Compactness of trajectories to some nonlinear second order evolution equations and applications.
Faouzia ALOUI
UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France.
[email protected] Imen BEN HASSEN
Institut pr´eparatoire aux ´etudes d’ing´enieur de Bizerte BP 64, 7021 Jarzouna, Bizerte, Tunisia.
imen [email protected] 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, Boˆıte courrier 187, 75252 Paris Cedex 05, France.
[email protected] Abstract
Under suitable growth and coercivity conditions on the nonlinear damping operator g, we establish boundedness or compactness properties of trajectories to the equation
¨
u(t) +g( ˙u(t)) +Au(t) =h(t), t∈R+,
whereA is a positive selfadjoint operator andg is a nonlinear damping operator. The compactness results are used to prove the existence of almost periodic solutions when h is almost periodic, and to generalize some recent results of Chergui and Ben Hassen- Chergui concerning convergence to equilibrium when a nonlinear term depending onu is added and h dies off sufficiently fast fortlarge.
AMS classification numbers: 34A34, 34D20, 35B40, 35L10, 35L90
Keywords: Second order equation, compactness, almost periodic , Lojasiewicz inequality.
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, V0) is a symmetric, positive, coercive operator , g ∈ C(V, V0) is monotone, f is a gradient operator satisfying some appropriate conditions and h is a forcing term. We are especially interested in the two following cases
1)f = 0 andh is almost periodic.
2) h tends to 0 sufficiently fast at infinity in t and f is the gradient of an analytic functional or more generally the gradient of a potential satisfying the Lojasiewicz gradient in 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 of decay 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 ≥ 0 is sufficient in the linear case and when h is time-periodic, the treatment of the general case has always required the existence of a precompact trajectory, and the problem is that it is difficult to distinguish between different trajectories at this level: if we were able to exhibit a precompact orbit without knowing anything about the others, it would mean that we can (by going to the positive ω-limit set) localize an almost periodic orbit and this is precisely what becomes impossible in the nonlinear case. Therefore we are condemned to prove compactness of all trajectories or nothing (note that for g tangent to 0 at the origine, extra regularity of the initial state or even of the forcing term will not help when t becomes large). Another difficulty is the following: if the existence of bounded trajectories can be proved by combining a coerciveness property of g “at infinity” with some growth restrictions, compactness requires a global, uniform kind of coerciveness. In the past more and more general compactness results have been obtained, but only when g is a Nemytskii type operator. When g is a non-local operator or involves differential operators in space, the theory 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 significative advances have been done, cf.e.g. [21, 16, 17, 18, 20, 11, 9, 10, 2, 3, 4], the other tool here being the Lojasiewicz gradient inequality [22, 23]. But here even the caseh= 0 is non-trivial since the set of equilibria needs not have any particular structure except for the restrictions induced by the existence of a Lojasiewicz inequality : we know for instance in advance that the potential energy is constant on continua inside the set of equilibria, a property which can fail for C∞ and even Gevrey potentials. The fact that precompactness of trajectories had been proved only for Nemytskii type damping operators limited until now the convergence results with non-linear damping to those damping operators. Therefore the second innovation of this paper is to contain the first convergence results in case 2) in presence of a non-local damping term.
The plan of the paper is as follows: In Section 2, we introduce the basic tools used in the statements and proofs of the main results. Section 3 is devoted to the initial value problem for (1.1). Sections 4 and 5 contain the statement and proof of the boundedness result and the compactness result, respectively. Sections 6 and 7 contain respectively the statement and proof of the asymptotic almost periodicity and the semilinear convergence result, respectively. Finally Section 8 is devoted to the application to PDE models with non-local damping terms.
2 Some useful tools.
In this section, we collect quite a few results of general interest which will reveal essential for the proofs of our main results. We also need to recall the definitions of some well known mathematical objects as well as their basic properties in the exact functional framework that shall 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 map A defined on a part D=D(A) with values in H is monotone if
∀(U,Uˆ)∈ D × D, (AU − AU , Uˆ −Uˆ)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) ∈ W1,1(0, T;H) there is a unique function U ∈ W1,1(0, T;H) with U(t) ∈ D(A)) for almost all t ∈(0, T) , U(0) = U0 and such that for almost all t∈(0, T)
U0(t) +AU(t) = F(t). (2.1)
In addition if for some Uˆ0 ∈ D(A) and Hˆ ∈ W1,1(0, T;H) we consider the solution Uˆ ∈ W1,1(0, T;H) with Uˆ(t)∈D(A)) for almost all t∈(0, T) , Uˆ(0) = ˆU0 of
Uˆ0(t) +AU(t) = ˆˆ F(t) then the difference satisfies the inequality
∀t∈[0, T], |U(t)−U(t)| ≤ˆ |U0−Uˆ0|+ Z t
0
|F(s)−Fˆ(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 thatU(0) =U0, cf. H. Brezis [7].
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 h,i. Then for any 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 orw 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, h|v|αv− |w|αw, v−wi ≥2−α|v−w|α+2 (2.3) with equality if and only if w=−v.
Proof. We set u =−w and P = h|v|αv− |w|αw, v −wi = h|u|αu+|v|αv, u+vi so that we are 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)hu, vi
=|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 hence
2P
|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 obtain
P
|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 ifu=tv. To conclude it is sufficient to observe thatf(t) = (1+t)1+tα+1α+1
is decreasing on (0,1) withf(1) = 2−α. Indeed setting h(t) = −lnf(t) we have
∀t∈(0,1), h0(t) = (α+ 1)(1−tα) (1 +t)(1 +tα+1) >0
2.3 Almost periodic functions
A typical almost periodic numerical function is the sum of two periodic functions with incommensurable periods. Such objects often appear in the mechanics of vibrating systems, and sometimes infinite sums naturally impose their presence, for instance when studying the energy 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 functionf :R→X is almost periodic iff the set of translates
[
α∈R
{f(.+α}=T(f)
is precompact in the space Cb(R, X) endowed with the topology of uniform convergence R→X.
It follows clearly from the definition that
i) For anyT > 0, any continuous T-periodic function: f :R→X is almost periodic with T(f) compact.
ii) For 3 complete metric spaces X, Y, Z, if f : R → X and g : R → Y are- almost periodic 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 almost periodic 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 precompact range.
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 → X when can we garantee that the ‘response ”, i.e. the general solution of the evolution equation
U0(t) +AU(t) =F(t)
adapts to F in the sense that it asymptotes an almost periodic function for t large? This problem is difficult even for ODEs and has only received a partial answer even in the case of equation (2.1), cf.e.g.[1, 5, 6, 12, 15, 14].
2.4 Stepanov spaces and generalized almost periodicity
Definition 2.5. Given a real Banach space X with norm k.k and an infinite interval J = [t0,+∞) we set
S1(J, X) = {f ∈L1loc(J, X), sup
t∈J
Z t+1
t
kf(s)kds <∞}
where “S” stands for Stepanov. It is immediate to check that S1(J, X) endowed with the semi-norm
kfkS1(J,X)= sup
t∈J
Z t+1
t
kf(s)kds
is a Banach space containing L∞(J, X). One similarly defines for any p≥1 Sp(J, X) ={f ∈Lploc(J, X), sup
t∈J
Z t+1
t
kf(s)kpds <∞}
and we may complete by setting S∞(J, X) = L∞(J, X). When X = R we simplify the notation:
∀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 response to an almost periodic forcing asymptotes an almost periodic function fort large, most of the time we can afford discontinuous locally integrable forcings thanks to the smoothing effect of integration.
Definition 2.6. We say that f ∈ Sp(R, X) is Sp- almost periodic : R→X if the function F(α) :f( +α) :=Tαf is almost periodic : R→Sp(R, X)
It is clear that usual (continuous) almost periodic functions are Sp- almost periodic for all 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+,
Z t+1
t
α(r)dr≥α0 >0 (2.4)
Let Φ∈L1loc(R+),Φ≥0 satisfy the differential inequality
Φ0(t) +α(t)Φ(t)≤F(t), a.e. on R+ (2.5)
Then Φ is bounded and we have
∀t∈R+, Φ(t)≤eA0{e−α0tΦ(0) + (1 + 1
α0)kFkS1} (2.6) with
A0 =α0+kαkS1
Proof. We observe that for any (s, t)∈R2 with 0≤s≤t we have Z t
s
α(r)dr ≥α0(t−s)−A0 Indeed setting t=s+n+ρ with n∈N,0≤ρ <1 we have
Z t
s
α(r)dr = Z s+n
s
α(r)dr+ Z t
s+n
α(r)dr≥nα0− kαkS1 ≥(t−s−1)α0− kαkS1 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], Z 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
Φ0(t) +α(t)Φ(t)≤F(t), a.e. on R+ (2.8) Then Φ is bounded and we have
∀t∈R+, Φ(t)≤eA0{e−α0tΦ(0) + (1 + 1
α0)kFkS1} (2.9) Proof. Introducing A(t) =Rt
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) + Z t
0
eA(s)F(s)ds hence
Φ(t)≤e−A(t)Φ(0) + Z t
0
eA(s)−A(t)F(s)ds
and by using (2.7), we find
Φ(t)≤eA0{e−α0tΦ(0) + Z t
0
eα0(s−t)F(s)ds}
The conclusion follows easily from the next simple lemma.
Lemma 2.9. Let
I(t) :=
Z t
0
eα0(s−t)F(s)ds Then I is bounded and we have
∀t ∈R+, I(t)≤(1 + 1−e−α0t
α0 )kFkS1 (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) :=
Z t
s
F(r)dr Then
∀t∈R+,∀s∈[0, t], g(s)≤(t−s+ 1)kFkS1 (2.11) and
eαtI(t) =− Z t
0
eαsg0(s)ds =−[eαsg(s)]t0+ Z t
0
αeαsg(s)ds =g(o) + Z t
0
αeαsg(s)ds and by (2.11) we obtain
eαtI(t)≤ kFkS1n
t+ 1 + Z t
0
α(t−s+ 1)eαsdso Finally, let
J(t) := t+ 1 + Z t
0
α(t−s+ 1)eαsds We find
J(t) =t+ 1 +eαt−1 + Z t
0
α(t−s)eαsds=t+eαt+ Z t
0
α(t−s)eαsds But integrating by parts again we obtain
Z t
0
α(t−s)eαsds = [(t−s)eαs]t0− Z t
0
(−1)×eαsds=−t+ Z t
0
eαsds=−t+ eαt−1 α
Finally
eαtI(t)≤ kFkS1h
eαt+ eαt−1 α
i
and the result follows.
The following more nonlinear Lemma will be useful for the proof of our main boundedness result (cf. Theorem 4.1).
Lemma 2.10. Let Ψ≥0 be a function in Wloc1,1(R+) which satisfies d
dtΨ≤ −µ(t)Ψ12 +F(t) (2.12) with µ∈S1(R+) and
∀t≥0,
Z t+1
t
µ(s)ds≥ρ >0 Then Ψ(t) :R+ →R+ is bounded.
Proof. We set
F∗ =kFkS1(R+); µ∗ =kµkS1(R+)
As a consequence of Gronwall’s inequality, Ψ(t) is bounded on [0,1] in terms of Ψ(0), µ∗and F∗. 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. Since
d
dtΨ≤µ−(τ)Ψ(τ) +F(τ) + 1
4µ−(τ).
By Gronwall’s inequality, we obtain:
Ψ(t+ 1) ≤K1exp(µ∗) + exp(µ∗)(F∗+ 1
4µ∗) =K2. (2.13) case 2: Assume that we have:
s∈[t,t+1]inf 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
Consequently
d
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≤1
Ψ(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 respectively denoted by k.k and |.|. We assume that V is densely and continuously embedded into H.
Identifying H with its dual H0, we obtain V ,→ H = H0 ,→ V0. We denote inner products by (.,.) and duality products by h·,·i; the spaces in question will be specified by subscripts.
The notation hf, ui without any subscript will be used sometimes to denote hf, uiV0,V. The duality map: V →V0will be denoted byA. We recall thatAis characterized by the property
∀(u, v)∈V ×V, hAu, viV0,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, V0) satisfies
∀(v, w)∈V ×V, hg(v)−g(w), v−wi ≥0 (3.2) We consider the (generally unbounded) operatorA defined on the Hilbert spaceH=V ×H by
D(A) = {(u, v)∈V ×V, Au+g(v)∈H}
and
∀(u, v)∈D(A), A(u, v) = (−v, Au+g(v))
Lemma 3.1. The operator A is maximal monotone.
Proof. Let U = (u, v) and ˆU = (ˆu,v) be two elements ofˆ D(A). We have
(AU− AU , Uˆ −Uˆ)H=−(u−u, vˆ −v)ˆ V + (Au+g(v)−Auˆ−g(ˆv), v−v)ˆ H
−(u−u, vˆ −ˆv)V +hAu+g(v)−Aˆu−g(ˆv), v−ˆviV0,V since Au+g(v)∈H and Aˆu+g(ˆv)∈H while v,vˆare in V . This reduces to
(AU − AU , Uˆ −Uˆ)H=hg(v)−g(ˆv), v−vˆiV0,V ≥0
Hence A is monotone. To prove that A is maximal monotone we are left to show that for any 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∈V0
But now the operator C ∈C(V, V0) defined by
∀v ∈V, Cv =Av+g(v) +v
is continuous and coercive: V →V0. Therefore by Corollary 14 p. 126 from H. Brezis [8], C is surjective. Finally A is maximal monotone as claimed.
As a consequence of Proposition 2.1, for anyh∈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 each compact interval [0, T] by approximating the initial data by elements of the domain, the forcing term h by C1 functions and passing to the limit: the limit is independent of the approximating elements so chosen. The next result shows that in fact the approximation can 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 ∈Wloc1,1(R+, V)∩ Wloc2,1(R+, H) of
¨
w+Aw+g( ˙w) = k(t); w(0) =w0, w(0) =˙ w1 satisfies
∀t≥0, ku(t)−w(t)k+|u(t)˙ −w(t)| ≤˙ δ
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,
Z n+1
n
|k(s)−h(s)|ds≤δ2−n−1
Choosing (w0, w1)∈D(A) such that kw0−u0k+|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
Z 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 the case of possibly non-local damping terms.
Theorem 4.1. Assume that (4.1) is satisfied and g ∈C(V, V0) satisfies the conditions
∃α >0, ∃C1 ≥0 ∀v ∈V, hg(v), vi ≥α|v|2−C1. (4.2)
∃τ >0, ∃C2 ≥0 ∀v ∈V, kg(v)k∗ ≤C2+τhg(v), vi. (4.3) Assume, also, that the following condition is fulfilled:
2τ h∗ <1 (4.4)
Then any solution u∈C(R+, V)∩C1(R+, H) of (3.1) is bounded on R+ in the sense that u has 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. we assume u∈Wloc1,1(R+, V)∩Wloc2,1(R+, H). The general case will follow by density. Let
E(t) = 1
2(|u|˙ 2+kuk2)
Under the regularity conditions [u0, v0] ∈ V × V, γ(0, v0) ∈ H and h ∈ Wloc1,1(R+, H), t →E(t) is absolutely continuous and we have ∀t∈R+:
d
dtE(t) = (h,u)˙ − hg( ˙u),ui.˙ (4.5)
In addition t→(u(t),u(t)) is absolutely continuous and˙ d
dt(u(t),u(t)) =˙ |u|˙ 2− kuk2− hg( ˙u), ui+ (h, u).
By using (4.3), we obtain:
d
dt(u(t),u(t))˙ ≤ |u|˙ 2− kuk2+kuk {P|h|+C2+τhg( ˙u),ui}˙ with P = sup{|u|, u∈V, kuk= 1}. Introducing Φ(t) = 2E(t), ∀t ≥0, we find:
d dt
n
(1 + Φ)−12(u,u)˙ o
≤ |u|˙ 2− kuk2
(1 + Φ)12 +τhg( ˙u),ui˙ +P|h|+C2+ P|u|kuk˙
(1 + Φ)32
dE dt
(4.6)
We have:
|u|˙ 2 − kuk2
(1 + Φ)12 = 2|u|˙ 2+ 1
(1 + Φ)12 − 1 +|u|˙ 2+kuk2 (1 + Φ)12
≤ 1 + 2|u| −˙ (1 + Φ)12 (4.7)
Then from (4.5), we obtain by Cauchy- Schwarz:
|u|kuk˙ (1 + Φ)32
dE dt
≤ 1
2(1 + Φ)12
dE dt
≤ |h|+ 1
2(1 + Φ)12hg( ˙u),ui −˙ (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
dt{τΦ + (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)
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|Φ12 + 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 2τ h∗ <1.In addition for K >0 large enough, Ψ is positive on R+ and we have:
Ψ≤τΦ + (1 + Φ)−12|u||u|˙ +P
2(1 + Φ)12 +K
≤τΦ +P(1 + Φ)−12kuk|u|˙ + P
2(1 + Φ)12 +K
≤τΦ + P
2(1 + Φ)−12Φ + P
2(1 + Φ)12 +K
≤τΦ +P(1 + Φ)12 +K We notice that for any η >0
P(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)−12kuk|u|˙ +P
2(1 + Φ)12 +K
≥τΦ−P
2(1 + Φ)−12Φ + P
2(1 + Φ)12 +K
≥τΦ−P(1 + Φ)−12Φ + 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)
Assuming that K−c(η)≥0 andη < 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 +ηΨ12 + 2τ|h|
τ−ηΨ12 + 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 +ηΨ12 + 2τ|h|
τ−ηΨ12 + 2P|h|+ 1 +C2+C1τ +C3. (4.15) We set
σ = 1 τ +η m(t) = 2τ|h(t)|
τ −η and
F(t) = 2P|h(t)|+ 1 +C2+C1τ +C3. We have
d
dtΨ≤ −σΨ12 +m(t)Ψ12 +F(t). (4.16) As a consequence of (4.4), for η small enough, we have τ−η2τ h∗ < τ+η1 =σ hence
m∗ = sup
t≥0
Z t+1
t
m(s)ds < σ (4.17)
and
F∗ = sup
t≥0
Z t+1
t
F(s)ds
= sup
t≥0
Z 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+. Hence by (4.14), Φ(t) and E(t) are bounded for t ≥ 0. The general case of weak solutions follows easily by density by using Lemma 3.2.
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 sup
→0t≥0
Z t+1
t
|h(s+)−h(s)|ds= 0. (5.1)
ii) g ∈C(V, V0) satisfies g(0) = 0 and the following conditions
∀(v, w)∈V ×V, hg(v)−g(w), v−wiV0,V ≥0 (5.2)
∀δ >0,∃C(δ)>0, ∀(v, w)∈V ×V, |v−w|2 ≤δ+C(δ)hg(v)−g(w), v−wiV0,V (5.3)
∃C >0, ∀v ∈V, kg(v)kZ0 ≤C(1 +hg(v), viV0,V) (5.4) where Z0 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 inZ0 by||| |||∗.
We observe that Z0 ⊂ V0 with continuous imbedding. From (5.5) and (5.6) it follows that
∀α >0, ∃c(α)≥0 such that
∀u∈V, |||u||| ≤αkuk+c(α)|u| (5.8) We prove first the result under the additional assumption u∈ Wloc1,1(R+, V)∩Wloc2,1(R+, H).
First w(t) =u(t+)−u(t) satisfies the equation:
¨
w+Lw+g( ˙u(t+))−g( ˙u(t)) = h(t+)−h(t) (5.9) where u(t) is the solution of equation (3.1).
We set
f(t) = h(t+)−h(t) (5.10)
and
∀t∈R+,∀w∈V, g( ˙u(t) +w)−g( ˙u(t)) = γ(t, w). (5.11) Then (5.9) becomes
¨
w+Lw+γ(t,w˙) = f(t), ∀t≥0 (5.12) As a consequence of section 2, we know that u(t) and ˙u(t) are bounded on R+. We denote by
E(t) = 1
2{|u(t˙ +)−u(t)|˙ 2+ku(t+ε)−u(t)k2}
= 1
2{|w˙(t)|2+kw(t)k2} the energy of the solution to (5.12).
Let us introduce, for some β >0 to be chosen later Φ = 1
2{|w˙|2+kwk2}+β(w,w˙).
Then, we have:
Φ0 ≤ −hw˙, γ(t,w˙)i+β|w˙|2−βkwk2−βhγ(t,w˙), wi+hf,w˙+βwi we obtain
Φ0 ≤ −βE− hw˙, γ(t,w˙)i+ 3β
2 |w˙|2−β
2kwk2+β|||w||| |||γ(t,w˙)|||∗+|f|
|w˙|+β|w|
If we impose β ≤√
λ1 , then Φ≥0 and from (5.8) we deduce Φ0 ≤ −βE− hw˙, γ(t,w˙)i+ 3β
2 |w˙|2− β
2kwk2+β
αkwk+c(α)|w|
|||γ(t,w˙)|||∗ +|f|
|w˙|+kwk
From (5.3) and |w˙|+kwk ≤2√
E, we have:
Φ0 ≤ −βE− hw˙, γ(t,w˙)i+3β
2 δ+ 3β
2 K(δ)hw˙, γ(t,w˙)i − β 2kwk2 +β
αkwk+c(α)|w|
|||γ(t,w˙)|||∗+ 2p E|f|
≤ −βE+ (3β
2 K(δ)−1)hw˙, γ(t,w˙)i −β
2kwk2+β
αkwk+c(α)|w|
|||γ(t,w˙)|||∗ + 2p
E|f|+ 3β 2 δ
From now on we fix δ >0 small enough and we choose β such that 3β
2 K(δ)−1≤0 which is equivalent to
β ≤ 2 3K(δ). Then
Φ0 ≤ −βE+β
αkwk+c(α)|w|
|||γ(t,w˙)|k|∗+ 2p
E|f|+3β 2 δ.
We have by Cauchy-Schwarz:
Φ = 1
2{|w˙|2+kwk2}+β(w,w˙)
≤ E+β2
2 |w|2+ 1 2|w˙|2. Then, for β ≤√
λ1, we have
Φ≤2E Also, if we assume the stronger condition β ≤ 12√
λ1, then Φ≥E−β2|w|2− 1
4|w˙|2 ≥ 1 2E Therefore , we have
1
2E(t)≤Φ(t)≤2E(t), ∀t ∈R (5.13)
From (5.13), we obtain Φ0 ≤ −βE+ 2√
E|f|+β
αkwk+c(α)|w|
|||γ(t,w˙)|||∗+ 3βδ2 (5.14) Then, we set
F(t) =β{
αkwk+c(α)|w|
|||γ(t,w˙)|||∗+ 3δ 2 } Then
Φ0 ≤ −β2Φ + 2√
E|f|+F(t)
By using (5.13), we find:
Φ0 ≤ −β2Φ + 2√ 2√
Φ|f|+F(t) By using the inequality 2√
Φ≤1 + Φ in the second term of the RHS and setting m(t) = √
2|f(t)|
we find
Φ0 ≤ −β2Φ +m(t)Φ +F(t) +m(t) Introducing
H(t) :=F(t) +m(t) we have:
Φ0(t) ≤ −(β2 −m(t))Φ(t) +H(t), ∀t ∈R+ (5.15) On the other hand:
Z t+1
t
m(s)ds =√ 2
Z t+1
t
f(s)ds=√ 2
Z t+1
t
|h(s+)−h(s)|ds.
with
lim sup
→0t≥0
Z t+1
t
|h(s+)−h(s)|ds= 0.
In addition Z t+1
t
F(s)ds = β Z t+1
t
{
αkw(s)k+c(α)|w(s)|
|||γ(s,w˙(s))|||∗+3δ 2}ds
≤ β(2αsup
t≥0
ku(t)k+c(α) sup
t≤s≤t+1
|w(s)|) Z t+1
t
|||γ(t,w˙(s))|||∗ds+3βδ 2 Now
γ(s,w˙(s)) =g( ˙u(s+))−g( ˙u(s)) From boundedness of the energy E(t) for t≥0,∃C > 0 such that
Z t+1
t
hg( ˙u(s)),u(s)i˙ dx ds≤C, ∀t≥0.
As a consequence of (5.4) we have
g( ˙u(s))∈L1(t, t+ 1, Z0)
and
Z t+1
t
|||g( ˙u(s))|||∗ds≤N(C+ 1) (5.16) Therefore
Z t+1
t
F(s)ds≤2N(C+ 1)β(2αsup
t≥0
ku(t)k+c(α) sup
t≤s≤t+1
|w(s)|) + 3βδ 2 We have
sup
t≤s≤t+1
|w(s)| ≤sup
s≥0
|u(s)|.˙ By choosing α small enough, we obtain
Z t+1
t
F(s)ds≤βδ+P sup
s≥0
|u(s)|˙ +3βδ 2 . Then for ≤2 small enough, we obtain:
sup
t≥0
Z t+1
t
F(s)ds ≤3βδ. (5.17)
Then by (5.17), we obtain:
sup
t≥0
Z t+1
t
H(s)ds≤3βδ+√ 2
Z t+1
t
|h(s+)−h(s)|ds Now we may fix
β =β0 ≤min{1 2
pλ1, 2 3K(δ)} Then by imposing that ε is small enough to ensure
√2 sup
t≥0
Z t+1
t
|h(s+)−h(s)|ds ≤min{βδ,β
4} (5.18)
we obtain
sup
t≥0
Z t+1
t
H(s)ds≤4βδ (5.19)
and (5.15) becomes
Φ0(t) ≤ −α(t)Φ(t) +H(t), ∀t∈R+ (5.20) where
Z t+1
t
α(s)ds ≥ β 4
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 easily and since this property is robust with respect to uniform convergence in the energy norm on R+, we obtain it for general weak solutions by using Lemma 3.2 . The compactness result now follows from the same argument as in [14], p.167-168: the main idea is that the average 1εRt+ε
t u(s)ds˙ of ˙uon a small time interval remains bounded (by a large constant) in V while the equation shows that the average 1εRt+ε
t u(s)dsof uremains bounded (by a large constant) in A−1(Z0) 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, hg(v)−g(w), v−wiV0,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 hg(v)−g(w), v−wiV0,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
U0(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 of an almost periodic solution. This property has been used in [14] to prove the existence of almost-periodic solutions of
( utt+g(ut)−∆u=h(t, x), in R+×Ω,
u= 0 onR+×∂Ω. (6.1)
where Ω is a bounded domain, g is the Nemytskii operator generated by an increasing function γ ∈ C(R) such that γ−1 is uniformly continuous and satisfying some dimension dependant 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-local dissipation terms. More precisely we have
Theorem 6.1. Assume that g satisfies the hypotheses of Theorem 5.1 and (4.3) with τ arbitrary. 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. In addition for any other solution u we have for some constant vector a∈V
t→∞lim(ku(t)−ω(t)−ak+|u(t)˙ −ω(t)|) = 0˙
In addition if g ∈ C(H, V0) 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 to a 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 h in S1(R, H) andω(.+an) converges to ω inS1(R, V) (The Vector (h, ω,ω) being˙ S1-almost periodic with values inH×V ×H), hence ω(.+an) converges toω also inCb(R, V) since ω is almost periodic in the usual sense. if u is any solution, precompactness of the range ofu implies that a subsequence ofu(.+an) converges uniformly on all compact subintervals with values inV to some limitz, while the same subsequence of ˙u(.+an) converges to ˙z uniformly on all compact subintervals with values in H. Since the energy norm of the differenceu−ω is non-increasing, it converges to some limit l ≥0. The energy norm of the difference z−ω is equal to l for all t. We finally show thatz−ω is constant (and equal to 0 ifg ∈C(H, V0)) by using the following Lemma
Lemma 6.2. LetJ = [a, b] be any compact interval ofR witha < b and letu, v be two weak solutions of
¨
u(t) +g( ˙u(t)) +Au(t) = h(t), t∈J
such asku(t)−v(t)k2+|u(t)−˙ v(t)|˙ 2 is constant on J. Thenu˙ = ˙v. In addition ifg ∈C(H, V0) then u=v.
Proof. The result is almost trivial if u andv are strong solutions. To prove it in the general case we approximatehand (u(a),u(a)),˙ (v(a),v(a)) by˙ hn∈C1(J, H) and (u0n, u1n),(vn0, vn1)∈ D(A)2. Let η >0 be an arbitrary small fixed number. We choose n in such a way that
sup{ku0n−u(a)k2+|u1n−u(a)|˙ 2,kv0n−v(a)k2+|vn1 −v(a)|˙ 2} ≤ η2 16 and
khn−hkL1(J) ≤ η 4 which implies
sup
J
{[kun(t)−u(t)k2+|u˙n(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, u˙n(a) =u1n. (6.3) Defining similarly the solution vn of
¨
vn(t) +g( ˙vn(t)) +Avn(t) =hn(t), t ∈J; vn(a) = vn0, v˙n(a) = vn1 (6.4) we find by the triangular inequality
[kun(a)−vn(a)k2+|u˙n(a)−v˙n(a)|2]1/2 ≤[ku(a)−v(a)k2+|u(a)˙ −v(a)|˙ 2]1/2+ η 2
= [ku(b)−v(b)k2 +|u(b)˙ −v(b)|˙ 2]1/2+ η
2 ≤[kun(b)−vn(b)k2+|u˙n(b)−v˙n(b)|2]1/2+η.
Hence by squaring and using boundedness of the sequence kun(b)−vn(b)k2+|u˙n(b)−v˙n(b)|2 we obtain
kun(a)−vn(a)k2+|u˙n(a)−v˙n(a)|2−[kun(b)−vn(b)k2+|u˙n(b)−v˙n(b)|2]≤η2+Cη which can be rewritten, assuming η <1, as
Z
J
hg( ˙un(t))−g( ˙vn(t)),u˙n(t)−v˙n(t)idt≤C0η.
By using the hypothesis (5.3) on g we deduce Z
J
|u˙n(t)−v˙n(t)|2dt ≤ |J|δ+C(δ) Z
J
hg( ˙un(t))−g( ˙vn(t)),u˙n(t))−v˙n(t)idt ≤ |J|δ+C0C(δ)η so that for η small enough we find
Z
J
|u˙n(t)−v˙n(t)|2dt ≤2|J|δ
As a first consequence ˙u= ˙v onJ. In addition ifg ∈C(H, V0) , since ˙un converges uniformly to ˙uonJ and ˙vnconverges uniformly to ˙vonJ with values inH, by considering the equation
¨
un(t)−v¨n(t) +g( ˙un(t))−g( ˙vn(t)) +A(un(t)−vn(t)) = 0 After integration on J we find that in the sense of V0
Z
J
A(un(t)−vn(t))dt →0 Hence if u−v ≡a we end up with|J|Aa = 0 and finallya= 0
The end of proof of Theorem 6.1 follows very easily by considering any interval J as in the Lemma applied with ureplaced by z and v replaced by ω.
Remark 6.3. 1) It does not seem easy to construct a counterexample in whicha 6= 0.
2) In order to have uniqueness of ω it is sufficient to assume a much weaker property, it suffices for instance that g be continuous fromH to X weak where X is a reflexive Banach space 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, Z0) is (Z, α)-admissible ifg(0) = 0 and for some positive constants c, C we have
∀(v, w)∈V ×V, hg(v)−g(w), v−wiV0,V ≥ckv−wkZα+2 (6.5)
∀(v, w)∈V ×V, kg(v)−g(w)kZ0 ≤C(kvkαZ+kwkαZ)kv−wkZ (6.6) The class of (Z, α)-admissible functions will be denoted by G(Z, α).
The following Lemma shows that the functions ofG(Z, α) satifies all the properties used in our main boundedness and compactness results.
Lemma 6.5. Let g ∈G(Z, α). Then g satisfies (5.21) with p=α+ 2hence (5.2), (5.3) and (4.2), (4.3) for any τ >0 and (5.4).
Proof. It is clear that (6.5) implies (5.21) with p = α+ 2 since the norm in Z dominates the 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 wheng is (Z, α)-admissible.
Proposition 6.6. Letg ∈G(Z, α).Then for anyJ = [a, b]compact interval ofRwitha < 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 have
˙
u∈Lα+2(J, Z); g( ˙u)∈Lα+2α+1(J, Z0)
¨
u(t) +g( ˙u(t)) +Au(t) =h(t) in L1(J, V0)
Proof. The result is obvious for a strong solution u since then ˙u ∈ C(J, V) and therefore g( ˙u) ∈ C(J, V0). Now let u be a weak solution and un be a sequence converging to u in C(J, V)∩C1(J, H) with un a strong solution of
¨
un(t) +g( ˙un(t)) +Aun(t) =hn(t) and
n→∞lim khn−hkL1(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, Z0). 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 of distributions from Int(J) with values in H, then also in L1(J, V0) on any bounded interval J. Then the equation gives Aω1 =Aω2 in the sense of L1(J, V0) and the conclusion follows easily
Finally the following result improves our main asymptotic theorem by giving a rate of con- vergence:
Theorem 6.8. Assume that g satisfies the hypotheses of Proposition 6.6. Then for any h which is S1-almost periodic: R→H the equation
¨
u(t) +g( ˙u(t)) +Au(t) =h(t), t ∈R+,
has a unique solutionω such that the vector(ω,ω)˙ is almost periodicR→V×H. In addition for any other solution u we have for some M ≥0
∀t ≥0, ku(t)−ω(t)k+|u(t)˙ −ω(t)| ≤˙ M(1 +t)−α1 (6.8) Proof. We extend the method of proof of [19], theorem 3.1 p. 200 (cf. also [15], theorem 7.5.1 p. 98) to the case of a non-local damping satisfying (6.5)- (6.6). Since the proof is quite similar we just sketch out the main steps for completeness. We start from 2 strong solutions uand 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, ku(t)−v(t)k+|u(t)˙ −v(t)| ≤˙ M(1 +t)−α1 (6.9) withM bounded in terms of the initial data and theS1 norm ofh. To this end we introduce z :=u−v and the function
E(t) := 1
2(|z(t)|˙ 2+kw(t)k2)
In particular, E is non-increasing and therefore bounded. Then we define Ψ(t) :=E(t)α2(z(t),z(t))˙
First we have
E0(t) =−hg( ˙u(t))−g( ˙v(t)),z(t)i˙ V0,V ≤ −ckz(t)k˙ Zα+2 Then we find
Ψ0(t) = α
2E(t)α2−1E0(t)(z(t), z0(t)) +E(t)α2h−Az(t)−g( ˙u(t)) +g( ˙v(t)), z(t)iV0,V
≤ −C1E0(t)−E(t)α2kz(t)k2 +CE(t)α2(ku(t)k˙ αZ +kv(t)k˙ αZ)kz(t)k˙ ZV ertz(t)k
It follows from the properties ofgthatr(t) :=ku(t)k˙ α+2Z +kv(t)k˙ α+2Z ∈S1(R+) Introducing k(t) := ku(t)k˙ αZ +kv(t)k˙ αZ a rather straightforward calculation yields for any δ >0
Ψ0(t)≤ −C1E0(t)−E(t)α2kz(t)k2+δkE(t)α2kz(t)k2+δkα+2α E(t)α+22 +C2δ−(α+1)kz(t)k˙ α+2Z Then setting
Fε:= (1 +C1ε)E+εΨ
