norm denoted by |.| and inner product by 〈, 〉. Then forany α > 0 we have

Lemma 2.2.

∀(v, w) ∈ X ×X, ||v|αv − |w|αw| ≤ (α + 1) max|v|, |w|α|v − w|. (2.2)

Proof. The inequality is trivial if v or w vanishes. Assuming |v| ≥ |w| > 0, we can write

||v|αv − |w|αw| ≤ (|v|α − |w|α)|w|+ |v|α|v − w|

Then we have 2 cases:

- If α ≥ 1, then

|v|α − |w|α ≤ α|v|α−1(|v| − |w|) ≤ α|v|α−1|v − w|

and then(|v|α − |w|α)|w| ≤ α|v|α−1|w|||v − w| ≤ α|v|α|v − w|

hence||v|αv − |w|αw| ≤ (α + 1)|v|α|v − w|

and the result is proved.

- If α < 1, then

|v|α − |w|α ≤ α|w|α−1(|v| − |w|) ≤ α|w|α−1|v − w|

and then

(|v|α − |w|α)|w| ≤ α|w|α−1|w|||v − w| = α|w|α|v − w| ≤ α|v|α|v − w|

leading to the same conclusion.

Lemma 2.3.

∀(v, w) ∈ X ×X, 〈|v|αv − |w|αw, v − w〉 ≥ 2−α|v − w|α+2 (2.3)

with equality if and only if w = −v.

4

Proof. We set u = −w and P = 〈|v|αv − |w|αw, v − w〉 = 〈|u|αu + |v|αv, u + v〉 so that weare left to prove P ≥ 2−α|u+ v|α+2 with equality if and only if u = v. We expand

P = |u|α+2 + |v|α+2 + (|u|α + |v|α)〈u, v〉

= |u|α+2 + |v|α+2 +1

2(|u|α + |v|α)(|u+ v|2 − |u|2 − |v|2)

=1

2(|u|α + |v|α)|u+ v|2 +

1

2(|u|α − |v|α)(|u|2 − |v|2).

If u+ v = 0 there is nothing to prove, otherwise

2P

|u+ v|α+2=|u|α + |v|α

|u+ v|α+

(|u|α − |v|α)(|u|2 − |v|2)

|u+ v|α+2

hence2P

|u+ v|α+2≥ |u|

α + |v|α

(|u|+ |v|)α+

(|u|α − |v|α)(|u|2 − |v|2)

(|u|+ |v|)α+2

with equality if and only if u and v are proportional with nonnegative ratio. But we have

|u|α + |v|α

(|u|+ |v|)α+

(|u|α − |v|α)(|u|2 − |v|2)

(|u|+ |v|)α+2

=(|u|α + |v|α)(|u|+ |v|)2) + (|u|α − |v|α)(|u|2 − |v|2)

(|u|+ |v|)α+2

=(|u|α + |v|α)(|u|+ |v|)) + (|u|α − |v|α)(|u| − |v|)

(|u|+ |v|)α+1= 2|u|α+1 + |v|α+1

(|u|+ |v|)α+1.

Assuming for instance |u| ≥ |v| (in particular u 6= 0) and setting t = |v||u| ≤ 1, we therefore

obtainP

|u+ v|α+2≥ |u|

α+1 + |v|α+1

(|u|+ |v|)α+1=

1 + tα+1

(1 + t)α+1

with equality if and only if u = tv . To conclude it is sufficient to observe that f(t) = 1+tα+1

(1+t)α+1

is decreasing on (0, 1) with f(1) = 2−α. Indeed setting h(t) = − ln f(t) we have

∀t ∈ (0, 1), h′(t) =(α + 1)(1− tα)

(1 + t)(1 + tα+1)> 0

5

2.3 Almost periodic functions

A typical almost periodic numerical function is the sum of two periodic functions withincommensurable periods. Such objects often appear in the mechanics of vibrating systems,and sometimes infinite sums naturally impose their presence, for instance when studying theenergy conservative vibrations of continuous media .

There are several equivalent definitions of almost periodicity, but in the theory of differ-ential equations, the most convenient criterion is Bochner’s functional definition:

Definition 2.4. Given a complete metric space X , a function f : R→ X is almost periodiciff the set of translates ⋃

α∈R

f(.+ α = T (f)

is precompact in the space Cb(R, X) endowed with the topology of uniform convergenceR→ X.

It follows clearly from the definition that

i) For any T > 0, any continuous T-periodic function: f : R→ X is almost periodic withT (f) compact.

ii) For 3 complete metric spaces X, Y, Z, if f : R → X and g : R → Y are- almostperiodic and C : X × Y → Z is uniformly continuous, the function h(t) = C(f(t), g(t))is almost periodic R → Z In particular if X is a Banach space, any finite sum of almostperiodic functions: R→ X is almost periodic: R→ X.

iii) Any uniform limit of almost periodic functions: R→ X is almost periodic: R→ X.

iv) Any almost periodic function: R → X is uniformly continuous with precompactrange.

In the theory of abstract differential equations an important problem is the following:given a nonlinear operator A : D → X and an exterior almost periodic force F : R → Xwhen can we garantee that the ‘response ”, i.e. the general solution of the evolution equation

U ′(t) +AU(t) = F (t)

adapts to F in the sense that it asymptotes an almost periodic function for t large? Thisproblem is difficult even for ODEs and has only received a partial answer even in the caseof equation (2.1), cf.e.g.[1, 5, 6, 12, 15, 14].

6

2.4 Stepanov spaces and generalized almost periodicity

Definition 2.5. Given a real Banach space X with norm ‖.‖ and an infinite interval J =[t0,+∞) we set

S1(J,X) = f ∈ L1loc(J,X), sup

t∈J

∫ t+1

t

‖f(s)‖ds <∞

where “S” stands for Stepanov. It is immediate to check that S1(J,X) endowed with thesemi-norm

‖f‖S1(J,X) = supt∈J

∫ t+1

t

‖f(s)‖ds

is a Banach space containing L∞(J,X). One similarly defines for any p ≥ 1

Sp(J,X) = f ∈ Lploc(J,X), supt∈J

∫ t+1

t

‖f(s)‖pds <∞

and we may complete by setting S∞(J,X) = L∞(J,X). When X = R we simplify thenotation:

∀p ∈ [1,∞], Sp(J,R) =: Sp(J)

For the theory of almost periodic functions the most important case is when t0 = −∞ i.e.J = R and p = 1. Indeed when we have a good adaptation result saying that the responseto an almost periodic forcing asymptotes an almost periodic function for t large, most of thetime we can afford discontinuous locally integrable forcings thanks to the smoothing effectof integration.

Definition 2.6. We say that f ∈ Sp(R, X) is Sp- almost periodic : R → X if the functionF (α) : f( + α) := Tαf is almost periodic : R→ Sp(R, X)

It is clear that usual (continuous) almost periodic functions are Sp- almost periodic forall p.

2.5 Boundedness via differential inequalities

Lemma 2.7. Let α ∈ S1(R+) and F ∈ S1(R+) be such that F ≥ 0 and

∀t ∈ R+,

∫ t+1

t

α(r)dr ≥ α0 > 0 (2.4)

Let Φ ∈ L1loc(R+),Φ ≥ 0 satisfy the differential inequality

Φ′(t) + α(t)Φ(t) ≤ F (t), a.e. on R+ (2.5)

7

Then Φ is bounded and we have

∀t ∈ R+, Φ(t) ≤ eA0e−α0tΦ(0) + (1 +1

α0

)‖F‖S1 (2.6)

withA0 = α0 + ‖α‖S1

Proof. We observe that for any (s, t) ∈ R2 with 0 ≤ s ≤ t we have∫ t

s

α(r)dr ≥ α0(t− s)− A0

Indeed setting t = s+ n+ ρ with n ∈ N, 0 ≤ ρ < 1 we have∫ t

s

α(r)dr =

∫ s+n

s

α(r)dr +

∫ t

s+n

α(r)dr ≥ nα0 − ‖α‖S1 ≥ (t− s− 1)α0 − ‖α‖S1

The result then relies on the following more general Lemma

Lemma 2.8. Let α ∈ L1loc(R+) and F ∈ S1(R+) be such that F ≥ 0 and

∀t ∈ R+,∀s ∈ [0, t],

∫ t

s

α(r)dr ≥ α0(t− s)− A0 (2.7)

for some α0 > 0, A0 ≥ 0. Let Φ ∈ L1loc(R+) satisfy the differential inequality

Φ′(t) + α(t)Φ(t) ≤ F (t), a.e. on R+ (2.8)

Then Φ is bounded and we have

∀t ∈ R+, Φ(t) ≤ eA0e−α0tΦ(0) + (1 +1

α0

)‖F‖S1 (2.9)

Proof. Introducing A(t) =∫ t

0α(r)dr we have almost-everywhere in t

d

dt(eA(t)Φ(t)) ≤ eA(t)F (t)

which by integration provides

eA(t)Φ(t) ≤ Φ(0) +

∫ t

0

eA(s)F (s)ds

hence

Φ(t) ≤ e−A(t)Φ(0) +

∫ t

0

eA(s)−A(t)F (s)ds

8

and by using (2.7), we find

Φ(t) ≤ eA0e−α0tΦ(0) +

∫ t

0

eα0(s−t)F (s)ds

The conclusion follows easily from the next simple lemma.

Lemma 2.9. Let

I(t) :=

∫ t

0

eα0(s−t)F (s)ds

Then I is bounded and we have

∀t ∈ R+, I(t) ≤ (1 +1− e−α0t

α0

)‖F‖S1 (2.10)

In addition this inequality is optimal for t ∈ N.

Proof. To simplify the formulas we write α instead of α0 and we set, for t fixed

g(t) :=

∫ t

s

F (r)dr

Then∀t ∈ R+,∀s ∈ [0, t], g(s) ≤ (t− s+ 1)‖F‖S1 (2.11)

and

eαtI(t) = −∫ t

0

eαsg′(s)ds = −[eαsg(s)]t0 +

∫ t

0

αeαsg(s)ds = g(o) +

∫ t

0

αeαsg(s)ds

and by (2.11) we obtain

eαtI(t) ≤ ‖F‖S1

t+ 1 +

∫ t

0

α(t− s+ 1)eαsds

Finally, let

J(t) := t+ 1 +

∫ t

0

α(t− s+ 1)eαsds

We find

J(t) = t+ 1 + eαt − 1 +

∫ t

0

α(t− s)eαsds = t+ eαt +

∫ t

0

α(t− s)eαsds

But integrating by parts again we obtain∫ t

0

α(t− s)eαsds = [(t− s)eαs]t0 −∫ t

0

(−1)× eαsds = −t+

∫ t

0

eαsds = −t+eαt − 1

α

9

Finally

eαtI(t) ≤ ‖F‖S1

[eαt +

eαt − 1

α

]and the result follows.

The following more nonlinear Lemma will be useful for the proof of our main boundednessresult (cf. Theorem 4.1).

Lemma 2.10. Let Ψ ≥ 0 be a function in W 1,1loc (R+) which satisfies

d

dtΨ ≤ −µ(t)Ψ

12 + F (t) (2.12)

with µ ∈ S1(R+) and

∀t ≥ 0,

∫ t+1

t

µ(s)ds ≥ ρ > 0

Then Ψ(t) : R+ → R+ is bounded.

Proof. We setF ∗ = ‖F‖S1(R+); µ∗ = ‖µ‖S1(R+)

As a consequence of Gronwall’s inequality, Ψ(t) is bounded on [0, 1] in terms of Ψ(0), µ∗ andF ∗.To show the boundedness of Ψ(t) on [0,+∞[, we select t ≥ 0 and we distinguish two cases.

case 1: Assume that ∃s ∈ [t, t+ 1] such that:

Ψ(s) ≤(F ∗

ρ

)2

= K1.

Sinced

dtΨ ≤ µ−(τ)Ψ(τ) + F (τ) +

1

4µ−(τ).

By Gronwall’s inequality, we obtain:

Ψ(t+ 1) ≤ K1 exp(µ∗) + exp(µ∗)(F ∗ +1

4µ∗) = K2. (2.13)

case 2: Assume that we have:

infs∈[t,t+1]

Ψ(s) ≥(F ∗

ρ

)2

. (2.14)

Then, for all s ∈ [t, t+ 1], we have:

1

Ψ(s)12

dΨ(s)

ds≤ −µ(s) +

F (s)

Ψ(s)12

10

Consequentlyd

ds(Ψ(s)

12 ) ≤ −µ(s)

2+

ρ

F ∗F (s)

2.

By integrating over [t, t+ 1] we obtain

Ψ(t+ 1) ≤ Ψ(t). (2.15)

Therefore in both cases 1 and 2 we have:

∀t ≥ 0, Ψ(t+ 1) ≤ max(K2, Ψ(t)

).

Finally, we conclude that:

∀t ≥ 0, Ψ(t) ≤ max(K2, sup

0≤ε≤1Ψ(ε)

). (2.16)

3 Existence and regularity of solutions.

3.1 Functional setting

Throughout this article we let H and V be two Hilbert spaces with norms respectivelydenoted by ‖.‖ and |.|. We assume that V is densely and continuously embedded into H.Identifying H with its dual H ′, we obtain V → H = H ′ → V ′. We denote inner productsby (.,.) and duality products by 〈·, ·〉; the spaces in question will be specified by subscripts.The notation 〈f, u〉 without any subscript will be used sometimes to denote 〈f, u〉V ′,V . Theduality map: V → V ′ will be denoted by A. We recall that A is characterized by the property

∀(u, v) ∈ V × V, 〈Au, v〉V ′,V = (u, v)V

3.2 Weak solutions in the purely dissipative case.

We consider the dissipative evolution equation:

u+ Au+ g(u) = h(t) (3.1)

where g ∈ C(V, V ′) satisfies

∀(v, w) ∈ V × V, 〈g(v)− g(w), v − w〉 ≥ 0 (3.2)

We consider the (generally unbounded) operator A defined on the Hilbert space H = V ×Hby

D(A) = (u, v) ∈ V × V, Au+ g(v) ∈ Hand

∀(u, v) ∈ D(A), A(u, v) = (−v,Au+ g(v))

11

Lemma 3.1. The operator A is maximal monotone.

Proof. Let U = (u, v) and U = (u, v) be two elements of D(A). We have

(AU −AU , U − U)H = −(u− u, v − v)V + (Au+ g(v)− Au− g(v), v − v)H

−(u− u, v − v)V + 〈Au+ g(v)− Au− g(v), v − v〉V ′,Vsince Au+ g(v) ∈ H and Au+ g(v) ∈ H while v, v are in V . This reduces to

(AU −AU , U − U)H = 〈g(v)− g(v), v − v〉V ′,V ≥ 0

Hence A is monotone. To prove that A is maximal monotone we are left to show that forany F = (ϕ, ψ) ∈ H the following equation

u− v = ϕ, Au+ g(v) + v = ψ

has a solution U = (u, v) ∈ D(A). This is equivalent to finding a solution v ∈ V of

Av + g(v) + v = ψ − Aϕ ∈ V ′

But now the operator C ∈ C(V, V ′) defined by

∀v ∈ V, Cv = Av + g(v) + v

is continuous and coercive: V → V ′. Therefore by Corollary 14 p. 126 from H. Brezis [8], Cis surjective. Finally A is maximal monotone as claimed.

As a consequence of Proposition 2.1, for any h ∈ L1loc(R+, H) and for each (u0, u1) ∈ V×H

there is a unique weak solution

u ∈ C(R+, V ) ∩ C1(R+, H)

of (3.1) such that u(0) = u0 and u(0) = u1. This solution is can be recovered on eachcompact interval [0, T ] by approximating the initial data by elements of the domain, theforcing term h by C1 functions and passing to the limit: the limit is independent of theapproximating elements so chosen. The next result shows that in fact the approximationcan even be made uniform on R+.

3.3 Regularity properties and density of strong solutions.

Lemma 3.2. For any h ∈ L1loc(R+, H) and for each (u0, u1) ∈ V × H and for each δ > 0

there exists (w0, w1) ∈ D(A) and k ∈ C1(R+, H) for which the solution w ∈ W 1,1loc (R+, V ) ∩

W 2,1loc (R+, H) of

w + Aw + g(w) = k(t); w(0) = w0, w(0) = w1

satisfies∀t ≥ 0, ‖u(t)− w(t)‖+ |u(t)− w(t)| ≤ δ

12

Proof. It suffices to use the last result of Proposition 2.1 by observing that for any h ∈L1loc(R+, H) we can find k ∈ C1(R+, H) such that

∀n ∈ N,∫ n+1

n

|k(s)− h(s)|ds ≤ δ2−n−1

Choosing (w0, w1) ∈ D(A) such that ‖w0 − u0‖+ |w1 − v1| ≤ δ2−1 the result follows imme-diately

4 A Boundedness result.

We consider the dissipative evolution equation (3.1). We say that h ∈ S1(R+, H) if

h ∈ L1loc(R+, H) and sup

t≥0

∫ t+1

t

|h(s)|ds = h∗ < +∞. (4.1)

Then we can state the following result which generalizes Theorem IV.2.1.1 from [14] to thecase of possibly non-local damping terms.

Theorem 4.1. Assume that (4.1) is satisfied and g ∈ C(V, V ′) satisfies the conditions

∃α > 0, ∃C1 ≥ 0 ∀v ∈ V, 〈g(v), v〉 ≥ α|v|2 − C1. (4.2)

∃τ > 0, ∃C2 ≥ 0 ∀v ∈ V, ‖g(v)‖∗ ≤ C2 + τ〈g(v), v〉. (4.3)

Assume, also, that the following condition is fulfilled:

2τh∗ < 1 (4.4)

Then any solution u ∈ C(R+, V )∩C1(R+, H) of (3.1) is bounded on R+ in the sense that uhas bounded range in V and u has bounded range in H.

Proof of Theorem: We start by an estimate in the case of a strong solutions, i.e. weassume u ∈ W 1,1

loc (R+, V ) ∩W 2,1loc (R+, H). The general case will follow by density. Let

E(t) =1

2(|u|2 + ‖u‖2)

Under the regularity conditions [u0, v0] ∈ V × V , γ(0, v0) ∈ H and h ∈ W 1,1loc (R+, H),

t→ E(t) is absolutely continuous and we have ∀t ∈ R+:

d

dtE(t) = (h, u)− 〈g(u), u〉. (4.5)

13

In addition t→ (u(t), u(t)) is absolutely continuous and

d

dt(u(t), u(t)) = |u|2 − ‖u‖2 − 〈g(u), u〉+ (h, u).

By using (4.3), we obtain:

d

dt(u(t), u(t)) ≤ |u|2 − ‖u‖2 + ‖u‖ P |h|+ C2 + τ〈g(u), u〉

with P = sup|u|, u ∈ V, ‖u‖ = 1. Introducing Φ(t) = 2E(t), ∀t ≥ 0, we find:

d

dt

(1 + Φ)−

12 (u, u)

≤ |u|

2 − ‖u‖2

(1 + Φ)12

+ τ〈g(u), u〉

+ P |h|+ C2 +P |u|‖u‖(1 + Φ)

32

∣∣∣∣dEdt∣∣∣∣ (4.6)

We have:

|u|2 − ‖u‖2

(1 + Φ)12

=2|u|2 + 1

(1 + Φ)12

− 1 + |u|2 + ‖u‖2

(1 + Φ)12

≤ 1 + 2|u| − (1 + Φ)12 (4.7)

Then from (4.5), we obtain by Cauchy- Schwarz:

|u|‖u‖(1 + Φ)

32

∣∣∣∣dEdt∣∣∣∣ ≤ 1

2(1 + Φ)12

∣∣∣∣dEdt∣∣∣∣

≤ |h|+ 1

2(1 + Φ)12

〈g(u), u〉 − (h, u)

= |h| − 1

2(1 + Φ)12

1

2

d

dt(1 + Φ)

= |h| − 1

2

d

dt[(1 + Φ)

12 ] (4.8)

Therefore, from (4.7) and (4.8) we deduce:

d

dt

(1 + Φ)−

12 (u, u) +

P

2(1 + Φ)

12

≤ −(1 + Φ)

12 + 2|u|+ 2P |h|

+ τ < g(u), u > +1 + C2.

(4.9)

By using (4.5) , (4.9) and (4.2) we obtain:

d

dtτΦ + (1 + Φ)−

12 (u, u) +

P

2(1 + Φ)

12 ≤ −(1 + Φ)

12 + 2(1 + τ |h|)|u|

− ατ |u|2 + 2P |h|+ 1 + C2 + C1τ.(4.10)

14

For K > 0 large enough, we set:

Ψ = τΦ + (1 + Φ)−12 (u, u) +

P

2(1 + Φ)

12 +K.

Therefore, we obtain:

d

dtΨ ≤ −(1 + Φ)

12 + 2τ |h|Φ

12 + 2|u| − ατ |u|2 + 2P |h|+ 1 + C2 + C1τ

≤ −(1− 2τ |h|)(1 + Φ)12 + 2|u| − ατ |u|2 + 2P |h|+ 1 + C2 + C1τ.

This differential inequality is verified when 2τh∗ < 1. In addition for K > 0 large enough, Ψis positive on R+ and we have:

Ψ ≤ τΦ + (1 + Φ)−12 |u||u|+ P

2(1 + Φ)

12 +K

≤ τΦ + P (1 + Φ)−12‖u‖|u|+ P

2(1 + Φ)

12 +K

≤ τΦ +P

2(1 + Φ)−

12 Φ +

P

2(1 + Φ)

12 +K

≤ τΦ + P (1 + Φ)12 +K

We notice that for any η > 0P (1 + Φ)

12 ≤ ηΦ + c(η). (4.11)

Then

Ψ ≤ (τ + η)Φ + c(η) +K

where η > 0 can be taken artibitrarily small. Setting c(η) +K =: Q, we obtain

Ψ ≤ (τ + η)Φ +Q (4.12)

Also, we have:

Ψ ≥ τΦ− (1 + Φ)−12 |u||u|+ P

2(1 + Φ)

12 +K

≥ τΦ− P (1 + Φ)−12‖u‖|u|+ P

2(1 + Φ)

12 +K

≥ τΦ− P

2(1 + Φ)−

12 Φ +

P

2(1 + Φ)

12 +K

≥ τΦ− P (1 + Φ)−12 Φ +

P

2(1 + Φ)

12 +K

≥ τΦ− P (1 + Φ)12 +K

By using (4.11) we obtain

Ψ ≥ (τ − η)Φ +K − c(η)

15

Assuming that K − c(η) ≥ 0 and η < τ , we deduce

Ψ ≥ (τ − η)Φ (4.13)

Hence by (4.12) and (4.13), we obtain for Q > 0 large enough:

(τ − η)Φ ≤ Ψ ≤ (τ + η)Φ +Q. (4.14)

Then:

d

dtΨ ≤ − 1

τ + ηΨ

12 +

2τ |h|τ − η

Ψ12 + 2|u| − ατ |u|2 + 2P |h|+ 1 + C2 + C1τ.

Since, by Cauchy-Schwarz , 2|u| − ατ |u|2 is less than a constant, we obtain:

d

dtΨ ≤ − 1

τ + ηΨ

12 +

2τ |h|τ − η

Ψ12 + 2P |h|+ 1 + C2 + C1τ + C3. (4.15)

We set

σ =1

τ + η

m(t) =2τ |h(t)|τ − η

andF (t) = 2P |h(t)|+ 1 + C2 + C1τ + C3.

We have

d

dtΨ ≤ −σΨ

12 +m(t)Ψ

12 + F (t). (4.16)

As a consequence of (4.4), for η small enough, we have 2ττ−ηh

∗ < 1τ+η

= σ hence

m∗ = supt≥0

∫ t+1

t

m(s)ds < σ (4.17)

and

F ∗ = supt≥0

∫ t+1

t

F (s)ds

= supt≥0

∫ t+1

t

(2P |h(s)|+ 1 + C2 + C1τ + C3)ds < +∞(4.18)

For the rest of proof, we apply lemma 2.10. We obtain that Ψ(t) is bounded on R+. Henceby (4.14), Φ(t) and E(t) are bounded for t ≥ 0. The general case of weak solutions followseasily by density by using Lemma 3.2.

16

5 Precompactness of bounded orbits.

The main result of this section is

Theorem 5.1. Let u ∈ C(R+, V ) ∩ C1(R+, H) be a solution of

u(t) + g(u(t)) + Au(t) = h(t), t ∈ R+,

with h ∈ S1(R+, H) such that u has bounded range in V and u has bounded range in H.Assume the following

i) h is S1-uniformly continuous with values in H in the sense that

lim supε→0 t≥0

∫ t+1

t

|h(s+ ε)− h(s)|ds = 0. (5.1)

ii) g ∈ C(V, V ′) satisfies g(0) = 0 and the following conditions

∀(v, w) ∈ V × V, 〈g(v)− g(w), v − w〉V ′,V ≥ 0 (5.2)

∀δ > 0,∃C(δ) > 0, ∀(v, w) ∈ V × V, |v − w|2 ≤ δ + C(δ)〈g(v)− g(w), v − w〉V ′,V (5.3)

∃C > 0, ∀v ∈ V, ‖g(v)‖Z′ ≤ C(1 + 〈g(v), v〉V ′,V ) (5.4)

where Z ′ is the dual of a reflexive Banach space such that

Z ⊂ H and the imbedding: Z → H is continuous (5.5)

V ⊂ Z and the imbedding: V → Z is compact (5.6)

V is everywhere dense in Z. (5.7)

Then u has precompact range in V and u has precompact range in H.

Proof. Let us denote the norm in Z by ||| ||| and the norm in Z ′ by ||| |||∗.We observe that Z ′ ⊂ V ′ with continuous imbedding. From (5.5) and (5.6) it follows that∀α > 0, ∃c(α) ≥ 0 such that

∀u ∈ V, |||u||| ≤ α‖u‖+ c(α)|u| (5.8)

We prove first the result under the additional assumption u ∈ W 1,1loc (R+, V ) ∩W 2,1

loc (R+, H).First wε(t) = u(t+ ε)− u(t) satisfies the equation:

wε + Lwε + g(u(t+ ε))− g(u(t)) = h(t+ ε)− h(t) (5.9)

where u(t) is the solution of equation (3.1).We set

fε(t) = h(t+ ε)− h(t) (5.10)

17

and∀t ∈ R+, ∀w ∈ V, g(u(t) + w)− g(u(t)) = γ(t, w). (5.11)

Then (5.9) becomeswε + Lwε + γ(t, wε) = fε(t), ∀t ≥ 0 (5.12)

As a consequence of section 2, we know that u(t) and u(t) are bounded on R+. We denoteby

Eε(t) =1

2|u(t+ ε)− u(t)|2 + ‖u(t+ ε)− u(t)‖2

=1

2|wε(t)|2 + ‖wε(t)‖2

the energy of the solution to (5.12).Let us introduce, for some β > 0 to be chosen later

Φ =1

2|wε|2 + ‖wε‖2+ β(wε, wε).

Then, we have:

Φ′ ≤ −〈wε, γ(t, wε)〉+ β|wε|2 − β‖wε‖2 − β〈γ(t, wε), wε〉+ 〈fε, wε + βwε〉

we obtain

Φ′ ≤ −βEε − 〈wε, γ(t, wε)〉+3β

2|wε|2 −

β

2‖wε‖2 + β|||wε||| |||γ(t, wε)|||∗ + |fε|

(|wε|+ β|wε|

)If we impose β ≤

√λ1 , then Φ ≥ 0 and from (5.8) we deduce

Φ′ ≤ −βEε − 〈wε, γ(t, wε)〉+3β

2|wε|2 −

β

2‖wε‖2 + β

(α‖wε‖+ c(α)|wε|

)|||γ(t, wε)|||∗

+ |fε|(|wε|+ ‖wε‖

)From (5.3) and |wε|+ ‖wε‖ ≤ 2

√Eε, we have:

Φ′ ≤ −βEε − 〈wε, γ(t, wε)〉+3β

2δ +

3β

2K(δ)〈wε, γ(t, wε)〉 −

β

2‖wε‖2

+ β

(α‖wε‖+ c(α)|wε|

)|||γ(t, wε)|||∗ + 2

√Eε|fε|

≤ −βEε + (3β

2K(δ)− 1)〈wε, γ(t, wε)〉 −

β

2‖wε‖2 + β

(α‖wε‖+ c(α)|wε|

)|||γ(t, wε)|||∗

+ 2√Eε|fε|+

3β

2δ

18

From now on we fix δ > 0 small enough and we choose β such that

3β

2K(δ)− 1 ≤ 0

which is equivalent to

β ≤ 2

3K(δ).

Then

Φ′ ≤ −βEε + β

(α‖wε‖+ c(α)|wε|

)|||γ(t, wε)|‖|∗ + 2

√Eε|fε|+

3β

2δ.

We have by Cauchy-Schwarz:

Φ =1

2|wε|2 + ‖wε‖2+ β(wε, wε)

≤ Eε +β2

2|wε|2 +

1

2|wε|2.

Then, for β ≤√λ1, we have

Φ ≤ 2Eε

Also, if we assume the stronger condition β ≤ 12

√λ1, then

Φ ≥ Eε − β2|wε|2 −1

4|wε|2 ≥

1

2Eε

Therefore , we have

1

2Eε(t) ≤ Φ(t) ≤ 2Eε(t), ∀t ∈ R (5.13)

From (5.13), we obtain

Φ′ ≤ −βEε + 2√Eε|fε|+ β

(α‖wε‖+ c(α)|wε|

)|||γ(t, wε)|||∗ + 3βδ

2(5.14)

Then, we set

F (t) = β(α‖wε‖+ c(α)|wε|

)|||γ(t, wε)|||∗ +

3δ

2

Then

Φ′ ≤ −β2Φ + 2

√Eε|fε|+ F (t)

19

By using (5.13), we find:

Φ′ ≤ −β2Φ + 2

√2√

Φ|fε|+ F (t)

By using the inequality 2√

Φ ≤ 1 + Φ in the second term of the RHS and setting

m(t) =√

2|fε(t)|

we find

Φ′ ≤ −β2Φ +m(t)Φ + F (t) +m(t)

IntroducingH(t) := F (t) +m(t)

we have:

Φ′(t) ≤ −(β2−m(t))Φ(t) +H(t), ∀t ∈ R+ (5.15)

On the other hand:∫ t+1

t

m(s)ds =√

2

∫ t+1

t

fε(s)ds =√

2

∫ t+1

t

|h(s+ ε)− h(s)|ds.

with

lim supε→0 t≥0

∫ t+1

t

|h(s+ ε)− h(s)|ds = 0.

In addition∫ t+1

t

F (s)ds = β

∫ t+1

t

(α‖wε(s)‖+ c(α)|wε(s)|

)|||γ(s, wε(s))|||∗ +

3δ

2ds

≤ β(2α supt≥0‖u(t)‖+ c(α) sup

t≤s≤t+1|wε(s)|)

∫ t+1

t

|||γ(t, wε(s))|||∗ds+3βδ

2

Nowγ(s, wε(s)) = g(u(s+ ε))− g(u(s))

From boundedness of the energy E(t) for t ≥ 0, ∃C > 0 such that∫ t+1

t

〈g(u(s)), u(s)〉 dx ds ≤ C, ∀t ≥ 0.

As a consequence of (5.4) we have

g(u(s)) ∈ L1(t, t+ 1, Z ′)

20

and ∫ t+1

t

|||g(u(s))|||∗ds ≤ N(C + 1) (5.16)

Therefore ∫ t+1

t

F (s)ds ≤ 2N(C + 1)β(2α supt≥0‖u(t)‖+ c(α) sup

t≤s≤t+1|wε(s)|) +

3βδ

2

We havesup

t≤s≤t+1|wε(s)| ≤ ε sup

s≥0|u(s)|.

By choosing α small enough, we obtain∫ t+1

t

F (s)ds ≤ βδ + Pε sups≥0|u(s)|+ 3βδ

2.

Then for ε ≤ ε2 small enough, we obtain:

supt≥0

∫ t+1

t

F (s)ds ≤ 3βδ. (5.17)

Then by (5.17), we obtain:

supt≥0

∫ t+1

t

H(s)ds ≤ 3βδ +√

2

∫ t+1

t

|h(s+ ε)− h(s)|ds

Now we may fix

β = β0 ≤ min1

2

√λ1,

2

3K(δ)

Then by imposing that ε is small enough to ensure

√2 supt≥0

∫ t+1

t

|h(s+ ε)− h(s)|ds ≤ minβδ, β4 (5.18)

we obtain

supt≥0

∫ t+1

t

H(s)ds ≤ 4βδ (5.19)

and (5.15) becomes

Φ′(t) ≤ −α(t)Φ(t) +H(t), ∀t ∈ R+ (5.20)

where ∫ t+1

t

α(s)ds ≥ β

4

21

Lemma 2.9 then provides

∀t ≥ 0, Φ(t) ≤ C[Φ(0) + (1 +4

β)4βδ]

where C is bounded in terms of the S1 norm of h. The uniform continuity follows easilyand since this property is robust with respect to uniform convergence in the energy normon R+, we obtain it for general weak solutions by using Lemma 3.2 . The compactnessresult now follows from the same argument as in [14], p.167-168: the main idea is that theaverage 1

ε

∫ t+εt

u(s)ds of u on a small time interval remains bounded (by a large constant) in

V while the equation shows that the average 1ε

∫ t+εt

u(s)ds of u remains bounded (by a largeconstant) in A−1(Z ′) which imbeds compactly in V . The uniform continuity of the vector(u(t), u(t)) shows that this vector is arbitrary close to its average on a small time interval.Finally precompactness follows from the total boundedness criterion.

Corollary 5.2. Assume that the conditions of Theorem 4.1 are all satisfied except (5.3)which is replaced by

∃p ≥ 2,∃η > 0, ∀(v, w) ∈ V × V, 〈g(v)− g(w), v − w〉V ′,V ≥ η|v − w|p (5.21)

Then the conclusion of Theorem 4.1 holds true.

Proof. Given δ > 0, we have

∀(v, w) ∈ V × V, |v − w|2 ≤ δ + C1(δ)|v − w|p ≤ δ +C1(δ)

η〈g(v)− g(w), v − w〉V ′,V

In particular (5.3) is fulfilled.

In the applications to non-local dissipations we shall use Lemma 2.3.

6 The purely dissipative almost periodic case.

Compactness of trajectories is a basic tool to prove the existence of almost periodic (weak)solutions to the equation

U ′(t) +AU(t) = F (t)

Indeed if A is maximal monotone on H and F : R → H is almost periodic, it follows from[12] or ISHII that the existence of a precompact trajectory is equivalent to the existence ofan almost periodic solution. This property has been used in [14] to prove the existence ofalmost-periodic solutions of

utt + g(ut)−∆u = h(t, x), in R+ × Ω,

u = 0 on R+ × ∂Ω.(6.1)

22

where Ω is a bounded domain, g is the Nemytskii operator generated by an increasingfunction γ ∈ C(R) such that γ−1 is uniformly continuous and satisfying some dimensiondependant growth conditions and h is S1-almost periodic :R→ L2(Ω).

6.1 A general result

We are now in a position to state and prove a more general result valid also for non-localdissipation terms. More precisely we have

Theorem 6.1. Assume that g satisfies the hypotheses of Theorem 5.1 and (4.3) with τarbitrary. Then for any h which is S1-almost periodic: R→ H the equation

u(t) + g(u(t)) + Au(t) = h(t), t ∈ R+, (6.2)

has at least one solution ω such that the vector (ω, ω) is almost periodic R → V × H. Inaddition for any other solution u we have for some constant vector a ∈ V

limt→∞

(‖u(t)− ω(t)− a‖+ |u(t)− ω(t)|) = 0

In addition if g ∈ C(H, V ′) the almost periodic solution is unique and the previous conver-gence result is satisfied with a = 0 for any solution u.

Proof. The existence follows from [14], Theorem IV.3.3.3 p.173. The uniqueness result up toa constant vector will be a consequence of the second part of the theorem since an almost-periodic vector whose norm tends to 0 is identically 0 (c.f. e.g. [1, 15]). For the last result,let us first select a common sequence an tending to +∞ for which h(. + an) converges to hin S1(R, H) and ω(.+ an) converges to ω in S1(R, V ) (The Vector (h, ω, ω) being S1-almostperiodic with values inH × V ×H), hence ω(.+ an) converges to ω also in Cb(R, V ) since ωis almost periodic in the usual sense. if u is any solution, precompactness of the range of uimplies that a subsequence of u(.+an) converges uniformly on all compact subintervals withvalues in V to some limit z, while the same subsequence of u(.+an) converges to z uniformlyon all compact subintervals with values in H. Since the energy norm of the difference u− ωis non-increasing, it converges to some limit l ≥ 0. The energy norm of the difference z − ωis equal to l for all t. We finally show that z−ω is constant (and equal to 0 if g ∈ C(H, V ′))by using the following Lemma

Lemma 6.2. Let J = [a, b] be any compact interval of R with a < b and let u, v be two weaksolutions of

u(t) + g(u(t)) + Au(t) = h(t), t ∈ J

such as ‖u(t)−v(t)‖2+|u(t)−v(t)|2 is constant on J. Then u = v. In addition if g ∈ C(H,V ′)then u = v.

23

Proof. The result is almost trivial if u and v are strong solutions. To prove it in the generalcase we approximate h and (u(a), u(a)), (v(a), v(a)) by hn ∈ C1(J,H) and (u0

n, u1n), (v0

n, v1n) ∈

D(A)2. Let η > 0 be an arbitrary small fixed number. We choose n in such a way that

sup‖u0n − u(a)‖2 + |u1

n − u(a)|2, ‖v0n − v(a)‖2 + |v1

n − v(a)|2 ≤ η2

16

and‖hn − h‖L1(J) ≤

η

4

which implies

supJ[‖un(t)− u(t)‖2 + |un(t)− u(t)|2]1/2 ≤ η

2

where un is the strong solution of

un(t) + g(un(t)) + Aun(t) = hn(t), t ∈ J ; un(a) = u0n, un(a) = u1

n. (6.3)

Defining similarly the solution vn of

vn(t) + g(vn(t)) + Avn(t) = hn(t), t ∈ J ; vn(a) = v0n, vn(a) = v1

n (6.4)

we find by the triangular inequality

[‖un(a)− vn(a)‖2 + |un(a)− vn(a)|2]1/2 ≤ [‖u(a)− v(a)‖2 + |u(a)− v(a)|2]1/2 +η

2

= [‖u(b)− v(b)‖2 + |u(b)− v(b)|2]1/2 +η

2≤ [‖un(b)− vn(b)‖2 + |un(b)− vn(b)|2]1/2 + η.

Hence by squaring and using boundedness of the sequence ‖un(b)− vn(b)‖2 + |un(b)− vn(b)|2we obtain

‖un(a)− vn(a)‖2 + |un(a)− vn(a)|2 − [‖un(b)− vn(b)‖2 + |un(b)− vn(b)|2] ≤ η2 + Cη

which can be rewritten, assuming η < 1, as∫J

〈g(un(t))− g(vn(t)), un(t)− vn(t)〉dt ≤ C ′η.

By using the hypothesis (5.3) on g we deduce∫J

|un(t)− vn(t)|2dt ≤ |J |δ +C(δ)

∫J

〈g(un(t))− g(vn(t)), un(t))− vn(t)〉dt ≤ |J |δ +C ′C(δ)η

so that for η small enough we find∫J

|un(t)− vn(t)|2dt ≤ 2|J |δ

24

As a first consequence u = v on J . In addition if g ∈ C(H, V ′) , since un converges uniformlyto u on J and vn converges uniformly to v on J with values in H, by considering the equation

un(t)− vn(t) + g(un(t))− g(vn(t)) + A(un(t)− vn(t)) = 0

After integration on J we find that in the sense of V ′∫J

A(un(t)− vn(t))dt→ 0

Hence if u− v ≡ a we end up with |J |Aa = 0 and finally a = 0

The end of proof of Theorem 6.1 follows very easily by considering any interval J as inthe Lemma applied with u replaced by z and v replaced by ω.

Remark 6.3. 1) It does not seem easy to construct a counterexample in which a 6= 0.

2) In order to have uniqueness of ω it is sufficient to assume a much weaker property, itsuffices for instance that g be continuous from H to X weak where X is a reflexive Banachspace such that V ⊂ X with continuous imbedding.

3) In the next subsection, we derive a better result for special kinds of damping terms.

6.2 A more precise result for a special class of damping operators.

In this section we consider a reflexive Banach space Z satisfying the conditions (5.5), (5.6)and (5.7).

Definition 6.4. Given α > 0, we say that g ∈ C(Z,Z ′) is (Z, α)-admissible if g(0) = 0 andfor some positive constants c, C we have

∀(v, w) ∈ V × V, 〈g(v)− g(w), v − w〉V ′,V ≥ c‖v − w‖Zα+2 (6.5)

∀(v, w) ∈ V × V, ‖g(v)− g(w)‖Z′ ≤ C(‖v‖αZ + ‖w‖αZ)‖v − w‖Z (6.6)

The class of (Z, α)-admissible functions will be denoted by G(Z, α).

The following Lemma shows that the functions of G(Z, α) satifies all the properties usedin our main boundedness and compactness results.

Lemma 6.5. Let g ∈ G(Z, α). Then g satisfies (5.21) with p = α+ 2 hence (5.2), (5.3) and(4.2), (4.3) for any τ > 0 and (5.4).

25

Proof. It is clear that (6.5) implies (5.21) with p = α + 2 since the norm in Z dominatesthe norm in H. (5.2) and (5.3) are immediate consequences. (4.2) follows by taking w = 0.Finally (4.3) for any τ > 0 and (5.4) follow easily from the combination of (6.5) and (6.6)applied with w = 0.

The next proposition clarifies the regularity of weak solutions when g is (Z, α)-admissible.

Proposition 6.6. Let g ∈ G(Z, α). Then for any J = [a, b] compact interval of R with a < b,any h ∈ L1(J,H) and u any weak solution of

u(t) + g(u(t)) + Au(t) = h(t), t ∈ J (6.7)

we haveu ∈ Lα+2(J, Z); g(u) ∈ L

α+2α+1 (J, Z ′)

u(t) + g(u(t)) + Au(t) = h(t) in L1(J, V ′)

Proof. The result is obvious for a strong solution u since then u ∈ C(J, V ) and thereforeg(u) ∈ C(J, V ′). Now let u be a weak solution and un be a sequence converging to u inC(J, V ) ∩ C1(J,H) with un a strong solution of

un(t) + g(un(t)) + Aun(t) = hn(t)

andlimn→∞

‖hn − h‖L1(J,H) = 0

It is an immediate consequence of (6.5) that un is a Cauchy sequence in Lα+2(J, Z), then

(6.6) shows that g(un) is Cauchy in Lα+2α+1 (J, Z ′). The result follows easily.

Corollary 6.7. In this case the almost periodic solution is unique

Proof. Indeed if ω1, ω2 are two such solutions, then ω1 = ω2 and ω1 = ω2 in the sense ofdistributions from Int(J) with values in H, then also in L1(J, V ′) on any bounded intervalJ . Then the equation gives Aω1 = Aω2 in the sense of L1(J, V ′) and the conclusion followseasily

Finally the following result improves our main asymptotic theorem by giving a rate of con-vergence:

26

Theorem 6.8. Assume that g satisfies the hypotheses of Proposition 6.6. Then for any hwhich is S1-almost periodic: R→ H the equation

u(t) + g(u(t)) + Au(t) = h(t), t ∈ R+,

has a unique solution ω such that the vector (ω, ω) is almost periodic R→ V ×H. In additionfor any other solution u we have for some M ≥ 0

∀t ≥ 0, ‖u(t)− ω(t)‖+ |u(t)− ω(t)| ≤M(1 + t)−1α (6.8)

Proof. We extend the method of proof of [19], theorem 3.1 p. 200 (cf. also [15], theorem7.5.1 p. 98) to the case of a non-local damping satisfying (6.5)- (6.6). Since the proof isquite similar we just sketch out the main steps for completeness. We start from 2 strongsolutions u and v of the same equation

u(t) + g(u(t)) + Au(t) = h(t), t ∈ J

and we try to derive the estimate

∀t ≥ 0, ‖u(t)− v(t)‖+ |u(t)− v(t)| ≤M(1 + t)−1α (6.9)

with M bounded in terms of the initial data and the S1 norm of h . To this end we introducez := u− v and the function

E(t) :=1

2(|z(t)|2 + ‖w(t)‖2)

In particular, E is non-increasing and therefore bounded. Then we define

Ψ(t) := E(t)α2 (z(t), z(t))

First we haveE ′(t) = −〈g(u(t))− g(v(t)), z(t)〉V ′,V ≤ −c‖z(t)‖Zα+2

Then we find

Ψ′(t) =α

2E(t)

α2−1E ′(t)(z(t), z′(t)) + E(t)

α2 〈−Az(t)− g(u(t)) + g(v(t)), z(t)〉V ′,V

≤ −C1E′(t)− E(t)

α2 ‖z(t)‖2 + CE(t)

α2 (‖u(t)‖αZ + ‖v(t)‖αZ)‖z(t)‖ZV ertz(t)‖

It follows from the properties of g that r(t) := ‖u(t)‖α+2Z +‖v(t)‖α+2

Z ∈ S1(R+) Introducingk(t) := ‖u(t)‖αZ + ‖v(t)‖αZ a rather straightforward calculation yields for any δ > 0

Ψ′(t) ≤ −C1E′(t)− E(t)

α2 ‖z(t)‖2 + δkE(t)

α2 ‖z(t)‖2 + δk

α+2α E(t)

α+22 + C2δ

−(α+1)‖z(t)‖α+2Z

Then settingFε := (1 + C1ε)E + εΨ

27

we have for all ε small enough

1

2E ≤ Fε ≤ 2E

and then we find

F ′ε ≤ −c‖z(t)‖Zα+2−εE(t)α2 ‖z(t)‖2+εδk

α+2α E(t)

α+22 +C2εδ

−(α+1)‖z(t)‖α+2Z +εδkE(t)

α2 ‖z(t)‖2

By choosing δ = (2C2εc

)1

α+1 we derive

F ′ε ≤ −c

2‖z(t)‖Zα+2 − εE(t)

α2 ‖z(t)‖2 + εδk

α+2α E(t)

α+22 + εδkE(t)

α2 ‖z(t)‖2

which, for ε sufficiently small, gives

F ′ε ≤ −ε(‖z(t)‖Zα+2 + E(t)α2 ‖z(t)‖2) +Kε1+ 1

α+1 (1 + kα+2

α )E(t)α+2

2

This inequality reduces to

F ′ε ≤ −ηεFα+2

2ε +K ′ε1+ 1

α+1 (1 + kα+2

α )Fα+2

2ε

Because h1(t) = 1 + kα+2

α is in S1(R+) we conclude by a direct application of Lemma 1.7from [19].

7 Semilinear perturbations with rapidly decaying source

terms.

7.1 An abstract convergence theorem

In this section we consider the equation

u(t) + g(u(t)) +M(u(t)) = h(t), t ∈ R+, (7.1)

where M = ∇E is the gradient operator of a C2 functional E on V and g ∈ C(V, V ′) is anonlinear damping operator such that there exists α ∈ (0, 1), ρ1 > 0 and ρ2 > 0 for which

∀v ∈ V, 〈g(v), v〉V ′,V ≥ ρ1‖v‖α+2H (7.2)

∀v ∈ V, ‖g(v)‖V ′ ≤ ρ2〈g(v), v〉α+1α+2

V ′,V . (7.3)

28

We assume that E satisfies a uniform Lojasiewicz gradient inequality near the set E =M−10 of equilibria with exponent θ ∈

(αα+1

, 12

]which means that for some ρ > 0 we have

for some constant C

∀u ∈ V, distV (u, E) ≤ ρ =⇒ ∀a ∈ E , |E(u)− E(a)|1−θ ≤ C‖Mu‖V ′ (7.4)

We also need an additional technical assumption as follows . Since E ∈ C2(V ) we have foreach u ∈ V

E ′′(u) =M′(u) ∈ L(V, V ′)

and thereforeA−1M′(u) ∈ L(V, V )

we require the slightly different condition

∀u ∈ V, A−1M′(u) ∈ L(H,H)

and moreover∀u ∈ V, ‖A−1M′(u)‖L(H,H) ≤ C(‖u‖) (7.5)

where C(R) is bounded on bounded subsets of R+. We obtain the following generalizationof a result due to Ben Hassen and Chergui [3].

Theorem 7.1. Let u ∈ W 1,1loc (R+, V ) ∩W 2,1

loc (R+, H) be a solution of (7.1))such that u hasprecompact range in V and u has precompact range in H. Assume in addition that

∃C ≥ 0, ∃δ > 0, ‖h(t, ·)‖H ≤C

(1 + t)1+δ+α, for all t ∈ R+. (7.6)

Then we have for some a ∈ E

limt→∞

(‖u(t)− a‖+ |u(t)|) = 0

Proof. First we prove that ‖u(t)‖H tends to 0 at infinity. Indeed introducing

E0(t) =1

2‖u(t)‖2

H + E(u(t))

we have, thanks to assumption (7.2)

E ′0(t) = −〈u, g(u)〉V,V ′ + 〈h, u〉H ≤ −ρ1‖u‖α+2H + ‖h‖H‖u‖H .

E ′0(t) ≤ −ρ1

2‖u‖α+2

H + C0‖h‖α+2α+1

H .

and therefore the bounded function

F(t) := E0(t) + C0

∫ ∞t

‖h(s)‖α+2α+1

H ds

29

is non-increasing and consequently convergent at infinity. Since F(t) − E0 tends to 0 atinfinity, it follows that E0(t) itself has a limit E. In addition we have

‖u‖α+2H ≤ − 2

ρ1

E ′0(t) +D0‖h‖α+2α+1

H .

hence ‖u(t)‖H ∈ Lα+2(R+) and in particular

u ∈ Lα+2(R+, V′)

But by the equation we also have

u ∈ L∞(R+, V′)

therefore limt→+∞

u(t) = 0 in V ′. By compactness,

limt→+∞

‖u(t)‖H = 0 (7.7)

From the definition of E0 we deduce

limt→+∞

E(u(t)) = E (7.8)

It is then rather straightforward to deduce the following property

limt→+∞

distV (u(t), E∗) = 0 (7.9)

whereE∗ = ϕ ∈ E , E(ϕ) = E

Indeed by compactness, u, being Lipschitz continuous with values in H, is uniformy contin-uous with values in V . In particular we have

limε→0

supt≥0‖1

ε

∫ t+ε

t

M(u(s))ds−M(u(t))‖V ′ = 0

From

E ′0(t) = −〈u, g(u)〉V,V ′ + 〈h, u〉H ≤ −1

2〈u, g(u)〉V,V ′ −

ρ1

2‖u‖α+2

H + ‖h‖H‖u‖H

it follows that 〈u, g(u)〉V,V ′ ∈ L1(R+) Then we observe that (7.3) implies the inequality

∀δ > 0,∀v ∈ V, ‖g(v)‖V ′ ≤ δ + C(δ)〈g(v), v〉V ′,VTherefore for each ε fixed

limt→+∞

‖1

ε

∫ t+ε

t

g(u(s))ds‖V ′ = 0

30

By integrating the equation over [t, t+ ε] and dividing through by ε we now obtain

1

ε

∫ t+ε

t

M(u(s))ds =1

ε

∫ t+ε

t

h(s)ds− 1

ε

∫ t+ε

t

g(u(s))ds− 1

ε(u(t+ ε)− u(t))

The right-hand side tends to 0 in V ′ as t→∞ for any fixed ε > 0. By choosing ε > 0 smallenough first and then letting t→∞ we finally see that

limt→+∞

‖M(u(t))‖V ′ = 0

and the result follow easily since for any limiting point ϕ of u(t) as t→∞ we haveM(ϕ) = 0and E(ϕ) = E.

Now let 0 < ε ≤ 1 be a real constant. We define the function :

H(t) =1

2‖u‖2

H + E(u)− E + ε‖u‖αV ′〈M(u), u〉V ′ +∫ +∞

t

〈h(s), u(s)〉H

+ε

2

(1 + α)2

1− α

∫ +∞

t

‖u‖αV ′‖h(s)‖2V ′ ds,

It is clear thatlimt→+∞

H(t) = 0

On the other hand H is differentiable at any point where u does not vanish and at thosepoints we have

H ′(t) = −〈u, g(u)〉V,V ′ + ε‖u‖αV ′〈M′(u)u, u〉V ′++ε‖u‖αV ′〈M(u), h(t)〉V,V ′ − ε‖u‖αV ′〈M(u), g(u)〉V ′ − ε‖u‖αV ′‖M(u)‖2

V ′+

εα‖u‖α−2V ′ 〈M(u), u〉V ′〈u, h〉V,V ′ − εα‖u‖α−2

V ′〈M(u), u〉2V ′−

εα‖u‖α−2V ′ 〈M(u), u〉V ′〈u, g(u)〉V,V ′ −

ε

2

(1 + α)2

1− α‖u‖αV ′‖h(s)‖2

V ′ .

By using Cauchy-Schwarz inequality, together with assumption (7.2) we obtain

H ′(t) ≤ −1

2〈u, g(u)〉V,V ′−

ρ1

2‖u‖α+2

H −ε(1−α)‖u‖αV ′‖M(u)‖2V ′+ε(1+α)‖u‖αV ′‖M(u)‖V ′‖h‖V ′+

ε‖u‖αV ′ |〈M′(u)u, u〉V ′|+ ε(1 + α)‖u‖αV ′‖M(u)‖V ′‖g(u)‖V ′ −ε

2

(1 + α)2

(1− α)‖u‖αV ′‖h‖2

V ′ .

It is not difficult to check that H is differentiable with derivative equal to 0 any point whereu = 0, therefore the above inequality is in fact valid everywhere. Since ‖M(u)‖V ′ is boundedand, by using Young’s inequality together with assumption (7.3) we get

‖g(u)‖V ′‖M(u)‖V ′ ≤ ρ2〈u, g(u)〉α+1α+2

V,V ′‖M(u)‖V ′ ≤ C1‖M(u)‖α+2V ′ + C2〈u, g(u)〉V,V ′ ,

31

where C1 and C2 are two positive constants.It is easy to see that

‖M(u)‖V ′‖h‖V ′ ≤1

2

1− α1 + α

‖M(u)‖2V ′ +

1 + α

1− α‖h‖2

V ′

.

Since limt→+∞

‖u(t)‖H = 0, there exists T such that for all t ≥ T we have ‖u‖H ≤ 1. Then we

get for all t ≥ T

H ′(t) ≤ (−ρ1

2+ ε(1 + α)C3)‖u‖α+2

H − ε1− α2‖u‖αV ′‖M(u)‖2

V ′+

ε‖u‖αV ′|〈M′(u)u, u〉V ′ |+ ε(1 + α)C4‖u‖αV ′‖M(u)‖α+2V ′ .

By assumption (7.5), and by choosing ε small enough we have for all t ≥ T

H ′(t) ≤ −εC5‖u‖αV ′(‖u‖2H + ‖M(u)‖2

V ′).

which also implies for all t ≥ T

H ′(t) ≤ −εC6‖u‖αV ′(‖u‖H + ‖M(u)‖V ′)2. (7.10)

From (7.10) we deduce that H is nonincreasing on [T,+∞[ and in particular H(t) ≥ 0 on R+.If for some t0 ≥ T it happens that H(t0) = 0, then H(t) vanishes on [t0,∞) and by (7.10)we conclude that ‖u‖αV ′‖u‖2

H = 0, hence u(t) is constant for t large and there is nothing toprove.

From now on we assume that for all t ≥ T we have H(t) > 0. Let δ > 0 be as in (7.6).Let θ ∈] α

α+1, 1

2] be the Lojasiewicz exponent and let β = θ − α(1− θ). We have

− 1

β

d

dt(H(t)β) =

−H ′(t)H(t)1−θ1+α

. (7.11)

By applying Cauchy-Schwarz inequality we obtain

H(t)1−θ ≤ C7

‖u‖2(1−θ)

H + |E(u)− E|1−θ + ‖u‖(α+1)(1−θ)V ′ ‖M(u)‖1−θ

V ′ +

+

(∫ +∞

t

|〈h, u〉H | ds)1−θ

+

(∫ +∞

t

‖h(s)‖2Hds

)1−θ.

and we observe that, assuming T large enough to insure distV (u, E∗) ≤ ρ for all t ≥ T , wehave |E(u)− E|1−θ ≤ C‖M(u)‖V ′ . Moreover by Young’s inequality and since ‖u‖H ≤ 1 forall t ≥ T we get

‖u‖(α+1)(1−θ)V ′ ‖M(u)‖1−θ

V ′ ≤ C7(‖u‖V ′ + ‖M(u)‖V ′).Now, let

E(t) =1

2‖u‖2

H + E(u)− E.

32

We have thanks to assumption (7.2)

E ′(t) = −〈u, g(u)〉V,V ′ + 〈h, u〉H ≤ −ρ1‖u‖α+2H + ‖h‖H‖u‖H .

By Young’s inequality we obtain

E ′(t) ≤ −ρ1

2‖u‖α+2

H + C8‖h‖α+2α+1

H .

By integrating we have∫ +∞

t

‖u‖α+2H ds ≤ C9E(t) + C10

∫ +∞

t

‖h‖α+2α+1

H ds.

Then we have

(

∫ +∞

t

|〈u, h〉H | ds)1−θ ≤ C11E(t)1−θ + C12(

∫ +∞

t

‖h‖α+2α+1

H ds)1−θ.

Now, as a consequence of (7.9) for t large enough we can use the Lojasiewicz gradientinequality. Combining with the last calculations and assumption (7.6), we obtain

H(t)1−θ ≤ C13(‖u‖H + ‖M(u)‖V ′ +1

(1 + t)ξ(1−θ)) (7.12)

where ξ = α+ 1 + δ(α+2α+1

). As in [3] we can replace, if necessary, θ by a smaller number still

greater than αα+1

, for which ξ(1 − θ) > 1, that we still call θ from now on. By combining(7.10), (7.11) and (7.12) we find

−C13d

dt(H(t)β) +

1

(1 + t)ξ(1−θ)≥ ‖u‖αV ′(‖u‖H + ‖M(u)‖V ′)2

(‖u‖H + ‖M(u)‖V ′ + 1(1+t)ξ(1−θ) )1+α

+1

(1 + t)ξ(1−θ).

By using lemma 4.2 in [3] this implies

−C13d

dt(H(t)β) +

1

(1 + t)ξ(1−θ)≥ C14‖u‖αV ′‖u‖1−α

H .

By using the embedding of H into V ′ we obtain

‖u‖V ′ ≤ −C15d

dt(H(t)β) +

C16

(1 + t)ξ(1−θ).

Hence by integrating, we get for all t ≥ T∫ t

T

‖u(s)‖V ′ ds ≤ C15(H(T )β) +C17

(1 + T )ξ(1−θ)−1.

This implies that limt→+∞

u(t) exists in V ′. By compactness, limt→+∞

u(t) exists in V.

Remark 7.2. It is easy to check that any g ∈ G(Z, α) for some Z satisfying the conditions(5.5), (5.6) and (5.7) verifies (7.2)and (7.3). Indeed (7.2) follows from (5.21) with p = α+ 2by taking w = 0 and (7.3) follows easily from the combination of (6.5) and (6.6) appliedwith w = 0.

33

7.2 A semilinear convergence theorem

In this section we consider the semilinear equation

u(t) + g(u(t)) + Au(t) + f(u(t)) = h(t), t ∈ R+, (7.13)

where A is as in the previous sections, g satisfies (7.2)-(7.3) and f = ∇F ∈ C(V, V ′) is thegradient operator of F ∈ C2(V ). We assume

i) There is a Banach space W ⊂ H such that V ⊂ W with compact imbedding for whichf : W −→ H is Lipschitz continuous on bounded sets of W .

ii) The functional E(u) := 12‖u‖2+F (u) satisfies a uniform Lojasiewicz gradient inequality

(7.4) with exponent θ ∈(

αα+1

, 12

]near the set of equilibria

E =M−10 = u ∈ V,Au+ f(u) = 0.

iii) For all u ∈ V we have A−1f ′(u) ∈ L(H,H) with

∀u ∈ V, ‖A−1f ′(u)‖L(H,H) ≤ C(‖u‖) (7.14)

where C(R) is bounded on bounded subsets of R+.

We obtain

Theorem 7.3. Let u ∈ W 1,1loc (R+, V ) ∩W 2,1

loc (R+, H) be a solution of (7.13) such that u hasbounded range in V and u has bounded range in H. Assume in addition that g satisfies (5.2),(5.3) and (5.4), and that h satisfies (7.6). Then we have for some a ∈ E

limt→∞

(‖u(t)− a‖+ |u(t)|) = 0

Proof. First we observe that under condition i) the function p(t) := f(u(t) is uniformlycontinuous: R+ → H . Indeed since u is bounded in W and Lipschitz continuous : R+ → H

|f(u(t+ a)− f(u(t)| ≤ C‖u(t+ a)− u(t)‖W ≤ ε‖u(t+ a)− u(t)‖V +K(ε)‖u(t+ a)− u(t)‖H

≤ ε‖u(t+ a)− u(t)‖V +MK(ε)|a| ≤ 2ε supt≥0‖u(t)‖V +MK(ε)|a|

and the result follows immediately. Then since h tends to 0 it is clear that k := h− p is S1-uniformly continuous. As a conserquence of Theorem 5.1, we deduce that u has precompactrange in V and u has precompact range in H. Then Theorem 7.3 becomes an immediateconsequence of Theorem 7.1.

34

8 Applications.

8.1 Some classes of admissible damping terms

When H = L2(Ω, dµ) with Ω, µ some finitely measured space, an important class of dampingterms satisfying the hypotheses of Theorems 4.1- 5.1 is the class of Nemytsckii operatorsassociated to a numerical non-decreasing function γ ∈ C(R,R) which means that

∀v ∈ V, g(v)(x) = γ(v(x)), µ− a.e in Ω

Assuming that γ satisfies

∃c0 > 0, ∃c1 ≥ 0 ∀s ∈ R γ(s)s ≥ c0s2 − c1

the coerciveness condition will be satisfied whatever be the space V as in subsection 3.1 withC1 = c1|Ω|. In this section we consider different kinds of damping operators of non local orsemi-local type. The first class that we consider corresponds to the gradient of the convexfunctional

Φ1(v) :=c

α + 2|A

β2 v|α+2

Proposition 8.1. Let α > 0, β ≥ 0, c > 0 Then the operator defined by

g(v) := c|Aβ2 v|αAβv

with β < 1 is (Z, α)-admissible with Z = D(Aβ2 ).

Proof. The result will in fact be a consequence of the following more general property.

Proposition 8.2. let Z be a reflexive banach space satisfying the conditions (5.5), (5.6) and(5.7). Let C ∈ L(Z,H) be one-to one. Then the operator defined by

g(v) := |Cv|αC∗Cv

is (Z, α)-admissible.

Proof. First we show that (6.5) is an immediate consequence of Lemma 2.3. Indeed we have

∀(v, w) ∈ V × V, 〈g(v)− g(w), v − w〉 = 〈|Cv|αC∗Cv − |Cw|αC∗Cw, v − w〉

= 〈C∗(|Cv|αCv − |Cw|αCw), v − w〉 = 〈|Cv|αCv − |Cw|αCw,Cv − Cw〉

≥ c(α)|Cv − Cw|α+2

by Lemma 2.3. By Banach’s theorem we have C−1 ∈ L(H,Z) and therefore

|Cv − Cw| ≥ δ‖v − w‖Z

35

hence the result follows immediately. Next we show that (6.6) is a consequence of Lemma

2.2. Indeed we have

∀(v, w) ∈ V × V, ‖g(v)− g(w)‖Z′ ≤ ‖C∗‖L(H,Z′)||Cv|αCv − |Cw|αCw|

≤ (α + 1)‖C∗‖L(H,Z′)(|Cv|α + |Cw|α)|Cv − Cw|and the result follows immediately since C ∈ L(Z,H) .

Another interesting class is the following class of ”semi-local” damping terms correspond-ing to to the gradient of the convex functional

Φ2(v) :=c

α + 2

∫Ω

|(Dv)(x)|α+2X dx

whereX is a finite dimensional Hilbert space with norm denoted by |.|X andD ∈ L(V, Lα+2(Ω, X)),cf also [15] for results of weak convergence involving this kind of damping term in presenceof almost periodic forcing.

Proposition 8.3. Let X be a Hilbert space with norm denoted by |.|X , Y = Lα+2(Ω, X),Z be as in the previous result and and D ∈ L(Z,Y) be be one-to one. Then the operatordefined by

g(v) := cD∗(|Dv|αXDv)

is (Z, α)-admissible.

Proof. We have, applying Lemma 2.3 in the Hilbert space X and using D−1 ∈ L(Y , Z)

∀(v, w) ∈ V × V, 〈g(v)− g(w), v − w〉 = 〈cD∗(|Dv|αXDv)− cD∗(|Dw|αXDw), v − w〉

= 〈|Dv|αXDv − |Dw|αXDw,Dv −Dw〉Y ′,Y = c

∫Ω

(|Dv|αXDv − |Dw|αXDw), Dv −Dw)Xdx

≥ c′∫

Ω

|Dv −Dw|α+2X dx ≥ c′′‖v − w‖α+2

Z

This proves (6.5). In order to check (6.6) we write, using Lemma 2.2 in the Hilbert space X

∀(v, w) ∈ V × V, ‖g(v)− g(w)‖Z′ ≤ ‖D∗‖L(Y ′,Z′)‖|Dv|αXDv − |Dw|αXDw‖Y ′

≤ K

∫Ω

||Dv|αXDv−|Dw|αXDw|α+2α+1

X dx

α+1α+2

≤ CK

∫Ω

(|Dv|αX+|Dw|αX)α+2α+1 |Dv−Dw|

α+2α+1

X dx

α+1α+2

≤ K ′∫

Ω

(|Dv|α+2X + |Dw|α+2

X )dx

αα+2∫

Ω

|Dv −Dw|α+2X )dx

1α+2

and the result follows easily since D ∈ L(Z,Y) .

36

8.2 Almost periodic forcing in presence of a non-local damping

In this Section, Ω denotes a bounded open domain of RNwith C2 boundary and α ≥ 0, c > 0.We consider the 5 following special cases of (6.2)

Example 1: the wave equation with nonlinear averaged dampingutt + c[

∫Ωut

2(t, x)dx]α2 ut −∆u = h(t, x), in R+ × Ω,

u = 0 on R+ × ∂Ω.(8.1)

Here V = H10 (Ω) and H = L2(Ω) = Z.

Example 2: A clamped plate equation with nonlinear structural averaged damp-ing

utt − c[∫

Ω|∇ut|2dx]

α2 ∆ut + ∆2u = h(t, x), in R+ × Ω,

u = |∇u| = 0 on R+ × ∂Ω,(8.2)

Here V = H20 (Ω), H = L2(Ω) and Z = H1

0 (Ω).

Example 3: A simply supported plate equation with nonlinear structural aver-aged damping

utt − c[∫

Ω|∇ut|2dx]

α2 ∆ut + ∆2u = h(t, x), in R+ × Ω,

u = ∆u = 0, on R+ × ∂Ω,(8.3)

Here V = H2 ∩H10 (Ω), H = L2(Ω)and Z = H1

0 (Ω).

Example 4: A clamped plate equation with a semi-local non linear dampingutt − c div(|∇ut|α∇ut) + ∆2u = h(t, x), in R+ × Ω,

u(t, x) = |∇u| = 0, on R+ × ∂Ω,(8.4)

Here V = H20 (Ω) and H = L2(Ω).

Example 5: A simply supported plate equation with a semi-local non lineardamping

utt − c div(|∇ut|α∇ut) + ∆2u = h(t, x), in R+ × Ω,

u(t, x) = ∆u = 0, on R+ × ∂Ω,(8.5)

Here V = H2 ∩H10 (Ω) and H = L2(Ω).

The following result is an immediate consequence of Theorem 6.8 and the propertiesestablished in Section 8.1.

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Theorem 8.4. Let α ≥ 0 be arbitrary with the restriction (N − 2)α < 4 in examples 4-5.Then for any h which is S1-almost periodic: R → H = L2(Ω) each equation above has aunique solution ω such that the vector (ω, ω) is almost periodic R → V × H . In additionfor any other solution u we have for some M > 0

∀t ≥ 0, ‖u(t)− ω(t)‖+ |u(t)− ω(t)| ≤M(1 + t)−1α

Proof. The only thing to check is that in all the examples, the operator g is (Z, α)- admissiblefor some relevant choice of Z. In examples 1,2, 3 we apply Proposition 8.2 and in examples4-5 we use Proposition 8.3 with Y = Lα+2(Ω,RN) and Z the closure of V in H2−ε for someε > 0. We skip the details.

8.3 Convergence in presence of a non-local damping

We now give some generalizations of the main infinite-dimensional result from [3]. Thespaces V and H are the same as in the corresponding examples in the previous subsection

Example 6: the wave equation with averaged damping

Under the following assumptions on f and h :

f : R→ R is analytic, (8.6)

there exists C ≥ 0 and η > 0 with (N − 2)η < 2 such that :

|f ′(s)| ≤ C(1 + |s|η) on R, (8.7)

we consider the equationutt + c[

∫Ωut

2(t, x)dx]α2 ut −∆u+ f(u) = h(t, x), in R+ × Ω,

u = 0 on R+ × ∂Ω.(8.8)

Example 7: A clamped plate equation with nonlinear structural averaged damp-ing

Under the following assumptions on f :

f : R→ R is analytic, (8.9)

there exists C ≥ 0 and η > 0 with (N − 4)η < 4 such that :

|f ′(s)| ≤ C(1 + |s|η) on R, (8.10)

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we consider the equationutt − c[

∫Ω|∇ut|2dx]

α2 ∆ut + ∆2u+ f(u) = h(t, x), in R+ × Ω,

u = |∇u| = 0, on R+ × ∂Ω,(8.11)

Example 8: A simply supported plate equation with with nonlinear structuralaveraged damping

Assuming that f satisfies (8.9) and (8.10) we consider the equationutt − c[

∫Ω|∇ut|2dx]

α2 ∆ut + ∆2u+ f(u) = h(t, x), in R+ × Ω,

u = ∆u = 0, on R+ × ∂Ω,(8.12)

Example 9: A clamped plate equation with a semi-local non linear damping

Assuming that f satisfies (8.9) and (8.10), assuming in addition (N−2)α < 4 we considerthe equation

utt − c div(|∇ut|α∇ut) + ∆2u+ f(u) = h(t, x), in R+ × Ω,

u = |∇u| = 0, on R+ × ∂Ω,(8.13)

Example 10: A simply supported plate equation with a semi-local non lineardamping

Assuming that f satisfies (8.9) and (8.10), assuming in addition (N−2)α < 4 we considerthe equation

utt − c div(|∇ut|α∇ut) + ∆2u+ f(u) = h(t, x), in R+ × Ω,

u = ∆u = 0, on R+ × ∂Ω,(8.14)

The following result is an immediate consequence of Theorem and the properties estab-lished in Section 8.1.

Theorem 8.5. Define V and H as in the examples of Section 8.2. Let u ∈ W 1,1loc (R+, V ) ∩

W 2,1loc (R+, H) be a solution of one of the above equations. Assume in addition that F (s) =∫ s

0f(s)ds is bounded from below, that α is small enough (depending on f ) and that h satisfies

∃K ≥ 0, ∃δ > 0, ‖h(t, ·)‖L2(Ω) ≤K

(1 + t)1+δ+α, for all t ∈ R+. (8.15)

Then we have for some ϕ ∈ V solution of the corresponding elliptic problem

−∆ϕ+ f(ϕ) = 0 (example 6)

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or∆2ϕ+ f(ϕ) = 0 (examples 7-10)

with the relevant boundary conditions

limt→∞

(‖u(t)− ϕ‖+ |u(t)|) = 0.

Proof. From the hypothesis on f it is easy to check that all solutions have a bounded energyand the set of stationary solutions is compact, hence the potential energy E satisfies (7.4).Then we apply Theorem 7.3. We skip the details which are rather classical.

Acknowledgments. This work was initiated during a one month sojourn of the secondauthor at the Laboratoire Jacques-Louis Lions, supported by the GDRE “CONEDP”.

References

[1] L. Amerio & G. Prouse, Uniqueness and almost-periodicity theorems for a non linearwave equation, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 8, 46 (1969) 1–8.

[2] I. Ben Hassen, Decay estimates to equilibrium for some asymptotically autonomous semi-linear evolution equations, .Asymptot. Anal. 69 (2010), no. 1-2, 31–44.

[3] I. Ben Hassen & L. Chergui, Convergence of global and bounded solutions of somenonautonomous second order evolution equations with nonlinear dissipation, J. Dynam.Differential Equations 23(2011), no. 2, 315-332.

[4] I. Ben Hassen & A. Haraux, Convergence and decay estimates for a class of secondorder dissipative equations involving a non-negative potential energy, J. Funct. Anal.260(2011), no. 10, 2933–2963.

[5] M. Biroli, Bounded or almost periodic solution of the non linear vibrating membraneequation, Ricerche Mat. 22 (1973), 190-202.

[6] M. Biroli, & A. Haraux, Asymptotic behavior for an almost periodic, strongly dissipa-tive wave equation. J. Differential Equations 38 (1980), no. 3, 422-440.

[7] H. Brezis, Operateurs maximaux monotones et semi-groupes de contractions dans lesespaces de Hilbert North-Holland Mathematics Studies, No. 5, Amsterdam-London(1973)183 pp.

[8] H. Brezis, Equations et inequations non lineaires dans les espaces vectoriels en dualiteAnn. Inst. Fourier (Grenoble) 18 (1968) fasc. 1, 115-175.

40

[9] L. Chergui, Convergence of global and bounded solutions of a second order gradient likesystem with nonlinear dissipation and analytic nonlinearity, J. Dynam. Diff. Equa. 20(2008), 643-652.

[10] L.Chergui, Convergence of global and bounded solutions of the wave equation with non-linear dissipation and analytic nonlinearity, J. Evol. Equ. 9 (2009), 405-418.

[11] R.Chill & M.A. Jendoubi, Convergence to steady states in asymptotically autonomoussemilinear evolution equations,, Nonlinear Anal. 53 (2000), no 7-8, 1017–1039.

[12] A. Haraux, Nonlinear evolution equations- Global behavior of solutions, Lecture Notesin Mathematics 841, Springer-Verlag, Berlin-New York, (1981). 313 pp.

[13] A. Haraux, Damping out of transient states for some semi-linear, quasi-autonomoussystems of hyperbolic type, Rc. Accad. Naz. Sci. dei 40 (Memorie di Matematica) 1017, fasc.7 (1983), 89–136.

[14] A. Haraux, Semi-linear hyperbolic problems in bounded domains, Mathematical reportsVol 3, Part 1 (1987), J. Dieudonne Editor, Harwood Academic Publishers, Gordon &Breach.

[15] A. Haraux, Systemes Dynamiques Dissipatifs et Applications, R.M.A.17 Masson, Paris(1991).

[16] A. Haraux & M.A. Jendoubi, Convergence of bounded weak solutions of the wave equa-tion with dissipation and analytic nonlinearity. Calc. Var. Partial Differential Equations9 (1999), no. 2, 95–124.

[17] A. Haraux & M.A. Jendoubi, Decay estimates to equilibrium for some evolution equa-tions with an analytic nonlinearity. Asymptotic Analysis 26 (2001), no. 1, 21–36.

[18] A. Haraux, M.A. Jendoubi & O. Kavian, Rate of decay to equilibrium in some semilinearparabolic equations, J. Evol. Equ. 3 (2003), no. 3, 463–484.

[19] A. Haraux, E. Zuazua, Decay estimates for some semilinear damped hyperbolic problems,Arch. Rational Mech. Anal.100 (1988), 191–206

[20] S.Z. Huang & P. Takac, Convergence in gradient-like systems which are asymptoticallyautonomous and analytic, Nonlinear Anal. 46 (2001), no. 5, Ser. A: Theory Methods,675-698.

[21] M.A. Jendoubi, Convergence of global and bounded solutions of the wave equation withlinear dissipation and analytic nonlinearity. J. Differential Equations 144 (1998), no. 2,302–312.

41

[22] S. Lojasiewicz, Une propriete topologique des sous-ensembles analytiques reels, Collo-ques internationaux du C.N.R.S.: Les equations aux derivees partielles, Paris (1962),Editions du C.N.R.S., Paris, 1963, pp. 87–89.

[23] S. Lojasiewicz, Ensembles semi-analytiques, Preprint, I.H.E.S. Bures-sur-Yvette, 1965.

[24] G. Prouse, Soluzioni quasi-periodiche dell’equazione non omogenea delle onde, con ter-mine dissipativo non lineare. I.-IV Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat.Natur. 8, 38-39 (1965)

[25] E. Zuazua, Stability and decay for a class of nonlinear hyperbolic problems, AsymptoticAnal. 1 (1988), no. 2, 161–185.

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