Global attractors for a semi-linear hyperbolic equation in viscoelasticity

doi:10.1006rjmaa.2001.7437, available online at http:rrwww.idealibrary.com on

Global Attractors for a Semilinear Hyperbolic Equation

in Viscoelasticity

Claudio Giorgi

Dipartimento di Matematica, Uni¨ersita di Brescia, I-25133 Brescia, Italy`

E-mail: giorgi@ing.unibs.it Jaime E. Munoz Rivera

˜

LNCC-Petropolis, 25651, Rio de Janeiro, Brazil´

E-mail: rivera@lncc.br

and

Vittorino Pata

Dipartimento di Matematica, Uni¨ersita di Brescia, I-25133 Brescia, Italy`

E-mail: pata@ing.unibs.it

Submitted by William F. Ames

Received June 26, 2000

A semilinear partial differential equation of hyperbolic type with a convolution term describing simple viscoelastic materials with fading memory is considered.

Ž .

Regarding the past history memory of the displacement as a new variable, the equation is transformed into a dynamical system in a suitable Hilbert space. The dissipation is extremely weak, and it is all contained in the memory term. Longtime behavior of solutions is analyzed. In particular, in the autonomous case, the existence of a global attractor for solutions is achieved. 䊚 2001 Academic Press

Key Words: viscoelasticity; memory kernel; absorbing set; global attractor.

83

0022-247Xr01 $35.00

Copyright䊚 2001 by Academic Press All rights of reproduction in any form reserved.

1. INTRODUCTION

We consider the following semilinear hyperbolic equation with linear memory in a bounded domain⍀ ; ⺢3_{, arising in the theory of isothermal}

Ž w x. viscoelasticity cf. 4, 25 , ⬁ _q ut ty k 0 ⌬u y

Ž .

_H

k⬘ s ⌬u t y s ds q g u s f

Ž .

Ž

.

Ž .

in ⍀ = ⺢ 0 u x , t

Ž

.

s 0, xg⭸ ⍀, t g ⺢

Ž

1.1

.

u x , t

Ž

.

s u x, t ,0

Ž

.

xg ⍀, t F 0 Ž . Ž . Ž . q

with k 0 , k⬁ ) 0, and k⬘ s F 0 for every s g ⺢ . Ž w x.

As is well known see 9 , this equation describes a homogeneous and isotropic viscoelastic solid. If the solid is not homogeneous and isotropic, the function k has to be replaced by a fourth order tensor depending on

Ž w x.

xg ⍀, satisfying certain additional hypothesis see, e.g., 8, p. 140 , and Ž .

the above Eq. 1.1 has to be written in divergence form. However, all the arguments presented here can be easily generalized to the non-homoge-neous non-isotropic case.

Ž . Ž .

Notice that if k⬘ ' 0, 1.1 reduces to the semilinear wave equation, where g represents some displacement-dependent body force density. Thus, neglecting the contributions of the nonlinearity, that is, taking g' 0, all the dissipation is contained in the convolution integral. In particular, the existence of a genuine memory induces a damping

mecha-Ž w x.

nism, and asymptotic stability is to be expected see 5, 8, 17 .

Ž . w x

A problem similar to 1.1 has been studied in 23 . In that paper, however, the situation was in some sense more favorable, because of the presence of an instantaneous damping term. Clearly, the trade off in considering a weaker dissipation is a much stronger requirement on the

w x

structure of the nonlinearity. Here, as in 23 , we introduce the new Ž w x.

variable see 5

␩t _{x , s s u x, t y u x, t y s . 1.2}

Ž

.

Ž

.

Ž

.

Ž

.

Ž . Ž . Ž . Ž .

We set for simplicity␮ s s yk⬘ s and k ⬁ s 1. In view of 1.2 , adding Ž .

and subtracting the term ⌬u, Eq. 1.1 transforms into the system

⬁

¡

_{ut ts ⌬u q}

_H

_{␮ s ⌬␩ s ds y g u q f}

_{Ž .}

_{Ž .}

_{Ž .}

~

¢

0

Ž

1.3

.

␩ s y␩ q u ,t s t

Ž .

conditions are then given by

¡

u x , t

Ž

.

s 0, xg⭸ ⍀, t G 0 t q ␩ x, s s 0,

Ž

.

Ž

x , s

.

g⭸ ⍀ = ⺢ , t G 0

~

u x , 0

Ž

.

s u x ,0

Ž .

xg ⍀

Ž

1.4

.

ut

Ž

x , 0

.

s_¨0

Ž .

x , xg ⍀ 0 q

¢

␩ x, s s ␩ x, s ,

Ž

.

0

Ž

.

Ž

x , s

.

g ⍀ = ⺢ having set

¡

u0

Ž .

x s u x, 00

Ž

.

~

¨0

Ž .

x s⭸ u x, tt 0

Ž

.

Nts0

¢

␩ x, s s u x, 0 y u x, ys .0

Ž

.

0

Ž

.

0

Ž

.

The memory kernel ␮ is required to satisfy the following hypotheses: Žh1. ␮ g C ⺢ l L ⺢1Ž q. 1Ž q. ᭙s g ⺢ ;q

Žh2. ␮ s G 0 and ␮⬘ s F 0 ᭙s g ⺢ ;Ž . Ž . q Žh3. H0⬁ ␮ s ds s k ) 0;Ž . 0

Žh4. ␮⬘ s q ␦␮ s F 0 ᭙s g ⺢ and some ␦ ) 0;Ž . Ž . q

Žh5. There exists s0G 0 such that ␮⬘ g L 0, s2ŽŽ 0.. and ␮⬘ s qŽ . Ž .

M␮ s G 0 ᭙s G s and some M ) 0.0

Ž .

We can clearly weaken h5 asking that ␮⬘ is square summable in a Ž . neighborhood of zero. This automatically implies, thanks to h1 , that

2

ŽŽ .. Ž . Ž .

␮⬘ g L 0, s , for every s ) 0. Conditions h4 ᎐ h5 , which are not0 0

needed in the existence and uniqueness result, imply the exponential decay

Ž . Ž .

of ␮ s . In particular, notice that ␮ 0 has to be finite. For instance, sums Ž . Ž .

of exponentials fulfill h1 ᎐ h5 . This rather strong decay rate of the kernel seems to be unavoidable in order to have the exponential decay of the

Ž w x.

associated linear problem see, e.g., 5, 8, 12, 17 . On the other hand, since all the dissipation of the system is contained only in the memory term, we

Ž . also have to require that ␮ k 0, and this explains h3 .

Concerning the nonlinear term, we assume that g is differentiable with bounded derivative. Clearly, this is a rather strong condition. Indeed, if we restrict our analysis to existence and uniqueness results, we may ask much

w x Ž w x.

weaker condition on g such as those in 23 see also 10 . Nonetheless, this condition is the best one to obtain, for instance, the absorbing set. We point out that in this work we are able to obtain the exponential decay of the associated linear homogeneous system via energy estimates. The

w x

the exponential decay is obtained employing semigroup techniques. Our approach allows us to improve the asymptotic analysis to the nonlinear case. However, due to the extremely weak dissipation of the linear semi-group, we cannot arrive at further Lipschitz nonlinearities.

Let us mention some others papers related to the problem we address. w x

For the nonlinear one-dimensional equation, Dafermos 6 , exploring the dissipative properties of the equation, showed that the system is well posed provided the initial data are small enough, whereas for the n-dimensional linear system the author proved the asymptotic stability of the solutions,

Ž w x.

but without providing an explicit rate of decay see 5 . For 3-dimensional w x isotropic and homogeneous materials, Dassios and Zafiropoulus 7 , using an asymptotic analysis, proved that the solution of the viscoelastic system of memory type has a uniform decay to zero provided that the relaxation kernel is the exponential function. This result was improved for more

w x

general relaxations functions in the articles 16, 18, 19, 21 . On the other hand, when the relaxation has a polynomial decay, it was proved that the solution of the corresponding model has a polynomial decay either, with

Ž w x.

the same rate of decay as the relaxation see 20 . Finally, for boundary w x

stabilization of viscoelastic plates see 15 . Unfortunately, the methods used to achieve uniform rate of decay in those works are based on second order estimates, which are time dependent in our problem. Thus these techniques fail in the case of semilinear problems, and a new asymptotic analysis has to be devised.

The main result of this paper is to show the existence of a global Ž .

attractor for the solution of Eq. 1.1 . The method we use introduces a new multiplier and applies the concept of strongly continuous semigroup of

w x operators, and the techniques developed in 23 .

The plan of the paper is as follows. In Section 2 we introduce the notation. In Section 3 we give the definition of weak solution, along with existence and uniqueness results. Section 4 is devoted to uniform energy estimates and to the existence of absorbing sets for the solutions. Finally, in Section 5, we prove that the semigroup associated to our problem admits a global attractor in the phase-space.

2. NOTATION

Let ⍀ ; ⺢3 _{be a bounded domain with smooth boundary. With usual}

notation, we introduce the spaces Hy1, L2_{, and H}1 _{acting on ⍀. The} 0

² : y1

H1_{. Exploiting the Poincare inequality}

´

0 < <2 _{< <}2 1 ␭0

H

¨ dxF

H

ⵜ_¨ dx , ᭙_¨g H ,0

Ž

2.1

.

⍀ ⍀

for some␭ ) 0, the norm in H1 _{is given by}

0 0 5 52 1 < <2 ¨ H0s

H

ⵜ¨ dx. ⍀ Ž . 2Ž q 1. 1

In view of h1 , let L_␮⺢ , H₀ be the Hilbert space of H -valued₀ functions on⺢q, endowed with the inner product

⬁

2 q 1

²␸, ␺:L Ž␮⺢ , H .0 s

H H

ž

␮ s ⵜ␸ s ⵜ␺ s ds dx.

Ž .

Ž .

Ž .

/

⍀ 0

Finally we introduce the Hilbert space H

Hs H01= L2= L2␮

_Ž

⺢q, H01

_.

.

To describe the asymptotic behavior of the solutions of our system we need also to introduce the space TT of L1 _{-translation bounded L}2_-valued

loc functions on⺢q, namely 1r2 ␰q1 2 1 q 2 5 5 < < T Ts f g L

½

loc

Ž

⺢ , L : f

.

TTs sup

H

ž

H

f t

Ž .

dx

/

dy- ⬁ .

5

␰ ⍀ ␰G0

3. EXISTENCE AND UNIQUENESS

We first formulate precisely the conditions on the nonlinearity. Let

1 Ž . gg C ⺢ and denote s G s

Ž .

s

_H

g y dy

Ž .

0 and G G u s G u x dx, for u g H1 .

Ž .

_H

Ž

Ž .

.

0 ⍀

The assumptions are as follows: there exist C₀) 0 and ⌫ ) 0 such that G y

Ž .

g1 lim inf G 0;

Ž

.

_y2 < <yª⬁ yg y

Ž .

y C G y0

Ž .

g2 lim inf G 0;

Ž

.

_y2 < <yª⬁ < < g3 g⬘ y F ⌫.

Ž

.

Ž .

Ž . Ž . Ž w x. The following inequalities are direct consequences of g1 ᎐ g2 cf. 10 ,

1 ₂ 1 < < GG

Ž .

u q

H

ⵜu dx G yC ,1 ᭙u g H ,0

Ž

3.1

.

4 ⍀ 1 ₂ 1 < < ug u dx

Ž .

y C GG

Ž .

u q ⵜu dx G yC , ᭙u g H , 3.2

Ž

.

H

0

H

2 0 2 ⍀ ⍀ for some C , C_{1 2}) 0. Ž . Ž .

The solution to the initial-boundary value problem 1.3᎐ 1.4 is defined in the following manner.

w x 1_Ž 2_.

DEFINITION 3.1. Set Is 0, T , for T ) 0, and let f g L I, L . We

Ž . Ž . Ž . Ž .

say a function zs u, u ,t ␩ g C I, HH is a solution to problem 1.3 ᎐ 1.4

Ž . Ž .

in the time interval I, with initial data z 0 s z s u ,0 0 ¨0,␩ g HH, pro-0

vided ⬁ ²u ,t t _¨

˜

:s y ⵜuⵜ

_H

_¨

˜

dxy

_{H H}

ž

␮ s ⵜ␩ s ds ⵜ

Ž .

Ž .

/

˜

_¨dx ⍀ ⍀ 0 y g u

_H

Ž .

¨

˜

dxq

_H

f_¨

˜

dx ⍀ ⍀ ⬁ ⬁ ␮ s ␩ s q ␩ s ⌬␩ s ds dx s u

Ž .

Ž

Ž .

Ž .

.

˜

Ž .

␮ s ⌬␩ s ds dx

Ž .

˜

Ž .

H H

ž

t s

/

H

t

ž

H

/

⍀ 0 ⍀ 0 1 2 Ž q 2 1.

for all ¨

˜

g H and0 ␩ g L ⺢ , H l H , and a.e. t g I.

˜

␮ 0

The proof of the next two theorems is omitted, since existence and w x

uniqueness are proved exactly like the analogous results of 23 , where the only difference is the presence of a damping term, which however plays a significant role only in time-independent estimates.

Ž . Ž . Ž . Ž . Ž .

THEOREM 3.2 Existence . Let h1 ᎐ h2 and g1 ᎐ g3 hold. Then,

Ž . Ž .

gi¨en any T) 0, problem 1.3 ᎐ 1.4 has a solution z in the time inter¨al

w x

Ž . Ž . Ž . Ž . Ž . THEOREM 3.3 Continuous Dependence . Let h1 ᎐ h2 and g1 ᎐ g3

4 Ž 1Ž 2..

hold. For is 1, 2, let z , f_{0 i i} z_{0 i}g HH and f g L I, L_i be two sets of Ž . Ž . data, and denote by z two corresponding solutions to problem 1.3i ᎐ 1.4 in

w x 5 5 5 5 1 2

the time inter¨al Is 0, T . Assume also that z0 i HH- R and fi L Ž I, L .- R,

for some R) 0. Then the following estimate holds,

5 52 _{5 5}2 _{5 5}2

1 2

z1y z2 HHF C TR

Ž

.

_Ž

z01y z02 HHq f y f1 2 L Ž I , L .

_.

Ž

3.3

.

Ž . Ž . Ž .

for some constant C TR ) 0. In particular, problem 1.3 ᎐ 1.4 has a

unique solution.

4. UNIFORM ENERGY ESTIMATES

Ž . In the sequel of the paper, we agree to denote the solution z t of Ž1.3. Ž᎐ 1.4 with initial data z by S t z . When the system is autonomous,. 0 Ž . 0

Ž .

namely, when f is independent of time, S t is a strongly continuous

Ž . Ž w x

semigroup of continuous nonlinear operators on HH see 24 for a .

detailed presentation of the theory . This follows directly from Theorem Ž .

3.2 and estimate 3.3 of Theorem 3.3. Ž .

The energy associated to 1.3 at time t is given by

1 _{2 2} < < < < E E

Ž .

t s

ž

H

ⵜu t

Ž .

dxq

H

u tt

Ž .

dx 2 _{⍀ ⍀} ⬁ _{t 2} < < q

H H

ž

␮ s ⵜ␩ s

Ž .

Ž .

ds dx .

/

/

Ž

4.1

.

⍀ 0

The main result of the section is

Ž . Ž . Ž . Ž .

THEOREM 4.1. Assume h1 ᎐ h5 and g1 ᎐ g3 , and let F ; TT be a

Ž .

bounded set. Then there exists positi¨e constants C, ⌳, ␧ depending on F such that the relation

E

E

Ž .

t F Cey␧ tEE

Ž .

0 q ⌳

Ž

4.2

.

holds for e_¨ery tG 0 and e_¨ery fg F. In particular, if g ' 0 and F reduces to

Ž .

the null function that is, the linear homogeneous case , then ⌳ s 0. Proof. Set

5 5 ⌽ s sup h .TT

Let fg F and denote 1r2 2 < < mf

Ž .

t s

ž

_H

f t

Ž .

dx

/

. ⍀ Clearly, ␰q1 sup

_H

mf

Ž

␰ d␰ F ⌽.

.

Ž

4.3

.

␰ ␰G0 Ž .

Taking the derivative with respect to t of 4.1 we have d E E

Ž .

t s

_H

ⵜu t ⵜu t dx q

Ž .

t

Ž .

_H

u t ut

Ž .

t t

Ž .

t dx dt ⍀ ⍀ ⬁ _{t t} q

_{H H}

ž

␮ s ⵜ␩ s ⵜ␩ s ds dx.

Ž .

t

Ž .

Ž .

/

⍀ 0 Ž .

Substituting 1.3 in the above inequality, we obtain

⬁ d 1 d ₂ t < < E E

Ž .

t s y

_{H H}

ž

␮ s

Ž .

ⵜ␩ s

Ž .

ds dx

/

dt 2 _⍀ 0 ds d y GG

Ž

u t

Ž .

.

q

_H

f t u t dx.

Ž .

t

Ž .

Ž

4.4

.

dt ⍀ Ž . Ž w x .

Integration by parts in s and h4 yield cf. 11 for the details

⬁ ⬁ 1 d ₂ ␦ ₂ t t < < < < y

_{H H}

ž

␮ s

Ž .

ⵜ␩ s

Ž .

ds dx

/

F y

_{H H}

ž

␮ s ⵜ␩ s

Ž .

Ž .

ds dx ,

/

2 _⍀ 0 ds 2 ⍀ 0

whereas the Holder inequality gives

_¨

1r2 2 < < f t u t dx

Ž .

Ž .

F m t

Ž .

u t

Ž .

dx .

H

t f

ž

H

t

/

⍀ ⍀ Ž .

Hence from 4.4 we get the estimate d E E

Ž .

t q GG

Ž

u t

Ž .

.

Ž

.

dt 1r2 ⬁ ␦ _< t _<2 _{< <}2 F y

H H

ž

␮ s ⵜ␩ s

Ž .

Ž .

ds dx

/

q m tf

Ž .

ž

H

u tt

Ž .

dx

/

2 ⍀ 0 ⍀ ⬁ ␦ _{t 2 1r2} < < F y

_{H H}

ž

␮ s ⵜ␩ s

Ž .

Ž .

ds dx

/

q m t EEf

Ž .

Ž .

t .

Ž

4.5

.

2 _⍀ 0

Introduce now the functional ⬁ t F F

Ž .

t s y u t

_H

t

Ž .

ž

_H

␮ s ␩ s ds dx.

Ž .

Ž .

/

Ž

4.6

.

⍀ 0

Throughout the proof, we will make extensive use of the Holder and

_¨

Ž . Ž .

Young inequalities, h3 , and 2.1 . Ž . The derivative with respect to t of FF t entails

⬁ d t FF

Ž .

t s y u t

_H

t t

Ž .

ž

_H

␮ s ␩ s ds dx

Ž .

Ž .

/

dt _⍀ 0 ⬁ _t y u t

_H

t

Ž .

ž

_H

␮ s ␩ s ds dx.

Ž .

t

Ž .

/

Ž

4.7

.

⍀ 0 Ž . Ž .

Exploiting h5 , and recalling h2 , we have

⬁ _t y u t

_H

t

Ž .

ž

_H

␮ s ␩ s ds dx

Ž .

t

Ž .

/

⍀ 0 ⬁ _t s y u t

H

t

Ž .

ž

H

␮ s u t y ␩ s ds dx

Ž .

Ž

t

Ž .

s

Ž .

.

/

⍀ 0 ⬁ 2 t < < s yk0

_H

u tt

Ž .

dxy

_H

u tt

Ž .

ž

_H

␮⬘ s ␩ s ds dx

Ž .

Ž .

/

⍀ ⍀ 0 < < s0 ␮⬘ s

Ž .

2 1r2 t < < < < < < F yk0

H

u tt

Ž .

dxq

H

u tt

Ž .

ž

H

_␮1r2 _s ␮

Ž .

s ␩ s ds

Ž .

Ž

.

⍀ ⍀ 0 0 ⬁ _t < < qM

_H

␮ s ␩ s ds dx

Ž .

Ž .

/

s0 < <2 F yk0

_H

u tt

Ž .

dx ⍀ 1r2

'

2 1 s0 _{2 1r2} < < q _␭ max

½

_␮1r2

_Ž

_s

_.

ž

_H

␮⬘ s

Ž .

ds

/

, Mk0

5

0 0 0 1r2 ⬁ _{t 2} < < < < ⭈

H

u tt

Ž .

ž

H

␮ s ⵜ␩ s

Ž .

Ž .

ds

/

dx ⍀ 0 ⬁ k_{0 2 2} t < < < < F y

_H

u tt

Ž .

dxq C3

_{H H}

ž

␮ s ⵜ␩ s

Ž .

Ž .

ds dx ,

/

Ž

4.8

.

2 _{⍀ ⍀} 0

Ž Ž . . Ž . where C₃s C₃ ␭ , k , ␮ s , M ; whereas, due to 1.3_{0 0 0}

⬁ t y u t

_H

t t

Ž .

ž

_H

␮ s ␩ s ds dx

Ž .

Ž .

/

⍀ 0 2 ⬁ _t ⬁ _t s ⵜu t

_H

Ž .

ž

_H

␮ s ⵜ␩ s ds dx q

Ž .

Ž .

/

_{H H}

ž

␮ s ⵜ␩ s ds dx

Ž .

Ž .

/

⍀ 0 ⍀ 0 ⬁ _t ⬁ _t q g u t

_H

Ž

Ž .

.

ž

_H

␮ s ␩ s ds dx y f t

Ž .

Ž .

/

_H

Ž .

ž

_H

␮ s ␩ s ds dx.

Ž .

Ž .

/

⍀ 0 ⍀ 0 4.9

Ž

.

Let us examine in detail the four terms appearing in the right-hand side of Ž4.9 . Concerning the first two, choosing. D ) 0 to be specified later, we get

2 ⬁ ⬁ t t ⵜu t

Ž .

␮ s ⵜ␩ s ds dx q

Ž .

Ž .

␮ s ⵜ␩ s ds dx

Ž .

Ž .

H

ž

H

/

H H

ž

/

⍀ 0 ⍀ 0 < < ⬁ D < <2 ⍀ < t <2 F

_H

ⵜu t

Ž .

dxq k 1 q0

ž

/

_{H H}

ž

␮ s ⵜ␩ s

Ž .

Ž .

ds dx.

/

2 _⍀ 2D _⍀ 0 4.10

Ž

.

Ž . By force of g3 , ⬁ _t g u t

Ž

Ž .

.

␮ s ␩ s ds dx

Ž .

Ž .

H

_ž

H

_/

⍀ 0 1r2 ⬁ ₂ 1r2 < < < < < t < F k0

_H

Ž

⌫ u t q g 0

Ž .

Ž .

.

ž

_H

␮ s ␩ s

Ž .

Ž .

ds

/

dx ⍀ 0 1r2 2 1r2 _⬁ D < <2 k0 ⌫ k0 < t <2 F

_H

ⵜu dx q

ž

1q

/

_{H H}

ž

␮ s ⵜ␩ s

Ž .

Ž .

ds dx

/

2 ⍀ 2␭0 D␭0 ⍀ 0 < <2_{< < 1r2} g 0

Ž .

⍀ k0 q .

Ž

4.11

.

2 Ž .

The last term of 4.9 is controlled as

⬁ t y f t

_H

Ž .

ž

_H

␮ s ␩ s ds dx

Ž .

Ž .

/

⍀ 0 1r2 1r2 _⬁ k_{0 2} t < < F _␭1r2mf

Ž .

t

ž

_{H H}

ž

␮ s ⵜ␩ s

Ž .

Ž .

ds dx

/

/

.

Ž

4.12

.

⍀ 0 0

Ž . Ž . Ž .

Collecting 4.8 ᎐ 4.12 , we obtain from 4.7 the estimate

d k_{0 2 2} < < < < F F

Ž .

t F y

H

u tt

Ž .

dxqD

H

ⵜu t

Ž .

dx dt 2 _{⍀ ⍀} ⬁ ₂ t 1r2 < < q C4

_{H H}

ž

␮ s ⵜ␩ s

Ž .

Ž .

ds dx

/

q C m t EE5 f

Ž .

Ž .

t q C6 ⍀ 0 4.13

Ž

.

Ž .

for some positive constants C₄s C₄ D , C , C . Notice that if g ' 0, then_{5 6} C₆s 0.

Finally we consider the equality

d ₂ < < u t u t dx

Ž .

Ž .

s u t

Ž .

dxq u t u

Ž .

Ž .

t dx.

H

t

H

t

H

t t dt _{⍀ ⍀ ⍀} Ž . Ž .

Thus, exploiting again 1.3 , and appealing to 3.2 , we have d u t u t dx

Ž .

Ž .

H

t dt _⍀ < <2 _{< <}2 s

_H

u tt

Ž .

dxy

_H

ⵜu t

Ž .

dx ⍀ ⍀ ⬁ _t y ⵜu t

_H

Ž .

ž

_H

␮ s ⵜ␩ s ds dx

Ž .

Ž .

/

⍀ 0 y u t g t dx q

_H

Ž .

Ž .

_H

u t f t dx.

Ž . Ž .

⍀ ⍀ 1 2 2 < < < < F

_H

u tt

Ž .

dxy

_H

ⵜu t

Ž .

dxy C G0G

Ž

u t

Ž .

.

q C2 4 ⍀ ⍀ ⬁ ₂ t 1r2 < < q k0

_{H H}

ž

␮ s ⵜ␩ s

Ž .

Ž .

ds dx

/

q m t EEf

Ž .

Ž .

t .

Ž

4.14

.

⍀ 0

To conclude the proof, for N) 0 and ␯ ) 0, introduce the functional LL

Ž .

t s NEE t q NG

Ž .

G

Ž

u t

Ž .

.

q NC q FF t q1

Ž .

␯ u t u t dx.

_H

Ž .

t

Ž .

⍀

Ž .

Recalling 3.1 , it is easy to check that, provided N is big enough and ␯ small enough, there exists two constants C₇) 1 and C ) 0 depending on₈

Ž .

N and␯ with C s 0 when g ' 0 such that₈ 1

E

E

Ž .

t F LL t F C EE t q C .

Ž .

7

Ž .

8

Ž

4.15

.

Ž . Ž . Ž .

Collecting 4.5 , 4.13 , and 4.14 , we end up with

d ␯ ₂ < < L L

Ž .

t q

ž

yD

/

_H

ⵜu t

Ž .

dx dt 4 _⍀ k_{0 2} ␯ ␯ < < q

ž

y␯

/

_H

u tt

Ž .

dxq GG

Ž

u t

Ž .

.

q FF

Ž .

t 2 ⍀ 8 8 N ⬁ N␦ ₂ t < < q

ž

y C y4 ␯ k0

/

_{H H}

ž

␮ s ⵜ␩ s

Ž .

Ž .

ds dx

/

2 ⍀ 0 ␯2 q

_H

u t u t dx

Ž .

t

Ž .

8 N _⍀ ␯ ␯ ␯2 F

ž

y␯ C G0

/

G

Ž

u t

Ž .

.

q FF

Ž .

t q

_H

u t u t dx

Ž .

t

Ž .

8 8 N 8 N _⍀ q N q C q␯ m t EE1r2 _{t q ␯ C q C . 4.16}

Ž

5

.

f

Ž .

Ž .

Ž

2 6

.

Ž

.

Ž . Ž .

At this point, it is easy to see that, due to 2.1 and 3.3 , and assuming without loss of generality N) 1 and␯ - 1.

␯ ␯ ␯2 y␯ C G0 G

Ž

u t

Ž .

.

q FF

Ž .

t q

_H

u t u t dx

Ž .

t

Ž .

ž

8

/

8 N 8 N _⍀ ␯ _{< <}2 _{< <}2 F

_H

ⵜu t

Ž .

dxq␯ C9

_H

u tt

Ž .

dx 16 ⍀ ⍀ ⬁ _{t 2} < < q␯ C9

_{H H}

ž

␮ s ⵜ␩ s

Ž .

Ž .

ds dx

/

q C ,10

Ž

4.17

.

⍀ 0

for some C , C_{9 10}) 0. Again, C s 0 when g ' 0. Choose now₁₀ ␯ small enough such that

k0 ␯

y␯ y C ␯ G ,9

2 8

Ž .

set D s ␯r16 which automatically fixes the value of C , and choose N₄ big enough such that

N␦ ␯

y C y4 ␯ k y ␯ C G .0 9

2 8

Then let ␧ s ␯r8 N. Denoting

C11s C71r2

Ž

Nq C q5 ␯

.

and

Žnotice that C₁₂s 0 when g ' 0 , from 4.15 ᎐ 4.17 we get the differen-. Ž . Ž . tial inequality d 1r2 L L

Ž .

t q␧ LL t F C m t LL

Ž .

11 f

Ž .

Ž .

t q C .12 dt Ž w x.

By virtue of a generalization of the Gronwall Lemma see, e.g., 22, 23 , Ž .

keeping in mind 4.3 , we obtain

2C12 e␧ y␧ t 2 L L

Ž .

t F 2 LL 0 e

Ž .

q q _{y␧ r2} ⌽ . ␧

Ž

1y e

.

Ž .

The proof is carried out applying once more 4.15 . The constants C and⌳ of the statement turn out to be

2C C_{7 12} C e₇ ␧

2 2

Cs 2C7 and ⌳ s 2C C q7 8 q _{y␧ r2} ⌽ .

␧

Ž

1y e

.

Ž 4.

Notice that when⌽ s 0 that is, F s 0 and g ' 0, then ⌳ s 0. Ž .

Remark 4.2. The uniform energy estimate 4.2 implies the existence of Ž .

a bounded absorbing set BB*; HH for S t , which is uniform as f is allowed to run in a bounded set F; TT. Indeed, if BB* is any ball of HH of radius

'

less than 2⌳ , for any bounded set BB; HH it is immediate to see that Ž .

there exists t BB G 0 such that

S t B

Ž .

B; BB* Ž .

for every tG t BB and every fg F. Moreover, if we define

B

B0s

D

S t B

Ž .

B*

tG0

it is clear that BB0 is still a bounded absorbing set which is also invariant

Ž . Ž .

for S t , that is, S t BB0; BB0 for every tG 0. In particular, if F is connected, then BB₀ is connected as well.

5. EXISTENCE OF A GLOBAL ATTRACTOR In this section we assume

Ž .

In this case, as mentioned before, S t is a strongly continuous semigroup Ž .

on HH. Our aim is to show S t admits a global attractor. Recall that the Žunique global attractor of S t acting on H. Ž . H is the compact set AA; HH enjoying the following properties:

Ž .1 AA is fully invariant for S t , that is, S t AŽ . Ž .As A for every t G 0; Ž .2 AA is an attracting set, namely, for any bounded set BB; HH,

lim␦ S t BHH

Ž

Ž .

B, AA

.

s 0,

tª⬁

where ␦ denotes the semidistance on HH._HH

w More details on the subject can be found in the classical books 1, 13, x

26 .

Actually, we could extend with minor modifications our analysis to some particular nonautonomous situations, introducing the notion of strongly

Ž w x.

continuous process of operators see 14 , and prove the existence of an attractor which is uniform as f belongs to the hull of a

translation-com-Ž w x.

pact function in a suitable space see, e.g., 2, 3 .

In the sequel, denote As y⌬, the Laplacian with Dirichlet boundary conditions. It is well known that A is a positive operator on L2 _with

Ž . 2 1 s

domain DD A s H l H . Moreover, one can define the powers A of A0

Ž s.

for sg ⺢. The space V s D2 s D A turns out to be a Hilbert space with the inner product

²u,¨:V2 ss A u, A² s s¨:.

In particular, V s Hy1, V s L2_{, V s H}1_{. The injection V ¨ V is}

y1 0 1 0 s₁ s₂

compact whenever s1) s . For further convenience, for s g ⺢, introduce2

the Hilbert space H

Hss V1qs= V = Ls 2␮

Ž

⺢q, V1qs

.

. Clearly, HH₀s HH.

Ž .

Let now z0s u ,0 ¨0,␩ g B0 B0, where BB0 is the invariant, connected, Ž .

bounded absorbing set of S t given by Remark 4.2. Ž w x.

Following a standard procedure cf. 23 we write the solution zs Žu,_¨,␩ to 1.3 ᎐ 1.4 as z s z q z , with z s u ,. Ž . Ž . _{L N L} Ž _{L ¨L},␩ and z s_L. _N Žu ,_N ¨_N,␩ , where z and z are the solutions in the sense of Defini-_N. _{L N} Ž

.

tion 3.1 to the problems

⬁

⭸ u s ⌬u qt t L L

_H

␮ s ⌬␩ s ds y g u q f

Ž .

L

Ž .

Ž .

0

⭸ ␩ s y⭸ ␩ q ⭸ ut L s L t L

and ⬁ ⭸ u s ⌬u qt t N N

_H

␮ s ⌬␩ s ds

Ž .

N

Ž .

0 ⭸ ␩ s y⭸ ␩ q ⭸ ut N s N t N zN

Ž .

0 s 0.

In the remaining of the section, let K denote a generic constant, which is independent of z₀g BB₀.

Ž . It is apparent from Theorem 4.1 that zL fulfills estimate 4.2 with ⌳ s 0; therefore 5_z

_{Ž .}

_t 5 _{F Ke}y␧ t _{᭙t g ⺢ .}q

_Ž

_5.1

_.

H H L Ž . Ž

Concerning z , for every tN G 0 there exists K t ) 0 independent of

. z0g BB0 , such that 5_z

_{Ž .}

_t 5 _{F K t}

_{Ž .}

_{᭙t g ⺢ .}q

_Ž

_5.2

_.

H H N 1r2 Ž . w x

Proof of 5.2 is obtained repeating the argument of Lemma 5.4 in 23 . The only difference is that here we get a time-dependent estimate,

w x Ž

whereas in 23 due to the presence of a weak damping in the equation for .

u_N the estimate is uniform as tG 0.

Ž w x.

Finally see Lemma 5.5 in 23 , we have the compact embedding C

C t s ␩t ¨ L2 _⺢q_{, H}1 _{. 5.3}

Ž .

D

N ␮

Ž

0

.

Ž

.

z0gB0

2_Ž q 1_.

Denote the closure of CC in L_␮⺢ , H0 by CC. With reference to Ž5.2. Ž᎐ 5.3 , for t G 0 let R. RŽ .t be the ball of V3r2= V1r2 of radius K tŽ . centered at zero and introduce the set

K

K

Ž .

t s RR

Ž .

t = CC; HH.

1 2 _{Ž . Ž .}

From the compact embedding V3r2= V1r2¨ H = L and 5.3 , K0 K t is Ž . Ž .

compact in HH. By construction, zN t g KK t for every tG 0. Ž .

Due to the compactness of KK t , for every fixed tG 0 and every

y␧ t _{Ž .}

d) Ke , there exist finitely many balls of HH of radius d such that z t belongs to the union of such balls, for every z0g BB0. This implies that

␣ S t BHH

Ž

Ž .

B0

.

F Key␧ t, ᭙t G 0,

Ž

5.4

.

Ž where ␣ is the Kuratowski measure of noncompactness, defined by cf.HH

w x.13

␣ BHH

Ž

B

.

s inf d : BBhas a finite cover of balls

4

of HHof diameter less that d .

Ž . Since the invariant, connected, bounded absorbing set BB₀ fulfills 5.4 ,

Ž exploiting a classical result of the theory of attractors of semigroups see w x.13 , we conclude that the ␻-limit set of BB₀, that is,

H H

␻ B

Ž

B0

.

s

F D

S s B

Ž .

B0 ,

tG0 sGt

Ž . is the connected and compact global attractor of S t .

ACKNOWLEDGMENTS

This work has been partially supported by the Italian MURST ’98 Research Projects ‘‘Mathematical Models for Materials Science’’ and ‘‘Problems and Methods in the Theory of Hyperbolic Equations’’ and by CNPq-Brazil Grant 305406r88-4.

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