large time behavious via the method of trajectories

# 2002 Elsevier Science USA Key Words: large time behavior; global attractor; exponential attractor; finite fractal dimension; ‘-trajectory; fluids with shear-dependent viscosity; power-la

doi:10.1006/jdeq.2001.4087, available online at http://www.idealibrary.com on

LargeTime Behavior via the Method of ‘-Trajectories1

Josef M!aalek and Dalibor Pra$zz!aak

Mathematical Institute, Charles University, Sokolovsk !aa 83, 18675 Prague 8, Czech Republic

E-mail: malek@karlin.mff.cuni.cz, prazak@karlin.mff.cuni.cz

Received August 19, 1999; revised June 11, 2001

The method of ‘-trajectories is presented in a general setting as an alternative approach to the study of the large-time behavior of nonlinear evolutionary systems.

It can be successfully applied to the problems where solutions suffer from lack of regularity or when the leading elliptic operator is nonlinear Here we concentrate on systems of a parabolic type and apply the method to an abstract nonlinear dissipative equation of the first order and to a class of equations pertinent to nonlinear fluid mechanics In both cases we prove the existence of a finite-dimensional global attractor and the existence of an exponential attractor # 2002 Elsevier Science (USA) Key Words: large time behavior; global attractor; exponential attractor; finite fractal dimension; ‘-trajectory; fluids with shear-dependent viscosity; power-law fluids.

INTRODUCTIONThe aim of this paper is to present a promising and powerful tool fordealing with the large-time behavior of nonlinear dissipative systems Thisnew approach, called the method of ‘-trajectories, is based on an observationthat the limit behavior of solutions to a dynamical system in an originalphase space can be equivalently captured by the limit behavior of

‘-trajectories; these are (continuous) parts of solution trajectories that areparametrized by time from an interval of the length ‘; ‘ > 0:

In this paper we focus on systems of partial differential equations of aparabolic type and apply the method of ‘-trajectories to

(1) an abstract nonlinear dissipative equation of the first order,(2) the system of equations describing the motion of a class

of non-Newtonian incompressible fluids (fluids with shear-dependentviscosity)

1 This research was supported by Grants MSM 113200007 and GACR 201/00/0768.

243

0022-0396/02 $35.00

# 2002 Elsevier Science (USA)

In both cases we prove the existence of a global attractor with finitefractal dimension and the existence of an exponential attractor Most of theresults are new.

The alternative way of describing the dynamics, which we aregoing to explain, allows us on one hand to weaken the requirements onthe regularity of the solution and on the other hand enables us to formulatethe results for a broader class of nonlinearities and, even more, to treat theproblems where the leading elliptic operator is nonlinear The reader cancompare our results with the theories presented in [1, 4, 7, 11, 24], forexample

Before explaining the main features of the method, we want to emphasizethat the method of ‘-trajectories is applicable not only to dissipativeequations of parabolic type: The same approach can be successfullyapplied to hyperbolic type problems As an example, the question of thefinite-dimensionality of the global attractor for the wave equation withnonlinear damping is addressed in the forthcoming paper [20], where theauthor proves that the fractal dimension of a global attractor for thedamped wave equation is finite provided that the damping function isstrictly increasing of the polynomial growth – a result which to ourknowledge does not have an analogy using other methods, such as that ofthe Lyapunov exponents, for example And last, but not least, it turns outthat the dynamical system of ‘-trajectories is also the natural description forstudying the dynamics of nonlinear dissipative systems with bounded delay;see [19]

Now, let us explain the essence of the method of ‘-trajectories Consider anonlinear system of differential equations written as an abstract evolu-tionary problem,

u0ðtÞ ¼ F ðuðtÞÞ in X ðt > 0Þ;

where X is an (infinite-dimensional) Banach space, F : X /X is a nonlinear operator, and u02 X : To give a brief characterization of the method of

‘-trajectories, let us assume for a while that solution operators fS tgt50 to

(0.1) defined by S t u0¼ uðtÞ form a semigroup and in addition that fS tgt50

possesses a global attractor A  X :

We describe such an arrangement of the dynamics in an equivalent way

by introducing two mappings The first mapping b adds to any u02 X the

‘-trajectory that begins at u0 (see Fig 1); i.e., we consider b as a mapping from X into a subset of X‘ ¼ L2ð0; ‘; X Þ defined by

fbðu ÞgðtÞ ¼ S u ;

The second mapping e assigns to any ‘-trajectory w its end point2 (seeFig 2); i.e.,

eðwÞ:¼ wð‘Þ:

Now, we use b to introduce a new semigroup fL tgt50 acting on the set of

‘-trajectories defined as (see Fig 3)

2Trajectories are supposed to be continuous at least in the weak topology of X; the value wð‘Þ

has then a clear meaning.

it usually turns out that A‘ is a global attractor related to fL tgt50: Thecomplete structure is drawn in Fig 4.

One might ask: What is the advantage of this alternative viewpoint on A?Clearly, instead of estimating the fractal dimension of A directly, we aregiven the possibility of estimating the fractal dimension of A‘ in the

topology of X‘; which is revealed to be an easier task.3After proving thefiniteness of the fractal dimension of A‘ one observes (see Lemma 1.2

below) that if e is Lipschitz (or at least a-H.oolder) continuous then the fractaldimension of A cannot increase (or increases at most 1=a times)

Note that the roles of b and e are different While b transfers the dynamics from X into X‘; the mapping e is responsible for delivering the properties of

A‘to A: The roles of e and its regularity are more important than the role

FIGURE 3

FIGURE 4

3 For example, as shown in this paper, the criterion stated in Lemma 1.3 below is applicable to

L with ease, while it can be applied to S only in special (quite regular) cases.

of b: It can happen that A or fS tgt50 is not defined in X : Such a case can

occur when the solution is not unique or when one does not have enough

regularity to construct A: Still, it might be possible to introduce fL tgt50: Inthis case the attractor A‘is constructed first, and after evaluating its fractal

dimension we set A to be eðA‘Þ: Not only does the defined set A have

properties of the global attractor in X ; but also its fractal dimension is finite (provided e is Lipschitz or H.oolder continuous).4

The paper is organized in the following way: Section 1 recalls thedefinitions of basic notions and provides general helpful assertions Section

2 presents the method of ‘-trajectories in a general framework In Section 3

we provide a class of evolutionary problems with a nonlinear (monotone)elliptic operator, for which the assumptions of the general scheme areverified directly Models from nonlinear fluid mechanics that include theSmagorinski model of isotropic turbulence and other shear-thickening fluidmodels in a three-dimensional setting and shear-thinning fluids in twodimensions are studied in Section 4; several new results are obtained bothfor the space periodic and the Dirichlet problems Conclusions andperspectives are presented in the last section, which also includesbibliographical notes

1 DEFINITIONS AND BASIC LEMMAS

In this section we recall several notions from the theory of dynamicalsystems

Let X be (a subset of) a normed space One parameter family of (nonlinear)mappings St: X/X ðt50Þ is called the semigroup provided that

Stþs¼ StSs for all t; s50 and S0¼ I:

A typical example is a semigroup formed by the solution operators for acertain evolutionary problem, defined on some suitable space of initialconditions for which there exists a unique global solution

The couple ðSt; XÞ is usually referred to as a dynamical system

A set A  X is called a global attractor to ðSt; XÞ if (i) A is compact in

X; (ii) S t A ¼ A for all t50; and (iii) for any B  X that is bounded,

distXðSt B; AÞ ! 0 as t ! 1; where distXðB; AÞ ¼ supb2Binfa2A jjb  ajjX:Note that a dynamical system can have at most one global attractor

A set C  X is called positively invariant w.r.t St if for all t50; S tC  C;and it is called uniformly absorbing w.r.t Stif for any B  X that is bounded

there exists t0¼ t0ðBÞ such that St B  C for all t5t0:

4 An example of such a situation is given in Section 4.1.

Lemma 1.1 Let ðSt; XÞ be a dynamical system Assume that there exists acompact set B1  X which is uniformly absorbing and positively invariantw.r.t St: Let moreover St be continuous on B1: Then ðSt; XÞ has a globalattractor.

Proof We simply set A to be the o-limit set of B1; cf [24] ]

Finally, the fractal dimension5 of a compact set C  X; denoted by

Proof Since jjFðuÞ  FðvÞjjY4cjju  vjjaX; it holds that

FðBXðu; ðe=cÞ1=aÞÞ  BYðFðuÞ; eÞ:

log NX

ðe=cÞ1=aðCÞlogð1=eÞ ¼

log NX

Z ðCÞ

a logð1=ZÞ  log c;

which leads to the conclusion letting e ! 0 ð) Z ! 0Þ: ]

5 The importance of the notion of a finite fractal dimension is illustrated by a result of Foias and

Olson [5]: if C is a compact metric space such that d C

in addition be an orthogonal projector.

6 By this sentence we simply mean that a-H oolder continuity is considered between metrics of X and Y respectively, though the mapping F can only be defined on a proper subset C of X:

Lemma 1.3 Let X; Y be normed spaces such that Y+ + X and C 

X be bounded Assume that there exists a mapping L such that LC  C andL: X/Y is Lipschitz continuous on C: Then dX

fðCÞ is finite

Proof Let k be a Lipschitz constant of L and N be the number of balls

in X of radii 1=4k necessary to cover the unit ball in Y: Let us choose R > 0 and u 2 C such that C  BXðu; RÞ: Then we have

Repeating the scheme inductively, we have NX

R=2k ðCÞ4N k: Now, for any

positive e4R there exists an integer k50 such that R=2 k 5e > R=2 kþ1: Then

log NX

e ðCÞlogð1=eÞ 4

is an exponential attractor w.r.t the dynamical system ðSt; B1Þ if

(i) E is compact,

(ii) E is positively invariant w.r.t St;

(iii) dX

fðEÞ is finite,

(iv) there exist c1; c2> 0 such that distXðStB1; EÞ4c1e c2t for all t50:

Note that necessarily A  E; so the basic idea behind the exponentialattractor is to enlarge the global attractor so that the rate of convergencebecomes exponential, yet keep the ‘‘good’’ properties (i)–(iii) The followinglemma resumes a criterion on the existence of the exponential attractorobtained in [4]

Lemma 1.4 Let X be a Hilbert space Let B1 X satisfy theassumptions of Lemma 1.1 Assume that there exists t > 0 such that

ðP1Þ St: X/X is Lipschitz continuous on B1;

ðP2Þ there exist W 2 ð0; 1=4Þ and a finite-dimensional orthogonal projector

P: X/X such that for all x1; x22 B1 there holds either

jjStx1 Stx2jjX4 ffiffiffi

2

p

jjP ðStx1 Stx2ÞjjXor

jjStx1 Stx2jjX4Wjjx1 x2jjX;and

/X defined by Gðx; tÞ :¼ S t x is on

B1 oolder continuous

Then the dynamical system ðSt; B1Þ possesses an exponential attractor.Proof See [4, Chaps 2 and 3] ]

We will also use this elementary lemma

Lemma 1.5 Let X; Y be normed spaces such that Y++ X; let moreover X

be a Hilbert space Then for a given e > 0 there exists a finite-dimensional subspace

Xn X such that, denoting by P the ortho-projector to Xn

;

jjðI  P ÞujjX4ejjujjY for any u 2 Y:

Proof Without loss of generality we assume that u 2 S ¼ fv 2 Y; jjvjjY¼

1g: But the set S  X is compact, and denoting by u1; ; u n its e-net, we seethat the space Xn spanned by u1; ; u n has the desired property ]

Finally, for the reader’s convenience we formulate the celebrated so-calledAubin–Lions lemma as it plays an important role in our paper

Lemma 1.6 Let p1 2 2 ½1; 1Þ: Let X be a Banach space and

Y ; Z be separable and reflexive Banach spaces such that Y ++ X + Z: Then

for any t 2 ð0; 1Þ;

fu 2 L p1ð0; t; Y Þ; u02 L p2ð0; t; ZÞg ++ L p1ð0; t; X Þ:

Proof See [23], for example ]

2 GENERAL SCHEMEThe method of ‘-trajectories can be used for various purposes in studyingthe large-time behavior of dynamical systems Depending on the purpose,

one needs certain assumptions to be satisfied In order to make thispresentation transparent we divide the general scheme, and correspondinglyalso the assumptions, into several subsections that emphasize their specificrole The titles of these subsections are:

(1) Dynamical system on the set of ‘-trajectories,

(2) A‘ – Attractor in the set of ‘-trajectories,

(3) Finite fractal dimension of A‘;

(4) A – Attractor in the original space,

(5) Finite fractal dimension of A;

(6) E‘ – Exponential attractor in the set of ‘-trajectories,

(7) E – Exponential attractor in the original space

(1) Dynamical System on the Set of ‘-Trajectories

The first assumptions concern the existence and uniqueness of thesolution to (0.1)

Let ðX ; jj  jj X Þ; ðY ; jj  jj Y Þ; and ðZ; jj  jj Z Þ be three Banach spaces, X being

reflexive and separable, such that

moreover, for any solution the estimates of jjujj Y T are uniform with respect to

jjuð0Þjj X:

(A2) There exists a bounded set B0 X with the following properties:

if u is an arbitrary solution to (0.1) with initial condition u02 X then (i) there exists t0¼ t0ðjju0jjX Þ such that uðtÞ 2 B0 for all t5t0 and (ii) if u02 B0 then

uðtÞ 2 B0 for all t50:

Now, let ‘ > 0 be an arbitrary fixed number By the ‘-trajectory we mean

by X‘ and equipped with the topology of X‘ ¼ L2ð0; ‘; X Þ: Note that since

trajectories On the other hand, it is not clear whether X‘ is closed in X‘

and hence X‘ in general is not a complete metric space

Since we do not require uniqueness of the solution, it is possible that more than

one trajectory will start from a point u02 X : We will impose a weaker condition:

(A3) Each ‘-trajectory has among all solutions unique continuation

In other words, from an end point of an ‘-trajectory there starts at most onesolution Combined with the assumption (A1) about the global existence ofsolutions this in particular implies that if w 2 X‘ and T > ‘ then there exists a

:7

Using (A3), we can define the semigroup L t on X‘ by

fL twgðtÞ:¼ uðt þ tÞ;

¼ w:

(2) A‘ – Attractor in the Set of‘-Trajectories

We define B0‘ as the set of all ‘-trajectories starting at any point of B0

from (A2) In symbols,

B0‘ :¼ fw 2 X‘; wð0Þ 2 B0g:

Observe that owing to (A2), B0‘ is positively invariant w.r.t L t: We add twomore assumptions:

(A4) For all t > 0; L t : X‘/X‘ is continuous on B0‘;

(A5) For some t > 0; L tðB0

‘ÞX‘  B0

‘:

7 This assumption is suited to cover the situation in fluid mechanics where inner points of

trajectories belong to a better space than X ; hence solutions starting from them are more regular

and consequently unique even in the wider class of weak solutions Similarly, (A3) would be satisfied if we work with equations containing terms delayed (in time) by ‘; at most; cf [19].

The assumption (A5) represents the crucial step in overcoming the problem

of incompleteness of X‘; since it asserts that the closure (¼ completion) of

L tðB0

‘Þ remains in X‘: Yet the assumption (A5) is naturally fulfilled provided

that B0 is (weakly) closed and we have the ‘‘compactness’’ of solutions; cf.(E2) in Section 3 or the proof of Theorem 4.1 in Section 4

Proof Consider a set B1‘  X‘defined in (2.3) Clearly, B1‘ is closed, and

by (A1), (A2) it is bounded in Y‘++ X‘; and hence compact Moreover, by

the continuity of L t – cf (A4) – and the positive invariance of B0‘; we have

‘ is uniformly absorbing Let B  X‘ be bounded by some

constant C: Then for w 2 B; one has R‘

0jjwðtÞjj2X d t4C and hence there is

X 4C= ffiffiffi

‘

p

: But then by (A2) there exists t0 > 0 such

that L tw  B0‘ for t5t0; t0 depending on C only.

By these considerations, the assumptions of Lemma 1.1 with St¼

L t; X ¼ X‘; and B1 ¼ B1‘ are satisfied and the existence of the globalattractor follows ]

(3) Fractal Dimension of A‘

The assumption which leads to the finiteness of the fractal dimension andwhich is also a key step in constructing the exponential attractor reads(A6) There exists a space W‘ with W‘++ X‘ and t > 0 such that

Theorem 2.2 Let (A1)–(A6) hold Then the fractal dimension of A‘in X‘

is finite

Proof We apply Lemma 1.3 with X ¼ X‘; Y ¼ W‘; L ¼ Lt and C ¼

A‘: Since A‘ X‘is compact, L tA‘ ¼ A‘; and (A6) holds, we see that allassumptions of the lemma are fulfilled ]

(4) A – Attractor in the Original Space

Now, we introduce a mapping e: X‘/X which to a given ‘-trajectory

assigns its end point In symbols,

eðwÞ ¼ wð‘Þ:

In this manner we construct a one-way bridge between the set X‘on one side

and the space X on the other side Note that the definition of e is meaningful

since, due to (A1), trajectories are weakly continuous We define

B1:¼ eðB1‘Þ:

Observe that by (A3) to a given initial condition u02 B1there corresponds a

unique solution to (0.1), hence solution operators S t are defined on B1:

Moreover, B1 is positively invariant w.r.t S t:

Next, supposing that

(A7) e : X‘/X is continuous on B1

‘

and defining

we obtain the following theorem

Theorem 2.3 Let (A1)–(A5) and (A7) hold Then A defined in (2.4) is a

global attractor to dynamical system ðS t ; B1Þ:

Proof Since A is a continuous image of a compact set, it is compact

Also, since L tðA‘Þ ¼ A‘; we have

S t ðAÞ ¼ S t ðeðA‘ÞÞ ¼ eðL tðA‘ÞÞ ¼ eðA‘Þ ¼ A:

To verify the attracting property of A; we proceed by contradiction: let

there exist sequences u n 2 B1; t n! 1; and a d > 0 such that

By the definition of B1 there exists a sequence fwng  B1

‘ such that

eðwn Þ ¼ u n: Since fwng is bounded and A‘ is an attractor w.r.t L t; we can

assume – coming to a subsequence if necessary – that L t nwn! w 2 A‘: But

by the continuity of e; S t n u n ¼ eðL t nwn Þ ! eðwÞ 2 A; which contradicts

(2.5) ]

Remark 2.1 The set A is also a global attractor to the dynamics of

(0.1) on the whole space X in the following sense: if B  X is bounded and

B t denotes the set of all values of all solutions to (0.1), starting from B; at time t; then

distX ðB t; AÞ ! 0 as t ! 1:

Indeed, by (A2), B t  B0 for t  t0; hence B t  B1for t  t0þ t and B t

S tðt0þtÞB1 for t sufficiently large.

(5) Finite Fractal Dimension of A

If we strengthen (A7) and require that

(A8) e : X‘/X is a-H.oolder continuous on B1

‘;

we come to the following assertion

Theorem 2.4 Let (A1)–(A6) and (A8) hold Then the fractal dimension

Proof The proof is a consequence of Theorem 2.2, (A8), and Lemma

1.2, where we take X ¼ X‘; Y ¼ X ; F ¼ e; and C ¼ A‘:

(6) E‘ – Exponential Attractor in the Set of‘-Trajectories

To construct an exponential attractor we will require

(A9) For all t > 0 the operators L t : X‘/X‘ are uniformly (with

respect to t 2 ½0; t]) Lipschitz continuous on B1‘;

(A10)

w 2 B1‘ and t1; t2 t1w  L t2wjjX‘4cjt1 t2jb:

Theorem 2.5 Let X be a Hilbert space and let assumptions (A1)–(A6)

and (A9)–(A10) hold Then ðL t; B1Þ possesses an exponential attractor E‘:

Proof We will apply Lemma 1.4 with X ¼ X‘; St ¼ L t and B1¼ B1

‘:

Note that since X is the Hilbert space, X‘is the Hilbert space as well Let usverify the assumptions (P1)–(P3) We fix a t > 0 for which (A6) holds Then(P1) follows from (A6) or from (A9)

Next, let k be the Lipschitz constant of the mapping Lt: X‘/W‘ on B1‘:

We apply Lemma 1.5 with X ¼ X‘; Y ¼ W‘; and e ¼ 1

8k; then there exists a

finite-dimensional ortho-projector P such that jjðI  P Þwjj X‘41=ð8kÞjjwjjW‘ for

all w 2 W‘: Thus we have

64jjw1 w2jj

2

X‘:

But whenever it holds that a þ b4c þ d then necessarily either a4c or b4d;

which is just (P2) with W ¼ ð4 ffiffiffi

2

p

Þ1:Finally, (A9), (A10) imply (P3) since

jjGðw1; t1Þ  Gðw2; t2ÞjjX‘ ¼ jjL t1w1 L t2w2jjX‘

4 jjL t1w1 L t1w2jjX‘þ jjL t1w2 L t2w2jjX‘

4#ccðjt1 t2jbþ jjw1 w2jjX‘Þ:

The proof of Theorem 2.5 is complete ]

(7) E – Exponential Attractor in the Original Space

Analogously to subsection (4) where we have obtained A from A‘via the

mapping e: X‘/X we obtain E as an image of E‘: We put

Theorem 2.6 Let (A1)–(A6), (A8)–(A10) hold Then E defined in (2.6) is

an exponential attractor to the dynamical system ðS t ; B1Þ:

Proof It immediately follows from the facts that E‘ is an exponential

attractor w.r.t ðL t; B1Þ and the mapping e is H.oolder continuous

We conclude by two lemmas that are often useful in verifying theassumptions (A4), (A8), (A9), and (A10).

Lemma 2.1 Let C‘  X‘be a set of trajectories and let C  X be defined

ðiiÞ the operator e: X‘/X is Lipschitz continuous on C‘:

Proof Note that by our assumptions, L tare defined on C‘: Let w1; w22

To prove (ii), note that eðw1Þ ¼ w1ð‘Þ ¼ S ‘sfw1

analogously for w2: Therefore

jjeðw1Þ  eðw2Þjj2X ¼ jjS ‘sfw1ðsÞg  S ‘sfw2ðsÞgjj2X 4cjjw1ðsÞ  w2ðsÞjj2X

obtain (ii) ]

Lemma 2.2 Assume (A3) holds Let C‘ X‘ be a set of trajectories

such that L tC‘  C‘ for all t50; and, let fw0; w 2 C‘g be bounded in space

3 APPLICATION TO AN ABSTRACT PARABOLIC EQUATION

In this section we consider a class of evolutionary problems of the type

u0ðtÞ þ N ðuðtÞÞ þ QðuðtÞÞ ¼ f ;

The choice of assumptions which we are going to impose on nonlinear

operators N and Q is twofold On one hand, they involve a class of nonlinear

strongly monotone elliptic operators As mostly the large time asymptotic isconsidered for the problems where the dissipative operator is linear, we seethat (3.1) covers a larger class of problems than those usually studied Onthe other hand, this class of problems is not too general so that we can verifythe assumptions (A1)–(A10) of the general framework of the method of

‘-trajectories in a rather direct way

Let X be a Hilbert space and Y be a Banach space such that Y ++ X and

We denote by ð; Þ the scalar product in X and by h; i the duality between

Yn

and Y so that for F 2 Yn

; v 2 Y we write FðvÞ ¼ hF; vi: The spaces X

and Y are regarded as subspaces of Yn

K½u 3ðjjujj X þ jjvjj X þ 1Þgðjjujj Y þ jjvjj Y þ 1Þb

holds for any j 2 Y for almost all t 2 ð0; T Þ:

Observe that (E1) implies, that

hN ðuÞ; ui 5 3c1

4 jjujj

2

Y *cc1;

jhN ðuÞ; jij 4*cc2ðjjujj Y þ 1ÞjjjjjY;

jhQðuÞ; jij 4*cc3ðjjujj Xþ 1Þbþgðjjujj Y þ 1ÞjjjjjY; ð3:5Þwhich in particular ensures that (3.4) makes sense for the functions (3.3);

; X Þ (see [6]), the initial condition

is meaningful

Further, we assume that the solutions to (3.1) are compact in the followingsense:

ðE2Þ

in X there exists a subsequence converging ð *Þ weakly in spaces

Finally, we assume that

ðE3Þ for arbitrary u0 2 X ; f 2 Yn

; and T > 0 there exists at least one

solution:

The reader familiar with this type of parabolic equation will note that byusing a priori estimates obtained below, (E2) could be derived solely on thebasis of the assumptions in (E1) Namely, the limiting process in the

equation can be done by compact embedding and monotonicity of N : Along

the same lines, the existence of a solution in (E3) can be obtained by theGalerkin method However, to avoid such a lengthy and technical procedure

we formulate (E2) and (E3) simply as assumptions

In the following theorem, we verify that the general scheme of theprevious section is applicable to Eq (3.1)

Theorem 3.1 Let (E1)–(E3) hold Then assumptions (A1)–(A9) aresatisfied Consequently, the dynamical system (3.1) has a global attractor withfinite fractal dimension

If moreover solutions to (3.1) satisfy

with estimates depending on jjuð0Þjj X; then (A10) is also satisfied and,consequently, the dynamical system (3.1) possesses an exponential attractor.Proof The general scheme of Section 2 will be applied with p1 ¼ p2¼ 2:Particular assumptions will be verified in corresponding steps

Step 1 The existential part of (A1) is just (E3) To prove the second part(the estimates in the spaces in (3.3)) note that by (3.5) we are justified to take

j ¼ uðtÞ in (3.4) We use (E1) and the fact that 2hu0ðtÞ; uðtÞi ¼ d

dt jjuðtÞjj2X (see[6]) to obtain that

12

(3.5) by the duality argument

Step 2 Using the embedding (3.2) and denoting *cc4:¼ 2 maxð*cc1þ c4;2