Global and exponential attractors for nonlinear reaction–diffusion systems in unbounded domains

Proceedings of the Royal Society of Edinburgh, 134A , 271{315, 2004

Global and exponential attractors for nonlinearreaction{di® usion systems in unbounded domains

M. EfendievMathematisches Institut A, Universit�at Stuttgart,Pfa¬enwaldring 57, 70569 Stuttgart, Germany

A. Miranville and S. ZelikUniversite de Poitiers, Laboratoire d’Applications desMathematiques{SP2MI, Boulevard Marie et Pierre Curie,Teleport 2, 86962 Chasseneuil Futuroscope Cedex, France

(MS received 2 December 2002; accepted 10 September 2003)

We study the long-time behaviour of solutions of autonomous and non-autonomousreaction{di® usion equations in unbounded domains of R3 . It is shown that, underappropriate assumptions on the nonlinear interaction function and on the externalforces, these equations possess compact global (uniform) attractors in thecorresponding phase space. Estimates for Kolmogorov’ s "-entropy of these attractorsin terms of Kolmogorov’ s entropy of the external forces are given. Moreover,(in¯nite-dimensional) exponential attractors with the same entropy estimate as thatof the corresponding global (uniform) attractor are also constructed.

1. Introduction

We consider the following system of reaction{di¬usion equations,

@tu ¡ a¢ xu + f (u; rxu) + ¶ 0u = g;

ujt= 0 = u0; uj@« = 0;

)

(1.1)

where « is an unbounded domain of R3 with a su¯ ciently regular boundary @«(see x 2), u = (u1; : : : ; uk) is an unknown vector-valued function, ¢ x is the Laplacianwith respect to x, f (u; rxu) and g = g(x) are given interaction function and exter-nal forces, respectively, and ¶ 0 > 0 and a > 0 are given constants. Such equationsappear, for instance, in turbulence when one considers convection-reaction{di¬usionequations of (generalized) Burgers type (see [7]),

@tu ¡ ¢ xu = juj¬ u + u a ¢ rxu;

where ¬ ; > 0 and a 2 RN , or when one considers models in population dynamics,as in [30],

@tu ¡ ¢ xu + ¶ 0u = · u ¬ ¡ ¸ jrxujD(jrxuj);

where ¶ 0 > 0; ¬ ; · and ¸ > 0 and where D(jruj) corresponds to a predator’sdensity. Equations of the form (1.1) are also important in the study of viscous

271

c® 2004 The Royal Society of Edinburgh

272 M. Efendiev, A. Miranville and S. Zelik

Hamilton{Jacobi equations (see, for instance, [5]). This class of problems also con-tains the case where the nonlinear term does not depend explicitly on rxu. Aparticularly important example is the Chafee{Infante equation (which appears inthe modelling of phase transitions (see [6])),

@tu ¡ a¢ xu + f (u) + ¶ 0u = 0;

where f (u) = au3 + bu2 + cu, a > 0.The long-time behaviour of the solutions of (1.1) is of a great current interest. It is

well known that, under appropriate assumptions on the nonlinear term f(u; rxu),on the external forces g and on the domain « , this behaviour can be described interms of the global and/or exponential attractors of the dynamical system generatedby (1.1) (see, for example, [1{4,14{16,18{20,23,24,26,28,29,31,33{36,38] and thereferences therein). In particular, when « is bounded, the global attractor A ofproblem (1.1) has usually nite Hausdor¬ and fractal dimensions (see [4, 31]). Inthat case, in nite-dimensional attractors can naturally appear only in the caseof non-autonomous equations (1.1) (e.g. with g = g(t)) and only when the timedependence of the external forces is described by an in nite number of parameters(e.g. in the case of almost-periodic external forces (see [9,11])). In contrast to this,in the case of unbounded domains, the attractor of problem (1.1) has usually in niteHausdor¬ and fractal dimensions, even in the autonomous case and, consequently,in that case, the in nite dimensionality of the attractor has an internal natureand can appear even when the external forces vanish (see [12,18,29,35,36,38], seealso [13,22,37] for analogous results for damped hyperbolic equations in unboundeddomains).

Nevertheless, there exist several particular cases of equations (1.1) in unboundeddomains « for which the associated dynamics is nite dimensional (and very sim-ilar to the case of bounded domains (see [1, 3, 8, 15, 16]). In such cases, the non-linear interaction function has a special form which allows energy income in abounded subdomain only (we will call this subdomain the e¬ective domain of inter-action « e¬ ). Furthermore, outside this subdomain, we only have energy dissipationand the in®uence of the processes acting inside « e¬ decays exponentially withrespect to the distance to « e¬ . As we will see below, this e¬ect explains the nitedimensionality of the attractors (see also remark 4.7).

In the present paper, we give a systematic study of one of the particular casesmentioned above, which is determined by the following assumption on the nonlin-earity f (u; rxu),

f (v; w) ¢ v > 0 for all v 2 Rk and w 2 R3k (1.2)

(here and below, u:v denotes the standard inner product in Rk), and assumingthat the external forces g(x) tend to zero in an appropriate sense as jxj ! 1.For instance, the Chafee{Infante equation satis es (1.2) for a proper choice of theconstants a, b and c. The case « = Rn and g belonging to a weighted Sobolevspace L2

¿ ¬(Rn), with ¿ ¬ (x) := (1 + jxj2) ¬ and ¬ > 0, was studied in [1{3,8,15] and

the case ¬ = 0 was considered in [16] (exponential attractors for (1.1) in weightedSobolev spaces W 2;2

¿ ¬(Rn) were constructed in [2,15] under additional strong growth

restrictions on the nonlinearity f (u; rxu)).

Global and exponential attractors 273

In this paper, we remove the growth restrictions on f (u; rxu) (see x 3 for the pre-cise conditions on f (u; rxu)) and weaken the assumptions on the external forces gby taking

g 2 _L2b( « ) :=

ng 2 L2

loc( « ); limjxj! 1

kgkL2( « \ B1x) = 0

o; (1.3)

where BRx denotes the open R-ball in R3 centred at x 2 R3.

We note that (1.3) requires no additional assumption on the rate of convergenceof g(x) ! 0 as jxj ! 1 and, consequently, the space _L2

b( « ) is larger than L2( « ).On the other hand, as elementary examples show (see [3,16]), the dimension of theattractor may be in nite (even under assumption (1.2)) if (1.3) is violated. Thusassumption (1.3) looks like a sharp border between nite- and in nite-dimensionaldynamics.

The following theorem gives the spatial asymptotics for the attractor A up toexponentially small terms as jxj ! 1 and clari es the nature of its nite dimen-sionality (see also [37] for an analogous result for nonlinear damped hyperbolicequations in an unbounded domain).

Theorem 1.1. Let the above assumptions hold and let z0 be an arbitrary equilib-rium of (1.1). Then there exists an e® ective radius Re¬ > 0, which is determinedby the rate of convergence in (1.3), such that

jv(x) ¡ z0(x)j 6 Ce¡ d i s t(x;BReff0 ) for all v 2 A; (1.4)

where the constants > 0 and C > 0 are independent of x 2 « , v 2 A and Re¬ .

Estimate (1.4) suggests that the fractal dimension of the attractor should be esti-mated in terms of the e¬ective radius Re¬ . The next theorem gives such estimatesfor the case of global A and exponential M attractors of problem (1.1).

Theorem 1.2. Let the above assumptions hold. Then there exists a ¯nite-dimens-ional exponential attractor M for problem (1.1) such that

dimF (A; C( « )) 6 dimF (M; C( « )) 6 C vol(« \ BReff

0 ); (1.5)

where the constant C is independent of Re¬ .

Moreover, examples of equations (1.1) for which estimate (1.5) is sharp withrespect to the radius Re¬ are also given (see x 7). We also note that the proof oftheorem 1.2 essentially uses the construction of exponential attractors in Banachspaces proposed in [19].

We then apply theorem 1.2 to the following reduction{di¬usion scheme (RDS)with a small di¬usion parameter ¸ ½ 1,

@tu = ¸ ¢ xu ¡ f (u) ¡ ¶ 0u + g; uj@« = 0; (1.6)

where, in contrast to (1.1), the interaction function f does not depend explicitlyon the gradient rxu. Indeed, after the rescaling x ! x0 ¸ 1=2 in (1.6), we obtainequation (1.6) with ¸ 0 = 1 and with external forces g (x0) := g(x0 ¸ 1=2). Applyingthen theorem 1.2 to this new equation and noting that Re¬ (g ¸ ) = ¸ ¡1=2Re¬ (g), weobtain the following result.


Corollary 1.3. Let the above assumptions hold and let, in addition, g belongto _L 1 ( « ) (which is de¯ned analogously to (1:3)). Then equation (1.6) possessesa uniform family of exponential attractors M ¸ whose fractal dimensions can beestimated by

dimF (A ¸ ; C( « )) 6 dimF (M ; C( « )) 6 C1 ¸ ¡3=2; (1.7)

where C1 is independent of ¸ , and this estimate is sharp (see x 7 for details).

We also consider the non-autonomous analogue of system (1.1),

@tu ¡ a¢ xu + f(u; rxu) + ¶ 0u = g(t);

ujt= 0 = u0; uj@« = 0;

)

(1.8)

where the external forces g(t) depend explicitly on t. Following the general schemeof construction of the (uniform) attractor for non-autonomous equations via theskew-product technique (see [9{11,24]), we introduce the hull g 2 C(R; _L2

b( « )),

H(g) := [Thg; h 2 R]Cloc(R; _L2b( « )); (Thg)(t) := g(t + h); (1.9)

where [¢]V denotes the closure in the space V . A standard assumption on the non-autonomous external forces g (see [6,7,9{11,38]) is that the hull H(g) be compactin the appropriate topology, namely,

H(g) »» Cloc(R; _L2b( « )): (1.10)

Some necessary and su¯ cient conditions to have such a compactness are given in x 4.As usual, in order to construct the (uniform) attractor for the non-autonomous

problem (1.8), we consider a family of equations of type (1.8) for all external forces¹ 2 H(g) and de ne the extended skew-product semigroup associated with prob-lem (1.8) by

St : © b £ H(g) ! © b £ H(g); St(u0; ¹ ) := (u ¹ (t); Ttg); (1.11)

where © b is an appropriate phase space for problem (1.8) (see x 3) and u ¹ (t) denotesthe solution of (1.8) with external forces ¹ 2 H(g) and with u ¹ (0) = u0. Then, byde nition, the projection A := ¦ 1A of the global attractor A of semigroup (1.11)onto the rst component is called the uniform attractor of the non-autonomousproblem (1.8) (see [9{11,24,38] and x 4 for details).

In contrast to the autonomous case, even in bounded domains « , the uniformattractor A of problem (1.8) has naturally in nite fractal and Hausdor¬ dimen-sions if the hull H(g) is in some sense in nite dimensional and, consequently, newquantitative characteristics are required in order to study such attractors. One pos-sible approach to handle this problem, which was suggested in [10], is to studyKolmogorov’s "-entropy of the uniform attractor.

We recall (see [25] for details) that, if K is a precompact set in a metric space M ,then, for every " > 0, it can be covered by a nite number of "-balls in M . LetN"(K; M ) be the minimal number of such balls. Then, by de nition, Kolmogorov’s"-entropy of K in M is the following number:

H"(K; M ) := log2 N"(K; M ): (1.12)


It is worth emphasizing that, in contrast to the fractal dimension, quantity (1.12) is nite for every precompact set K in M and, in particular, it is nite for the uniformattractor A of problem (1.8).

The entropy of in nite-dimensional uniform attractors of problem (1.8) in bound-ed domains « was studied in [9{11]. The case of autonomous RDEs in Rn wasconsidered in [12,33]. The entropy of attractors of autonomous and non-autonomousRDEs in unbounded domains was investigated in [35,36,38]. Analogous results fordamped hyperbolic equations were obtained in [37].

We now recall that an exponential attractor M (if it exists) always containsthe global (uniform) attractor (see, for example, [2, 14, 19{21]). Consequently, a¯nite-dimensional exponential attractor M cannot exist for problem (1.8) if thedimension of the associated uniform attractor is in nite. Therefore, it is natural,following [20], to introduce the notion of an in nite-dimensional exponential attrac-tor and use Kolmogorov’s "-entropy in order to control its `size’. It is also natural toseek for an exponential attractor with the same type of asymptotics of the "-entropyas that of the uniform attractor,

H"(A; © b) ¹ H"(M; © b):

In nite-dimensional exponential attractors for non-autonomous reaction{di¬usionequations in bounded domains « were constructed in [20]. In the present paper, weextend this result to the case of unbounded domains « .

Theorem 1.4. Let the above assumptions hold. Then there exists an in¯nite-di-mensional exponential attractor M for the non-autonomous problem (1.8), whichpossesses the following entropy estimate

H"(A; C( « )) 6 H"(M; C( « ))

6 C vol(« \ BReff

0 ) log2

1

"

+ H"=K

µH(g)j[0;L log2(1=")]£ « ; C

µ·0; L log2

1

"

¸; _L2

b( « )

¶¶;

(1.13)

where the constants C, K and L are independent of " and the e® ective radius Re¬

is determined in the same way as in theorem 1.1.

Although the exponential attractor M constructed in theorem 1.4 is a prioriin¯nite dimensional, estimate (1.13) guarantees that it is ¯nite dimensional if thehull H(g) is nite dimensional; in particular, this is the case for quasi-periodicexternal forces.

Corollary 1.5. Let the above assumptions hold and let the external forces g(t)be quasi-periodic with respect to t, with m rationally independent frequencies. Thenproblem (1.8) possesses a ¯nite-dimensional exponential attractor M such that

dimF (A; C( « )) 6 dimF (M; C( « )) 6 C vol(« \ BReff ) + m; (1.14)

where C is the same as in (1.5).


As in the autonomous case, applying corollary 1.5 to the non-autonomous versionof problem (1.6), we obtain the following result.

Corollary 1.6. Let the above assumptions hold and let, in addition, the externalforces g(t) be quasi-periodic with respect to t, with m rationally independent frequen-cies, and belong to the space Cb(R £ « ). Then (1.8) possesses a uniform family ofnon-autonomous exponential attractors M ¸ and the following estimate holds,

dimF (A ¸ ; C( « )) 6 dimF (M ¸ ; C( « )) 6 C1 ¸ ¡3=2 + m; (1.15)

where the constant C1 is independent of ¸ and m.

Finally, in x 6, we also verify the sharpness of (1.15) and indicate examples ofnonlinearities f for which we have

C1 ¸ ¡3=2 + m 6 dimF (A ¸ ; C( « )) 6 dimF (M ¸ ; C( « )) 6 C2 ¸ ¡3=2 + m; (1.16)

where C1 and C2 are independent of ¸ and m.This paper is organized as follows. The de nitions of the functional spaces, which

are necessary to study problem (1.8) in an unbounded domain, and their basicproperties are brie®y indicated in x 2. Analytic properties of the solutions of prob-lem (1.8) in a nite time-interval, such as existence and uniqueness, regularity andsmoothing property, are established in x 3. The global (uniform) attractor for prob-lems (1.1) and (1.8) is constructed in x 4. Estimate (1.13) for the "-entropy of theglobal attractor A is obtained in x 5. The construction of an exponential attractorM, which satis es estimate (1.13), is given in x 6. Finally, in x 7, we apply the aboveresults to equation (1.6) and indicate analogous results for more general classes ofreaction{di¬usion equations with non-scalar di¬usion matrices a.

The results of the present paper have been partially announced in [19].

2. Functional spaces

In this section, we introduce several classes of Sobolev spaces in unbounded domainsand brie®y recall some of their properties which will be essential in the sequel. Fora detailed study of these spaces, see [16,37,38].

Definition 2.1. A function ¿ 2 L 1loc(Rn) is called a weight function with growth

rate · > 0 if the condition

¿ (x + y) 6 C¿ e · jxj ¿ (y); ¿ (x) > 0; (2.1)

is satis ed for every x; y 2 Rn.

Remark 2.2. It is not di¯ cult to deduce from (2.1) that

¿ (x + y) > C¡1¿ e¡ · jxj ¿ (y); (2.2)

is also satis ed for every x; y 2 Rn.

The following example of weight functions are of fundamental signi cance for ourpurposes:

¿ f g;x0(x) = e¡ jx¡x0 j; 2 R; x0 2 Rn:

(Obviously, this weight has the growth rate j j.)


Definition 2.3. Let « » Rn be some (unbounded) domain and let ¿ be a weightfunction with growth rate · . We set

Lp¿ ( « ) =

½u 2 D0( « ) : ku; « kp

¿ ;0;p ²Z

«

¿ (x)ju(x)jp dx < 1¾

:

Analogously, the weighted Sobolev space W l;p¿ ( « ), l 2 N, is de ned as the space of

distributions whose derivatives up to order l belong to Lp¿ ( « ).

In order to simplify the notation, we will write throughout the paper W s;pf g( « )

instead of W s;pe ¡ jxj( « ).

We also de ne another class of weighted Sobolev spaces as follows:

W l;pb;¿ ( « ) =

nu 2 D0( « ) : ku; « kp

b;¿ ;l;p = supx0 2 «

f¿ (x0)ku; « \ B1x0

kpl;pg < 1

o:

Here and below, we denote by BRx0

the ball in Rn of radius R centred at x0 andku; V kl;p stands for kukW l;p(V ).

We will write W l;pb ( « ) instead of W l;p

b;1( « ).

Proposition 2.4.

(1) Let u belong to Lp¿ ( « ), where ¿ is a weight function with growth rate · . Then,

for any 1 6 q 6 1, the following estimate holds:

µZ

«

¿ (x0)q

µZ

«

e¡ jx¡x0jju(x)jp dx

¶q

dx0

¶1=q

6 C

Z

«

¿ (x)ju(x)jp dx; (2.3)

for every > · , where the constant C depends only on , · and the constantC ¿ in (2.1) (and is independent of « ).

(2) Let u belong to L 1¿ ( « ). Then, for every > · , the following analogue of

estimate (2.3) is valid:

supx0 2 «

n¿ (x0) sup

x 2 «fe¡ jx¡x0jju(x)jg

o6 C sup

x 2 «f ¿ (x)ju(x)jg: (2.4)

The proof of this proposition can be found in [16,38].In order to study reaction{di¬usion equations (1.1) and (1.8), we need to impose

some regularity assumptions on the unbounded domain « » Rn, which are assumedto be valid throughout the paper.

We assume that there exists a positive number R0 > 0 such that, for every pointx0 2 « , there exists a smooth domain Vx0 » « such that

BR0x0

\ « » Vx0» BR0 + 1

x0\ « : (2.5)

Moreover, we also assume that there exists a di¬eomorphism ³ x0 : B20 ! BR0 + 2

x0

such that ³ x0 (x) = x0 + px0 (x), ³ x0 (B10) = Vx0 and

kpx0kCN + kp¡1

x0kCN 6 K; (2.6)

where the constant K is independent of x0 2 « and N is large enough. For sim-plicity, we assume from now on that (2.5) and (2.6) hold for R0 = 2.


We note that, in the case where « is bounded, conditions (2.5) and (2.6) are equiv-alent to the following: the boundary @« is a smooth manifold. Now, for unboundeddomains, the sole smoothness of the boundary is not su¯ cient to obtain the regularstructure of « as jxj ! 1, since some uniform (with respect to x0 2 « ) smooth-ness conditions are required. It is, however, more convenient to formulate theseconditions in the form (2.5) and (2.6).

Proposition 2.5. Let the domain « satisfy conditions (2.5) and (2.6) and theweight function ¿ satisfy condition (2.1) and let R be some positive number. Thenthe following estimates hold:

C1

Z

«

¿ (x)ju(x)jp dx 6Z

«

¿ (x0)

Z

« \ BRx0

ju(x)jp dxdx0 6 C2

Z

«

¿ (x)ju(x)jp dx:

(2.7)

Proof. We have

Z

«

¿ (x0)

Z

« \ BRx0

ju(x)jp dxdx0 =

Z

«

ju(x)jpµZ

«

À « \ BRx

(x0) ¿ (x0) dx0

¶dx: (2.8)

Here, À « \ BRx

denotes the characteristic function of the set « \ BRx . It follows from

inequalities (2.1) and (2.2) that

C1 ¿ (x) 6 infx0 2 BR

x

¿ (x0) 6 supx0 2 BR

x

¿ (x0) 6 C2 ¿ (x); (2.9)

and assumptions (2.5) and (2.6) imply that

0 < C1 6 mes(« \ BRx ) 6 C2; (2.10)

uniformly with respect to x 2 « .Estimate (2.7) is an immediate corollary of estimates (2.8){(2.10) and proposi-

tion 2.5 is proved.

Corollary 2.6. Let (2.5) and (2.6) hold. Then the following norm is equivalent

to the usual norm in W l;p¿ ( « ):

ku; « k ¿ ;l;p =

µZ

«

¿ (x0)ku; « \ BRx0

kpl;p dx0

¶1=p

: (2.11)

In particular, all the norms (2.11) are equivalent, for R > 0.

To study equation (1.1), we also need weighted Sobolev spaces of fractional orders 2 R + (and not for s 2 Z only). We rst recall (see [32] for details) that, if V isa bounded domain, a classical norm in the space W s;p(V ), s = [s] + l, 0 < l < 1,[s] 2 Z+ , can be de ned by

ku; V kps;p = ku; V kp

[s];p +X

j ¬ j = [s]

Z

V

Z

V

jD ¬ u(x) ¡ D ¬ u(y)jpjx ¡ yjn + lp

dxdy: (2.12)


It is not di¯ cult to prove, arguing as in proposition 2.5 and using this representa-tion, that, for any bounded domain V with a su¯ ciently smooth boundary,

C1ku; V kps;p 6

Z

V

ku; V \ BRx0

kps;p dx0 6 C2ku; V kp

s;p: (2.13)

This justi es the following de nition.

Definition 2.7. We de ne the space W s;p¿ ( « ), for s 2 R + , as the space of distri-

butions whose norm (2.11) is nite.

It is not di¯ cult to check that these norms are also equivalent for di¬erent R > 0.We now note that the weight functions

¿ f g;x0= e¡ jx¡x0j (2.14)

satisfy conditions (2.1) uniformly with respect to x0 2 Rn and, consequently, allthe estimates obtained above for arbitrary weights will be valid for family (2.14)uniformly with respect to x0 2 Rn. Since these estimates are of fundamental signif-icance for what follows, we write them explicitly in several corollaries formulatedbelow.

Corollary 2.8. Let u belong to Lpf ¯ g( « ), for 0 < ¯ < . Then the following

estimate holds uniformly with respect to y 2 Rn :

µZ

«

e¡q¯ jx0¡yjµZ

«

e¡ jx¡x0jju(x)jp dx

¶q

dx0

¶1=q

6 C ;¯

Z

«

e¡ ¯ jx¡yjju(x)jp dx:

(2.15)Moreover, if u 2 L 1

f ¯ g( « ), ¯ 6 , then

supx0 2 «

ne¡ ¯ jx0¡yj sup

x2 «fe¡ jx¡x0jju(x)jg

o6 C ;¯ sup

x 2 «fe¡ ¯ jx¡yjju(x)jg: (2.16)

Corollary 2.9. Let u belong to W l;pb;¿ ( « ) and ¿ be a weight function with growth

rate · < . Then

C1ku; « kpb;¿ ;l;p 6 sup

x0 2 «

½¿ (x0)

Z

x 2 «

e¡ jx¡x0jku; « \ B1xkp

l;p dx

¾

6 C2ku; « kpb;¿ ;l;p: (2.17)

For the proof of this corollary, see [38].We will essentially use the subspaces of W l;p

b ( « ) which consist of functions decay-ing when jxj ! 1 below.

Definition 2.10. We de ne the space _W l;pb ( « ) as follows:

_W l;pb ( « ) :=

nu 2 W l;p

b ( « ) : limjx0 j! 1

ku; « \ B1x0

kl;p = 0o

: (2.18)

The following proposition gives simple compactness criteria for sets in _W l;pb ( « ).

Proposition 2.11. A set B » _W l;pb ( « ) is compact if and only if the following hold.


(1) For every x0 2 « , the restriction BjVx0of the set B to Vx0

is compact inW l;p(Vx0 ).

(2) The set B possesses a uniform tail’ estimate, i.e. there exists a continuousfunction RB(z) : R + ! R + such that limz ! 1 RB(z) = 0 and

ku; « \ B1x0

kl;p 6 RB(jx0j) 8u 2 B: (2.19)

This proposition can be easily proved by using the Hausdor¬ criterion.The next proposition will be useful in order to verify that a function belongs to

the space _W l;pb ( « ). In order to formulate it, we need the following de nition.

Definition 2.12. For every > 0 and every u 2 W l;pb ( « ), we introduce the func-

tion Rl;p(u; z), z 2 R + , as follows:

Rl;p(u; z) := sup

x 2 «fe¡ d is t(x;RnnBz

0 )ku; « \ B1xkl;pg: (2.20)

Proposition 2.13.

(1) Let u 2 W l;pb ( « ). Then the following estimate holds for every z 2 R + and

> 0 :

Rl;p(u; z) 6 max

nku; « kb;l;pe¡ z=2; sup

jxj>z=2

fku; « \ B1xkl;pg

o: (2.21)

In particular, u 2 _W l;pb ( « ) if and only if

Rl;p(u; z) ! 0 as jzj ! 1: (2.22)

(2) In addition, let a function v 2 W l1;p1

b ( « ) satisfy the estimate

kv; « \ B1x0

kl1;p16 C sup

x 2 «fe¡ jx¡x0 jku; « \ B1

xkl;pg; (2.23)

for an appropriate > 0. Then

Rl1;p1

(v; z) 6 CRl;p(u; z); z 2 R + ; (2.24)

where the constant C is independent of l, p, l1, p1, > 0, z 2 R + , u and v.In particular, if u 2 _W l;p

b ( « ), then v 2 _W l1;p1

b ( « ).

Proof. The proof of estimate (2.21) is straightforward and we omit it here. In orderto prove estimate (2.24), it remains to observe that the weight function

’ ;z(x) := e¡ d i s t(x;RnnBz0 ) (2.25)

has the growth rate and satis es inequality (2.1) with C’ ;z² 1. Multiplying now

inequality (2.23) by ¿ ;z(x0), applying the operator supx0 2 « and using inequal-ity (2.4), we obtain estimate (2.24) and proposition 2.13 is proved.


3. A priori estimates. Existence and uniqueness of solutions

In this section, we consider the following parabolic boundary-value problem,

@tu ¡ a¢ xu + f(u; rxu) + ¶ 0u = g(t);

ujt= 0 = u0; uj@« = 0;

)

(3.1)

in an unbounded domain « » R3 which satis es conditions (2.5) and (2.6).We recall that u = (u1; : : : ; uk), a; ¶ 0 > 0, f = (f 1; : : : ; f k), g = (g1; : : : ; gk) and

that the nonlinear term f satis es the following conditions.

Assumptions 3.1. For every (v; p) 2 Rk £ R3k and for some monotonic functionQ : R+ ! R + , we have

(1) f 2 C1(Rk £ R3k; Rk);

(2) f (v; p) ¢ v > 0;

(3) jf(v; p)j 6 jvjQ(jvj)(1 + jpjr), r < 2;

(4) jf 0v(v; p)j 6 jvjQ(jvj)(1 + jpjr);

(5) jf 0p(v; p)j 6 jvjQ(jvj)(1 + jpjr¡1).

We further assume that the right-hand side g(t) = g(t; x) belongs to the space

§ := Cb(R; _L2b( « )) (3.2)

(we will assume below that g is translation-compact in § , endowed with the localtopology (see x 4)). The phase space © b for problem (3.1) is de ned as

© b := W 2¡ ¯ ;2b ( « ) \ fu0j@« = 0g; (3.3)

i.e. u0 2 © b, where the xed exponent ¯ > 0 is such that ¯ < minf1=r ¡ 12 ; 1

2g. A

solution of equation (3.1) is de ned as a function u belonging to the space

L 1 (R + ; © b) \ Cb([0; 1); L2b( « ));

which satis es equation (3.1) in the sense of distributions.The main aim of this section is to derive several useful a priori estimates for the

solutions of (3.1) and to prove, based on these estimates, the unique solvability ofthe problem under consideration. We start by formulating the following L 1 -boundson the solution u.

Theorem 3.2. Let u be a solution of (3.1). Then the following estimate holds,uniformly with respect to x0 2 « ,

ju(T; x0)j2 6 C supx 2 «

fe¡ jx¡x0jju(0; x)j2ge¡ ® T

+ C supt2 [0;T ]

½e¡ ® (T ¡t)

Z

«

e¡ jx¡x0jjg(t; x)j2 dx

¾; (3.4)

for some ® > 0 and su± ciently small > 0.


Inequality (3.4) follows from a standard application of the maximum principle tothe function w(t; x) = u(t; x) ¢ u(t; x) (see [16,27] for details).

The next theorem gives a dissipative estimate for the solution u(t) in the phasespace © b.

Theorem 3.3. Let u be a solution of problem (3.1). Then the following estimate isvalid, uniformly with respect to x0 2 « ,

ku(T ); « \ B1x0

k2¡ ¯ ;2

6 e¡ ® T Q1(ku0k © b ) supx 2 «

fe¡ jx¡x0jku0; « \ B1xk2¡ ¯ ;2g

+ Q1(kgk§ ) supt2 [0;T ]

ne¡ ® (T ¡t) sup

x 2 «fe¡ jx¡x0jkg(t); « \ B1

xk0;2go

; (3.5)

for some monotonic function Q1 and small constants ® > 0 and > 0.

Proof. We rewrite equation (3.1) as a linear equation,

@tu ¡ a¢ xu + ¶ 0u = h(t) := g(t) ¡ f (u(t); rxu(t)): (3.6)

The following analogue of parabolic regularity theorems for weighted Sobolev spacesis proved in [16],

Z

«

e¡2 jx¡y0jku(T ); Vxk22¡ ¯ ;2 dx

6 Ce¡ ® T

Z

«

e¡2 jx¡y0jku0; Vxk22¡ ¯ ;2 dx

+ C supt2 [0;T ]

½e¡ ® (T ¡t)

Z

«

e¡2 jx¡y0jkh(t); Vxk20;2 dx

¾; (3.7)

for some positive constant C and small positive constants ® and which areindependent of y0 2 « and Vx de ned in (2.5) and (2.6). Applying the operatorsupy0 2 « e¡ jx0¡y0j to both sides of (3.7) and using (2.17), we obtain the followingestimate,

supx2 «

fe¡ jx¡x0jku(T ); Vxk22¡ ¯ ;2g

6 C 0e¡ ® T supx 2 «

fe¡ jx¡x0jku0; Vxk22¡ ¯ ;2g

+ C 0 supt2 [0;T ]

ne¡ ® (T ¡t) sup

x2 «fe¡ jx¡x0jkg(t); Vxk2

0;2go

+ C 0 supt2 [0;T ]

ne¡ ® (T ¡t) sup

x2 «fe¡ jx¡x0jkf(u(t); rxu(t)); Vxk2

0;2go

; (3.8)

where C 0 and the small positive constants ® and are also independent of x0.Thus there only remains to estimate the last integral in the right-hand side

of (3.8). According to assumptions 3.1, we have

kf (u(t); rxu(t)); Vxk20;2 6 Cku(t); Vxk2

0; 1 Q2(ku(t)kL1 ( « ))(ku; Vxk2r1;2r + 1): (3.9)


Using a standard interpolation inequality (see [32]), we obtain

ku; Vxk1;2r 6 Cku; Vxk1¡ ³0; 1 ku; Vxk ³

2¡ ¯ ;2; (3.10)

where ³ = 1=(2 ¡ ¯ ) 2 (0; 1) (we also note that, due to assumptions (2.5) and (2.6),the constant C in (3.10) is independent of x). It follows from our assumptions on ¯that 2r³ < 2 and, consequently, (3.9) and (3.10) imply

kf (u(t); rxu(t)); Vxk20;2 6 ku(t); Vxk2

0; 1 Q · (ku(t)kL1 ( « )) + · ku(t); Vxk22¡ ¯ ;2:

(3.11)Here, · > 0 is an arbitrary positive number and Q · is an appropriate monotonicfunction, depending on · . According to theorem 3.2, we have

Q · (ku(t)kL1 ( « )) 6 e¡ ® t ~Q · (ku0kL1 ( « )) + ~Q · (kgk § ); (3.12)

for an appropriate new function ~Q · . Consequently, due to (3.4), (3.8), (3.11) and(3.12), we obtain

Zx0 (T ) 6 e¡ ® T Q · (ku0k© b) sup

x 2 «fe¡ jx¡x0jku0; Vxk2

2¡ ¯ ;2g

+ Q · (kgk § ) supt 2 [0;T ]

ne¡ ® (T ¡t) sup

x 2 «fe¡ jx¡x0 jkg(t); Vxk2

0;2go

+ C· supt2 [0;T ]

fe¡ ® (T ¡t)Zx0 (t)g;

(3.13)

where Zx0 (t) := supx2 « fe¡ jx¡x0jku(t); Vxk22¡ ;2g and where · > 0 can be chosen

arbitrarily small. We now x · > 0 such that C· < 12 . Then (3.13) implies that

Zx0 (T ) 6 C1e¡ ® T Q(ku0k© b ) supx 2 «

fe¡ jx¡x0jku0; Vxk22¡ ¯ ;2g

+ C1Q(kgk § ) supt 2 [0;T ]

ne¡ ® (T ¡t) sup

x 2 «fe¡ jx¡x0 jkg(t); Vxk2

0;2go

; (3.14)

for an appropriate monotonic function Q (see [16]). Estimate (3.14), together withthe obvious estimate

C¡1 supx 2 «

fe¡ jx¡x0 jkv; « \ B1xkl;pg 6 sup

x2 «fe¡ jx¡x0jkv; Vxkl;pg

6 C supx 2 «

fe¡ jx¡x0 jkv; « \ B1xkl;pg; (3.15)

imply (3.5) and theorem 3.3 is proved.

Corollary 3.4. Let the assumptions of theorem 3.3 hold. Then the following esti-mate is valid,

ku(t)k© b6 Q1(ku0k © b )e¡ ® t + Q1(kgk § ); (3.16)

for some monotonic function Q1 and positive constant ® .

Indeed, applying supx0 2 « to (3.5), we derive (3.16).


Corollary 3.5. Let u be a solution of (3.1) and let the assumptions of theorem 3.3be satis¯ed. Then

kf (u(T ); rxu(T )); « \ B1x0

k20;2

6 e¡ ® T Q2(ku0k © b ) supx 2 «

fe¡ jx¡x0jku0; « \ B1xk2

2¡ ¯ ;2g

+ Q2(kgk § ) supt2 [0;T ]

ne¡ ® (T ¡t) sup

x2 «fe¡ jx¡x0jkg(t); « \ B1

xk20;2g

o;

(3.17)

for some monotonic function Q2 and positive constants ® and .

Indeed, equation (3.17) follows from estimates (3.4), (3.5) and (3.9).

Corollary 3.6. Let u be a solution of (3.1) and let the assumptions of the previoustheorem be valid. Then u 2 C1¡ ¯ =2(R + ; L2

b( « )) and

kuk2C1 ¡ ¯ =2([T;T + 1];L2( « \ B1

x0))

6 e¡ ® T Q3(ku0k © b ) supx 2 «

fe¡ jx¡x0jku0; « \ B1xk2

2¡ ¯ ;2g

+ Q3(kgk § ) supt2 [0;T ]

ne¡ ® (T ¡t) sup


xk20;2g

o:

(3.18)

Indeed, rewriting equation (3.1) in the form (3.6) and applying the parabolicregularity theorem, together with estimate (3.17), we derive estimate (3.18) (seealso [16] for details).

We now derive the smoothing property for equation (3.1).

Corollary 3.7. Let u be a solution of (3.1) and let us assume that the assumptionsof theorem 3.3 hold. Then, for every ¯xed 0 < ¯ 1 < ¯ ,

u 2 C([1; 1); W 2¡ ¯ 1 ;2b ( « ));

and, for arbitrary T > 1, the following estimate is satis¯ed,

ku(T ); « \ B1x0

k22¡ ¯ 1 ;2

6 e¡ ® T Q4(ku0k © b) sup

x 2 «fe¡ jx¡x0jku0; « \ B1

xk22¡ ¯ ;2g

+ Q4(kgk § ) supt2 [0;T ]

ne¡ ® (T ¡t) sup


xk20;2g

o;

(3.19)

where the monotonic function Q4 depends on ¯ 1, but is independent of x0 2 « .

Proof. We introduce the function w(t) = (t ¡ T +1)u(t). Then this function satis es

@tw ¡ a¢ xw + ¶ 0w = (t ¡ T + 1)g ¡ (t ¡ T + 1)f (u; rxu) + u ² ~h(t);

wjt = T ¡1 = 0; wj@« = 0:

)

(3.20)


Applying the parabolic regularity theorem to the linear equation (3.20), we derive(3.19), analogously to (3.8), but replacing ¯ by ¯ 1, using estimates (3.8) and (3.17)in order to estimate the right-hand side of (3.20) and taking into account thatw(T ¡ 1) = 0.

The next corollary shows that the trajectory u(t) is H�older continuous withrespect to t 2 [1; 1) in the space © b.

Corollary 3.8. Let the assumptions of theorem 3.3 hold and let u be a solutionof (3.1). Then the following estimate is valid, for T > 1,

kukCs([T;T + 1];© b) 6 Q5(ku0k © b )e¡ ® T + Q5(kgk § ); (3.21)

where s := 12( ¯ ¡ ¯ 1)((2 ¡ ¯ )=(2 ¡ ¯ 1)), Q5 is an appropriate monotonic function

and ® > 0 is a constant.

Indeed, according to (3.18) and (3.19), we have

kukC1 ¡ ¯ =2([T;T + 1];L2b ( « )) + kuk

L1 ([T;T + 1];W2 ¡ ¯ 1 ;2b ( « ))

6 Q(ku0k © b )e¡ ® T + Q(kgk § ):

Interpolating between C1¡ ¯ =2(L2b) and L 1 (W 2¡ ¯ 1;2

b ), we obtain estimate (3.21).The next theorem shows that, under the assumptions of theorem 3.2, equa-

tion (3.1) generates a Lipschitz continuous dynamical system in © b.

Theorem 3.9. Let the assumptions of theorem 3.3 hold. Then, for every u0 2 © b,equation (3.1) possesses a unique solution u(t) 2 © b, t > 0, i.e. this equationgenerates a dynamical process fUg(t; ½ ); t > ½ ; ½ 2 Rg in the phase space © b,

Ug(t; ½ ) : © b ! © b; Ug(t; ½ )u( ½ ) := u(t); t > ½ : (3.22)

Moreover, if u1(t) and u2(t) are two solutions of (3.1) with right-hand sides g1 andg2, respectively, then

ku1(T ) ¡ u2(T ); « \ B1x0

k22¡ ¯ ;2

6 CeKT supx 2 «

fe¡ jx¡x0jku1(0) ¡ u2(0); « \ B1xk2

2¡ ¯ ;2g

+ C supt2 [0;T ]

neK(T ¡t) sup

x 2 «fe¡ jx¡x0jkg1(t) ¡ g2(t); « \ B1

xk20;2g

o; (3.23)

where the constants C > 0 and K > 0 depend on kui(0)k © b and kgik § , but areindependent of x0 2 « .

Proof. The existence of a solution for problem (3.1) can be veri ed, based on esti-mate (3.16), in a standard way (e.g. by approximating the unbounded domain « bybounded ones « n, solving (3.1) in « n by the Leray{Schauder principle and nallypassing to the limit n ! 1 (see [16] for details)).

So there only remains to verify estimate (3.23). Let u1(t) and u2(t) be two solu-tions of (3.1) with right-hand sides g1(t) and g2(t), respectively. Then the functionv(t) := u1(t) ¡ u2(t) satis es the equation

@tv = a¢ xv ¡ ¶ 0v ¡ L1(t; x)v ¡ L2(t; x)rxv+h(t); vjt= 0 = u1(0) ¡ u2(0); (3.24)


where h(t) := g1(t) ¡ g2(t) and

L1(t; x) :=

Z 1

0

f 0u(su1(t) + (1 ¡ s)u2(t); srxu1(t) + (1 ¡ s)rxu2(t)) ds;

L2(t; x) :=

Z 1

0

f 0rxu(su1(t) + (1 ¡ s)u2(t); srxu1(t) + (1 ¡ s)rxu2(t)) ds:

9>>>=

>>>;

(3.25)We note that, due to H�older’s inequality, assumptions 3.1 on the derivatives of fand Sobolev’s embeddings,

W 2¡ ¯ ;2 » W 1;3r » W 1;6(r¡1) and W 2¡ ¯ ;2 » C (3.26)

(we recall that n = 3 and ¯ < 1=r ¡ 12 ), we have

kL1(t); Vxk0;3 + kL2(t); Vxk0;6

6 (ku1(t); Vxk2¡ ¯ ;2 + ku2(t); Vxk2¡ ¯ ;2)Q(ku1(t); Vxk2¡ ¯ ;2 + ku2(t); Vxk2¡ ¯ ;2);(3.27)

where the monotonic function Q is independent of x 2 « . Consequently, accordingto (3.16),

kL1(t); Vxk0;3 + kL2(t); Vxk0;6 6 M; (3.28)

where the constant M depends on kuik© b and kgik § , i = 1; 2, but is independentof x.

We derive estimate (3.23) in two steps. In a rst step, we obtain an estimate ofthe L2-norm of v, similar to (3.23).

Lemma 3.10. Let the assumptions of theorem 3.9 hold. Then, for every x0 2 « ,the following estimate is satis¯ed,

supx0 2 «

fe¡ jx¡x0 jkv(T ); Vxk20;2g

6 CM eKM T supx0 2 «

fe¡ jx¡x0jkv(0); Vxk20;2g

+ CM supt 2 [0;T ]

neKM (T ¡t) sup

x0 2 «fe¡ jx¡x0 jkh(t); Vxk2

0;2go

; (3.29)

where > 0 is a small constant and constants CM and KM depend on the constantM in (3.28), but are independent of x0.

Proof of lemma 3.10. We multiply equation (3.24) by ve¡2 jx¡y0j, integrate over «and use the obvious inequality

jrxe¡2 jx¡y0 jj 6 2 e¡2 jx¡y0j: (3.30)


Then we obtain, for a su¯ ciently small > 0,

@t

Z

«

e¡2 jx¡y0jjv(t; x)j2 dx

+ ¶ 0

Z

«

e¡2 jx¡y0jjv(t; x)j2 dx + a

Z

«

e¡2 jx¡y0jjrxv(t; x)j2 dx

6Z

«

e¡2 jx¡y0jjL1(t; x)j ¢ jv(t; x)j ¢ jv(t; x)j dx

+

Z

«

e¡2 jx¡y0jjL2(t; x)j ¢ jv(t; x)j ¢ jrxv(t; x)j dx

+

Z

«

e¡2 jx¡y0jjh(t; x)j2 dx:

(3.31)

We estimate the rst two terms in the right-hand side of (3.31) by using repre-sentation (2.17), H�older’s inequality, classical Sobolev’s embedding theorems andestimate (3.28). Indeed, the rst term can be estimated as follows:

I1(t) 6 C1

Z

«

e¡ jx¡x0 jkjL1(t)j ¢ jv(t)j ¢ jv(t)j; Vxk0;1 dx

6 C2

Z

«

e¡2 jx¡y0jkL1(t); Vxk0;3 ¢ kv(t); Vxk0;2 ¢ kv(t); Vxk1;2 dx

6 C3M 2

Z

«

e¡2 jx¡y0 jjv(t; x)j2 dx + 14a

Z

«

e¡2 jx¡y0jjrxv(t; x)j2 dx: (3.32)

Analogously, using, in addition, the interpolation inequality

kv; Vxk0;3 6 Ckv; Vxk1=20;2 ¢ kv; Vxk1=2

1;2 ;

we nd

I2(t) 6 C

Z

«

e¡2 jx¡y0 jkjL2(t)j ¢ jrxv(t)j ¢ jv(t)j; Vxk0;1 dx

6 C1

Z

«

e¡2 jx¡y0jkL2(t); Vxk0;6 ¢ kv(t); Vxk0;3 ¢ kv(t); Vxk1;2 dx

6 C2

Z

«

e¡2 jx¡y0jkL2(t); Vxk0;6 ¢ kv(t); Vxk1=20;2 ¢ kv(t); Vxk3=2

1;2 dx

6 C3M 4

Z

«

e¡2 jx¡y0 jjv(t; x)j2 dx + 14a

Z

«

e¡2 jx¡y0jjrxv(t; x)j2 dx: (3.33)

Inserting (3.32) and (3.33) into (3.31), we have

@t

Z

«

e¡2 jx¡y0 jjv(t; x)j2 dx

¡ KM

Z

«

e¡2 jx¡y0jjv(t; x)j2 dx + 12a

Z

«

e¡2 jx¡y0jjrxv(t; x)j2 dx

6Z

«

e¡2 jx¡y0jjh(t; x)j2 dx;

(3.34)


where KM := ¡ ¶ 0 +C3M 2(1+M2). Applying Gronwall’s lemma to relation (3.34),we obtain

kv(T ); Vx0 k20;2 6 CMeKM T

Z

«

e¡2 jx¡y0jjv(0; x)j2 dx

+

Z T

0

eKM (T ¡t)

Z

«

e¡2 jx¡y0jjh(t; x)j2 dxdt: (3.35)

Multiplying (3.35) by e¡ jx0¡y0j, taking the supremum over y0 2 « of both sidesof the obtained inequality and using inequalities (2.17), we nally nd (3.29) andlemma 3.10 is proved.

We are now in a position to complete the proof of the theorem. Indeed, applyingthe parabolic regularity theorem to equation (3.24) (see [16]), we obtain, analo-gously to (3.7) and (3.8),

supx0 2 «

fe¡ jx¡x0jkv(T ); Vxk22¡ ¯ ;2g

6 Ce¡ ® T supx 2 «

fe¡ jx¡x0jkv(0); Vxk22¡ ¯ ;2g

+ C supt2 [0;T ]

ne¡ ® (T ¡t) sup

x 2 «fe¡ jx¡x0jkh(t); Vxk2

0;2go

+ C supt2 [0;T ]

ne¡ ® (T ¡t) sup

x 2 «fe¡ jx¡x0j(kL1(t)v(t); Vxk2

0;2

+ kL2(t)rxv(t); Vxk20;2)g

o:

(3.36)

Thus it remains to estimate the last supremum in the right-hand side of (3.36). Tothis end, we note that, due to H�older’s inequality, Sobolev’s embedding theoremsand estimate (3.28),

kL1(t)v(t); Vxk20;2 + kL2(t)rxv(t); Vxk2

0;2 6 C· kv(t); Vxk20;2 + · kv(t); Vxk2

2¡ ¯ ;2;(3.37)

which is valid for every · > 0 (see also (3.32) and (3.33)). Inserting these estimatesinto (3.36) and using (3.23), we nd

Zx0 (T ) 6 C 0· eK 0

M T supx 2 «

fe¡ jx¡x0 jkv(0); Vxk22¡ ¯ ;2g

+ C 0 supt 2 [0;T ]

neK 0

M (T ¡t) supx 2 «

fe¡ jx¡x0jkh(t); Vxk20;2g

o

+ C· supt 2 [0;T ]

fe¡ ® (T ¡t)Zx0 (t)g; (3.38)

where

K 0M := maxf¡ ® ; KM g and Zx0 (t) := sup

x 2 «fe¡ jx¡x0jkv(t); Vxk2

2¡ ¯ ;2g:

Fixing · < 1=(2C) in (3.38), we obtain (3.23) as in (3.13) and theorem 3.9 isproved.


Remark 3.11. We note that the constant K = K 0M in (3.23) is de ned by

K 0M = maxf¡ ® ; ¡ ¶ 0 + CM 2(1 + M 2)g; (3.39)

where M is de ned in (3.28) and ® and C are independent of M . Therefore, theexponent K 0

M is negative, if M is small enough. This will be essential for the nextsections.

The next corollary is an analogue of corollary 3.7 for equation (3.24).

Corollary 3.12. Let u1(t) and u2(t) be two solutions of (3.1) with right-handsides g1(t) and g2(t), respectively. Then, for every ¯xed 0 < ¯ 1 < ¯ and for anarbitrary T > 1, the following estimate holds,

ku1(T ) ¡ u2(T ); « \ B1x0

k22¡ ¯ 1;2

6 CeKT supx 2 «

fe¡ jx¡x0jku1(0) ¡ u2(0); Vxk22¡ ¯ ;2g

+ C supt2 [0;T ]

neK(T ¡t) sup

x 2 «fe¡ jx¡x0 jkg1(t) ¡ g2(t); Vxk2

0;2go

; (3.40)

where the constant C depends on ¯ 1, but is independent of x0 2 « .

Indeed, we set w(t) = (t ¡ T + 1)v(t). Then

@tw ¡ a¢ xw + ¶ 0w

= v(t) + (t ¡ T + 1)h(t) ¡ (t ¡ T + 1)(L1(t)v(t) + L2(t)rxv(t))

² ~h(t);

wjt= T ¡1 = 0; wj@« = 0:

9>>>>=

>>>>;

(3.41)

Applying the parabolic regularity theorem to the linear equation (3.41), we obtain(3.40), analogously to (3.8), but replacing ¯ by ¯ 1, using estimates (3.23) and (3.37)in order to estimate the right-hand side of (3.41) and taking into account thatw(T ¡ 1) = 0.

4. The global (uniform) attractor

In this section, we prove that equation (3.1) possesses the global (uniform) attractorin the phase space © b. To this end, following [9{11,21,38], we consider a family ofequations of type (3.1), generated by all time translations of the initial equationand their limits in the corresponding topology. To be more precise, we consider thefollowing family of problems,

@tu = a¢ xu ¡ ¶ 0u ¡ f (u; rxu) + ¹ (t); ¹ 2 H(g); (4.1)

where H(g) is the hull of the initial external forces g, de ned via

H(g) := [Thg; h 2 R]§ loc ; (Thg)(t) := g(t + h); (4.2)

§ loc := Cloc(R; _L2b( « )) and [¢] § loc denotes the closure in the space § loc.


In order to construct the attractor for family (4.2), we further assume that thehull (4.2) is compact in the space § loc,

H(g) »» § loc (4.3)

(following [11], such functions are called translation-compact in § loc). We nowintroduce the function

R0;2(H(g); z) := sup

t 2 Rsup

¹ 2 H (g)

R0;2( ¹ (t); z); (4.4)

where > 0 is a su¯ ciently small parameter which will be speci ed below andthe function R

0;2( ¹ ; z) is de ned by (2.20). Then, due to (4.3) and due to proposi-tions 2.11 and 2.13, we have

R0;2(H(g); z) ! 0 as z ! 1: (4.5)

We now de ne the semigroup fSh; h > 0g in the extended phase space © b £ H(g)via

Sh : © b £ H(g) ! © b £ H(g); Sh(u0; ¹ ) := (U ¹ (h; 0)u0; Th ¹ ); (4.6)

where U¹ (t; ½ ) is the solving operator of problem (3.1) with the external forces greplaced by ¹ 2 H(g).

We recall that a set A » © b £ H(g) is the global attractor of semigroup (4.6) ifthe following conditions hold.

(1) The set A is compact in © b £ H(g).

(2) The set A is strictly invariant, i.e. ShA = A.

(3) A is an attracting set for the semigroup Sh, i.e. for every neighbourhoodO(A) of the attractor A in the space © b £ H(g) and every bounded subsetB » © b £ H(g), there exists T = T (O; B) such that

ShB » O(A) for h > T (4.7)

(see, for example, [9, 11] for details). If the global attractor A exists, thenthe projection A := ¦ 1A onto the rst component of © b £ H(g) is called theuniform attractor of equation (3.1).

Theorem 4.1. Let the assumptions of theorem 3.3 hold and let, in addition, theexternal forces g be translation-compact in § loc (see (4:3)). Then the semigroup Sh

possesses the global attractor A in the space © b £ H(g). Moreover, the associateduniform attractor A belongs to the space

_© b := _W 2¡ ¯ ;2b ( « ) (4.8)

and has the following structure,

A =[

¹ 2 H (g)

K ¹ jt= 0; (4.9)

where K ¹ , ¹ 2 H(g), is the set of all solutions u(t) of (4.1) de¯ned and bounded forall t 2 R (u 2 L 1 (R; © b)).


Proof. We rst prove that Sh possesses a locally compact attractor in the space© b £H(g), i.e. an attractor that is a priori only bounded in © b £H(g), but compactin the local topology of © loc £ H(g), where

© loc := W 2¡ ¯ ;2loc ( ·« ); (4.10)

and attracts bounded subsets of © b £ H(g) also in the local topology (i.e. theneighbourhood O(A) in (4.7) must be understood in the space © loc £ H(g) (see, forexample, [38])). We recall that, due to the abstract attractor’s existence theorem(see [4]), it su¯ ces, in order to obtain such an attractor, to verify the followingconditions.

(1) The operators St are © loc £ H(g)-continuous on every © b £ H(g)-boundedsubset B and every xed t > 0.

(2) The semigroup St possesses an attracting set K, bounded in © b £ H(g) andcompact in © loc £ H(g).

Let us verify these conditions for our semigroup. The rst condition is an imme-diate corollary of estimate (3.23). Applying the operator supx0 2 « to both sidesof (3.19) and using (2.4), we have, for T > 1,

ku(T )kW

2 ¡ ¯ 1;2b ( « )

6 e¡ ® T Q(ku0k © b ) + Q(kgk§ ); (4.11)

for an appropriate monotonic function Q and xed constants 0 < ¯ 1 < ¯ and ® > 0.Estimate (4.11), together with (4.3) and the obvious inequality

k ¹ k § 6 kgk§ for every ¹ 2 H(g); (4.12)

imply that the set

K := fu0 2 W 2¡ ¯ 1;2b ( « ) : ku0k

W2 ¡ ¯ 1;2b

6 2Q(kgk § )g £ H(g) (4.13)

is indeed an absorbing set for the semigroup Sh. We note that the embeddingW 2¡ ¯ 1;2

b ( « ) » © loc is compact (since ¯ 1 < ¯ ). Thus the second condition is also sat-is ed and, consequently, the semigroup Sh possesses a locally compact attractor A.Relation (4.9) then follows from the general global attractor’s description theorem(see [4,11]).

Let us now prove that the obtained attractor is in fact compact in the uniformtopology of © b £H(g). To this end, we note that, according to (3.5), every completebounded solution u(t) 2 K ¹ of equation (4.1) satis es the estimate

ku(T ); « \ B1x0

k22¡ ¯ ;2

6 Q(kgk§ ) supt2 (¡1 ;T ]

ne¡ ® (T ¡t) sup

x 2 «fe¡ jx¡x0jk ¹ (t); « \ B1

xk20;2g

o: (4.14)

Proposition 2.13 now implies that

ku(T ); « \ B1x0

k2¡ ¯ 1;2 6 Q(kgk § )R0;2(H(g); x0); (4.15)

for some function Q and, consequently,

kA; B1x0

\ « k2¡ ¯ 1;2 6 Q(kgk§ )R0;2(H(g); x0): (4.16)


Estimate (4.16), together with condition (4.5), implies that A » _© b and possessesa uniform `tail’ estimate, i.e.

limjx0j! 1

kA; « \ B1x0

k2¡ ¯ ;2 = 0: (4.17)

We also note that, by construction, A is compact in © loc. Thus, due to proposi-tion 2.11, the set A is compact in _© b, and therefore A is compact in © b £ H(g).

So it remains to verify that A is an attracting set for Sh in the uniform topologyof © b £ H(g). Let us assume the converse. Then there exist a sequence ¹ n 2 H(g) ofexternal forces, a sequence of solutions un(t) of (4.1) with ¹ replaced by ¹ n, un(0)belonging to some bounded subset B » © b and a sequence tn ! 1 such that

dist© b

(un(tn); A) > 0 > 0: (4.18)

Since A is a locally compact attractor, then there exists u0 2 A such that

kun(tn) ¡ u0; « \ BR0 k2¡ ¯ ;2 ! 0 for every R > 0: (4.19)

It follows from (4.6) and proposition 2.13 that

kun(tn); « n BR0 k2¡ ¯ ;2 6 e¡ ® tn Q(kBk © b ) + Q(kgk § )R

0;2(H(g); R); (4.20)

and consequently, due to (4.16) and (4.20),

lim supn ! 1

kun(tn) ¡ u0; « n BR0 k2¡ ¯ ;2 6 Q(kgk § )R

0;2(H(g); R): (4.21)

Convergences (4.5), (4.19) and (4.21) contradict (4.18). Therefore, A is an attractingset in the uniform topology of © b £ H(g) and theorem 4.1 is proved.

Remark 4.2. The uniform attractor A of equation (3.1) can be de ned withoutusing the extended semigroup Sh and the skew-product technique. Namely, the setA is called the uniform attractor of equation (3.1) if the following hold.

(1) A is compact in © b.

(2) A is a uniformly attracting set for the process fUg(t; ½ ); t > ½ ; ½ 2 Rg, i.e. forevery neighbourhood O(A) in © b and for every bounded subset B » © b, thereexists T = T (O; B) such that, for every ½ 2 R,

Ug( ½ + t; ½ )B » O(A) for t > T: (4.22)

(3) A is minimal among the closed sets enjoying properties (1) and (2).

The equivalence of the two de nitions is proved in [11].

Remark 4.3. We note that every periodic, quasi-periodic and almost-periodic ex-ternal force with values in _L2

b( « ) is obviously translation-compact in § loc and,consequently, equation (3.1) with right-hand sides from these classes possesses theuniform attractor A. We also note that the class of translation-compact functionsis larger than the class of almost-periodic functions. Indeed, it is not di¯ cult toverify, using the Arzela{Ascoli theorem, that every

g 2 C ¬b (R; _W ¬ ;2

b ( « )); ¬ > 0; (4.23)


is translation-compact in L 1loc(R; _L2

b( « )). So the translation compactness can beinterpreted as some regularity assumption on g. Moreover, it is possible to verifythat g is translation-compact in § loc if and only if

g 2 [C1b (R; C1

00(R3)j « )]Cb(R;L2b( « )); (4.24)

where C100(R3) denotes the subspace of C1(R3) consisting of functions with nite

support in R3 (see [38] for details).

Now, in order to study Kolmogorov’s entropy of the obtained attractor, we needsome estimates on the di¬erence of solutions belonging to a neighbourhood of theattractor A. First, we construct a special neighbourhood of A in © b. To this end,analogously to (4.4), we introduce, for every u0 2 © b, the function

R© b

(u0; z) := R2¡ ¯ ;2(u0; z); (4.25)

where the constant > 0 is the same as in (3.5). Then, due to (3.5), (2.23)and (2.24), we have the following `tail’ estimate for the solution u(t) of equa-tion (4.1):

R© b

(u(t); z) 6 Q(ku(0)k © b)e¡ ® tR

© b(u(0); z) + Q(kgk § )R

0;2(H(g); z): (4.26)

We nally set, for every R > 0

R0;2(H(g); z; R) := 2Q(kgk § )

(R

0;2(H(g); z); z < R;

R0;2(H(g); R); z > R;

(4.27)

and

V R := fu0 2 © b : R© b

(u0; z) 6 R0;2(H(g); z; R) 8z 2 R + g: (4.28)

Then, on the one hand, it follows from (4.26) that (4.28) is a uniformly absorbingset for the family of equations (4.1). On the other hand, equation (4.26) impliesthat, for every u0 2 V R, the corresponding solution u(t) of (4.1) satis es

R© b

(u(t); z) 6 ~Q(kgk § )R0;2(H(g); z; R); (4.29)

where ~Q is independent of R.

Theorem 4.4. Assume that the assumptions of theorem 4.1 hold. Then there existsR = Re¬ > 0 such that, for solutions u1(t) and u2(t) of (4.1) with right-hand sidesg1 2 H(g) and g2 2 H(g), respectively, and with initial data u1(0); u2(0) 2 V Reff ,

u1(t) ¡ u2(t) = L u1;u2 (t) + N u1;u2 (t); (4.30)

where the ¯rst term in (4.30) satis¯es

kL u1;u2 (t)k © b6 Ce¡ ® tku1(0) ¡ u2(0)k © b + CeKtkg1 ¡ g2kC([0;t];L2

b) (4.31)

and the second one can be estimated by

kN u1;u2 (t); « \B1x0

k2¡ ¯ 1;2 6 C exp(Kt ¡ dist(x0; BReff

0 ))ku1(0) ¡ u2(0)k © b ; (4.32)


where the constant Re¬ can be found from

R0;2(H(g); Re¬ ) = · ; · := · (f; a; ¶ 0; kgk § ) (4.33)

and the positive constants C, ® , , · , K and ¯ 1 < ¯ depend only on kgk § , but areindependent of x0, Re¬ and u1(0); u2(0) 2 V Reff .

Proof. We set v(t) := u1(t) ¡ u2(t) and h(t) := g1(t) ¡ g2(t). Then the function vobviously satis es equation (3.24). Let us split the function v(t) into a sum of threefunctions vi(t), i = 1; 2; 3,

v(t) = v1(t) + v2(t) + v3(t); (4.34)

where v1(t) is solution of

@tv1 = a¢ xv1 ¡ ¶ 0v1 ¡ L1(t)v1 ¡ L2(t)rxv1 + h(t); v1jt = 0 = 0 (4.35)

(the functions L1(t) and L2(t) are de ned by (3.25)), the function v2(t) is solutionof

@tv2 = a¢ xv2 ¡ ¶ 0v2 ¡ (1 ¡ À R)L1(t)v2 ¡ (1 ¡ À R)L2(t)rxv2; v2jt= 0 = v(0); (4.36)

À R = À R(x) being the characteristic function of the R-ball BR0 in Rn, and the

function v3(t) is solution of

@tv3 = a¢ xv3 ¡ ¶ 0v3 ¡ L1(t)v3 ¡ L2(t)rxv3 + hR(t); v3jt = 0 = 0; (4.37)

with hR(t) := ¡ À R(x)(L1(t)v2(t) + L2(t)rxv2(t)).We note that equation (4.35) is very similar to (3.24) and, consequently, esti-

mate (3.23) implies that

kv1(t)k © b6 CeKtkhkC([0;t];L2

b( « )); (4.38)

for appropriate constants C and K, depending only on kgk § . Let us then investigateequation (4.36). To this end, we observe that, due to (3.27) and (4.29), we have

k(1 ¡ À R)L1(t); « \ B1x0

k0;3 + k(1 ¡ À R)L2(t); « \ B1x0

k0;6

6 ~Q(kgk § )R0;2(H(g); R) := MR; (4.39)

for some monotonic function ~Q. We note that (4.36) is also of the form (3.24) and,consequently, the function v2(t) satis es the following analogue of (3.23),

kv2(t)k © b 6 CeKMR tkv(0)k © b ; (4.40)

where C depends only on kgk § and the coe¯ cient KM satis es

KM := C1M 2R(1 + M 2

R) ¡ C2 ¶ 0: (4.41)

Since R0;2(H(g); R) ! 0 as R ! 1, then it is possible to x R = Re¬ such

that KM 6 ¡ ® := ¡ 12 C2 ¶ 0. Moreover, the minimal number R which possesses this

property obviously satis es (4.33). In that case, (4.40) reads

kv2(t)k © b6 Ce¡ ® tkv(0)k © b : (4.42)


So there only remains to study equation (4.37). As above, equation (4.37) is of theform (3.24). Applying the smoothing property (3.19) and using estimates (3.37)and (3.40), we deduce that, for some 0 < ¯ 1 < ¯ and for all t > 1,

kv3(t); « \ B1x0

k2¡ ¯ 1;2 6 C sups 2 [0;t]

neK(t¡s) sup

x 2 «fe¡ jx¡x0 jkh(s); « \ B1

x0k0;2g

o

6 C1eKtkv2(t)k © b supjxj6R

fe¡ jx¡x0jg

6 C2 exp(Kt ¡ dist(x0; BR0 ))kv(0)k © b : (4.43)

We now set

L u1;u2 (t) := v1(t) + v2(t); N u1;u2 (t) := v3(t): (4.44)

Then estimates (4.38), (4.42) and (4.43) give (4.31) and (4.32), and theorem 4.4 isproved.

The next theorem shows that the `size’ of the attractor Aj « \ B1x0

decays expo-nentially as jx0j ! 1.

Theorem 4.5. Let the assumptions of theorem 4.4 hold and let u1(t) and u2(t),t 2 R, be two arbitrary solutions belonging to the set K ¹ (de¯ned in (4.9)), forsome ¹ 2 H(g). Then the following estimate is valid,

ku1(t) ¡ u2(t); « \ B1x0

k2¡ ¯ ;2 6 C exp( ¡ dist(x0; BReff

0 )); t 2 R; (4.45)

where the constant C is independent of u1; u2 2 K ¹ , t 2 R and ¹ 2 H(g) and theconstants > 0 and Re¬ are the same as in theorem 4.4.

Proof. We set v(t) := u1(t) ¡ u2(t). Then this function satis es the equation

@tv = a¢ xv ¡ ¶ 0v ¡ (1 ¡ À Reff )L1(t)v ¡ (1 ¡ À Reff )L2(t)rxv = hv(t); t 2 R; (4.46)

where the functions L1 and L2 are de ned in (3.25) and

hv(t) := À Reff (x)L1(t)v(t) + À Reff (x)L2(t)rxv(t): (4.47)

Since the attractor A is bounded in © b, then

khv(t); B1x0

kL2 6 CÀ Reff + 1(x0); (4.48)

where the constant C is independent of t, u1, u2 and ¹ . We now recall that we x the e¬ective radius Re¬ such that (see (4.41)) equation (4.46) (or, equivalently,equation (4.36)) is exponentially stable and, consequently, according to (3.23), wehave

kv(t); « \ B1x0

k22¡ ¯ ;2 6 C sup

s2 (¡1 ;t]

ne¡ (t¡s)

nsupx 2 «

e¡ jx¡x0jkhv(s); « \ B1xk2

0;2

oo;

(4.49)where > 0 and C are independent of t, u1, u2 and ¹ . Inserting estimate (4.48)into the right-hand side of (4.49), we nally obtain inequality (4.45). This nishesthe proof of theorem 4.5.


Corollary 4.6. Let the assumptions of theorem 4.4 hold and let, in addition,equation (3.1) be autonomous,

g(t; x) := g(x) 2 _L2b( « ): (4.50)

We further assume that z0 2 _W 2;2( « ) is an arbitrary equilibrium of equation (3.1),

a¢ xz0 ¡ ¶ 0z0 ¡ f (z0; rxz0) = g; z0j@« = 0: (4.51)

Then the following estimate holds,

kA ¡ z0; « \ B1x0

k2¡ ¯ ;2 6 C exp( ¡ dist(x0; BReff

0 )); (4.52)

where > 0 and C > 0 are independent of z0.

Indeed, taking u2(t) := z0 in (4.45), we nd (4.52).

Remark 4.7. Estimate (4.52) describes the asymptotic behaviour of the attractorA as jxj ! 1 in the autonomous case. Moreover, this estimate shows that, outsidethe e¬ective domain « e¬ := « \ BReff

0 , the attractor A coincides with the equilib-rium z0 up to exponentially small terms as jxj ! 1. That is the reason why we canpredict that the dynamics generated by (1.1) in an unbounded domain « can bemore or less well approximated by equations (1.1) in the bounded domain « e¬ . Inthe next section, we partially prove this conjecture by showing that A has indeeda nite fractal dimension, which can be estimated from above in terms of Re¬ ,

dimF (A; © b) 6 C vol(« e¬ ); (4.53)

where C is independent of Re¬ . We also note that the right-hand side of (4.53)coincides with the typical asymptotics for the dimension of the attractor A ofequations (1.1) in the bounded domain « e¬ (see, for example, [4]).

5. Kolmogorov’s "-entropy of the global (uniform) attractor

This section is devoted to the study of Kolmogorov’s "-entropy of the global (uni-form) attractor A of equation (3.1). First, we brie®y recall the de nition of Kol-mogorov’s "-entropy and give some classical examples (see [25, 32] for a moredetailed study).

Definition 5.1. Let K be a (pre) compact set in a metric space M . Then, dueto Hausdor¬’s criteria, it can be covered by a nite number of "-balls in M . LetN"(K; M ) be the minimal number of "-balls that cover K. Then Kolmogorov’s"-entropy of K is de ned as follows:

H"(K; M ) := log2 N"(K; M ): (5.1)

We now give several examples of typical asymptotics for the "-entropy.

Example 5.2. We assume that K = [0; 1]n and M = Rn (more generally, K is ann-dimensional compact Lipschitz manifold of a metric space M ). Then

H"(K; M ) = (n + ··o(1)) log2

1

"as " ! 0: (5.2)


This example justi es the de nition of the fractal dimension.

Definition 5.3. The fractal dimension dimF (K; M) is de ned as

dimF (K; M ) := lim sup" ! 0

H"(K; M )

log2 1=": (5.3)

The following example shows that, for sets that are not manifolds, the fractaldimension may be non-integer.

Example 5.4. Let K be a standard ternary Cantor set in M = [0; 1]. Then

dimF (K; M ) =ln 2

ln 3< 1:

The next example gives the typical behaviour of the entropy in classes of functionswith nite smoothness.

Example 5.5. Let V be a smooth bounded domain of Rn and let K be the unitball in the Sobolev space W l1;p1 (V ) and M be another Sobolev space W l2;p2 (V )such that the embedding W l1;p1 » W l2;p2 is compact, i.e.

l1 > l2 > 0;l1n

¡ 1

p1>

l2n

¡ 1

p2:

Then the entropy H"(K; M ) has the following asymptotics (see [32]):

C1

µ1

"

¶n=(l1¡l2)

6 H"(K; M) 6 C2

µ1

"

¶n=(l1¡l2)

: (5.4)

Finally, the last example shows the typical behaviour of the entropy in classes ofanalytic functions.

Example 5.6. Let V1 » V2 be two bounded domains of Cn. We assume that K isthe set of all analytic functions ¿ in V2 such that k ¿ kC(V2) 6 1 and that M = C(V1).Then

C3

µlog2

1

"

¶n+ 1

6 H"(KjV1; M ) 6 C4

µlog2

1

"

¶n + 1

(5.5)

(see [25]).

We now state the main result of this section.

Theorem 5.7. Let the assumptions of theorem 4.1 hold. Then the entropy of theattractor A satis¯es (for " > 0 small enough)

H"(A; © b) 6 C vol(« \ BReff + C1

0 ) log2

1

"

+ H"=K(H(g)j[0;L log2 1="]£ « ; Cb([0; L log2 1="]; L2b( « ))); (5.6)

where Re¬ is de¯ned in theorem 4.4 and constants C, C 0, L and K depend only onthe left-hand side of equation (3.1) and on kgk § and are independent of Re¬ .


Proof. We adapt the method suggested in [20, 38] to our situation. Let V R » © b

be the same as in theorem 4.4. Then, according to (4.31), (4.32) and (4.26), thereexists a time T = T (kgk§ ) such that

U ¹ ( ½ + T; T ) V R » V R; ½ 2 R; ¹ 2 H(g); (5.7)

and, for every u1(0); u2(0) 2 V R, decomposition (4.30) is valid,

kL u1;u2 (T )k © b6 1

16ku1(0) ¡ u2(0)k © b + K1kg1 ¡ g2kCb([0;T ];L2

b );

kN u1 ;u2 (T )k© eff6 L1ku1(0) ¡ u2(0)k © b ;

)

(5.8)

where © e¬ := W 2¡ ¯ 1;2b;¿ eff

( « ), ¿ e¬ (x) := exp(+ dist(x; BReff

0 )), ¯ 1 < ¯ , and the con-stants K1 and L1 depend only on kgk§ . It is extremely important here that theembedding © e¬ » © b be compact. Let B = B(R; u0; © b), u0 2 V R, be a © b-ball ofradius R centred at u0 such that A » B (such a ball exists, since A is boundedin © b). Our task is to construct, starting with the initial ball B, an R=2k-coveringof the attractor A, for every xed k. Estimating then the number of balls in thesecoverings, we will deduce estimate (5.6).

Let k 2 N be xed. We x the minimal (R=8K1)(1=2k)-covering of the setH(g)j[0;(k + 1)T ] for the metric of C([0; (k + 1)T ]; L2

b( « )) (it is possible to do so,since g is translation-compact; the reason for taking the interval [0; (k+1)T ] insteadof [0; kT ] will be clari ed in the next section, which is devoted to the constructionof an exponential attractor). Let G := f ¹ 1; : : : ; ¹ N g » H(g) be the centres of thiscovering. Then, obviously,

log2 N = log2 N(k) := HR=(2k+3K1)(H(g)j[0;(k + 1)T ]; C([0; (k +1)T ]; L2b( « ))): (5.80)

Let us also x the minimal covering of the unit ball B(1; 0; © e¬ ) for the © e¬ -normby a nite number P of 1=(8L1)-balls for the metric of © b (such a covering exists,since © e¬ »» © b) and

log2 P := H1=(8L1)(B(1; 0; © e¬ ); © b): (5.9)

It is important that, for every r > 0 and v 2 © b, the ball B(r; v; © e¬ ) of radius r in© e¬ centred at v can be covered by the same number P of balls of radius r=(8L1)for the © b-norm (this covering can be constructed from the initial covering of theunit ball B(1; 0; © e¬ ) by shifting and homotetion).

We now set U j0 := fu0g » V R, j = 1; : : : ; N , where u0 is the centre of our initial

R-ball and we de ne the sets U jl » V R by induction.

Let us assume that the sets U jl » V R, j = 1; : : : ; N , are already de ned for some

l < k. We now de ne the sets W jl+ 1 as

W jl+ 1 := U ¹ j ((l + 1)T; lT ) U j

l : (5.10)

We consider, for every j 2 [1; N ], the system of L1R=2l-balls for the © e¬ -normcentred at the points of W j

l . We cover each of these balls by a nite number P ofR=(2l+ 3)-balls for the metric of © b and we denote by U 0

l + 1 the set of all centres ofthese new balls. We nally construct the sets U j

l + 1 from U 0l + 1 by the following rule: if

ul 2 U 0l+ 1 and V R \ B(R=2l + 2; ul; © b) 6= ?, then we add one (arbitrary) point from


this intersection to U jl+ 1; if this intersection is empty, then we add nothing. Thus,

obviously,

#U jl + 1 6 # U 0

l + 1 6 P #U jl ; l = 0; : : : ; k ¡ 1:

Thus we have de ned by induction the sets U jl , for every 0 6 l 6 k and j 2

[1; : : : ; N ]. We also note that the number of points in U jl is given by

# U jl 6 P l: (5.11)

Lemma 5.8. The R=2k-balls in © b centred at the points of the set U k :=SN

j = 1 U jk

cover the attractor A.

Proof. Let w be an arbitrary point of A. Then, according to theorem 4.1, thereexist ¹ 2 H(g) and a complete bounded solution u(t) of (4.1) such that u(kT ) = w.

According to our construction, there exist j ¤ 2 [1; : : : ; N ] and u ¤0 := u0 2 U j¤

0

such that

k ¹ ¡ ¹ j¤ kC([0;(k + 1)T ];L2b( « )) 6 R

8K1

1

2k; ku(0) ¡ u ¤

0k © b6 R: (5.12)

Let us assume that we have already proved that, for some l < k, there existsu ¤

l 2 U j¤

l such that

ku(lT ) ¡ u ¤l k © b

6 R

2l: (5.13)

We now set vl + 1 := U¹ j¤ ((l + 1)T; lT )u ¤l 2 W j¤

l + 1. Then, due to (5.8),

u((l + 1)T ) 2 OR=2l+3

µB

µL1

R

2l; vl + 1; © e¬

¶; © b

¶; (5.14)

where O · (V; © b) is a · -neighbourhood of the set V in the space © b. Thus, due toour construction, there exists u0

l + 1 2 U 0l+ 1 such that

u((l + 1)T ) 2 OR=2l+3

µB

µR

2l+ 3; u0

l+ 1; © b

¶¶= B

µR

2l + 2; u0

l + 1; © b

¶:

We note that V R \ B(R=2l + 2; ul + 1; © b) 6= ?, since it contains u(lT ). Consequently,by the de nition of U j¤

l + 1, there exists u ¤l + 1 2 U j¤

l + 1 such that

ku((l + 1)T ) ¡ u ¤l + 1k© b

6 R

2l + 1: (5.15)

Thus, by induction, we conclude that there exists u ¤k such that

ku(kT ) ¡ u ¤kk© b

6 R

2k:

Since w = u(kT ) 2 A is arbitrary, lemma 5.8 is proved.

Therefore, we have constructed the R=2k-covering of the attractor A (centred atthe points of U k :=

SNj = 1 U j

k). Consequently,

NR=2k (A; © b) 6 NP k: (5.16)


Taking the logarithm of both sides of (5.16) and writing the explicit expression ofN = N(k), we obtain, for every k 2 N,

HR=2k (A; © b) 6 k log2 P + HR=(2k+2L1)(H(g)j[0;(k + 1)T ]£« ; C([0; (k + 1)T ]; L2b( « ))):

(5.17)Estimate (5.6) is a simple corollary of (5.17). Indeed, let R > " > 0 be xed andlet k be such that

R

2k¡1> " >

R

2k: (5.18)

Then, noting that the "-entropy is a non-increasing function of ", it follows fromequation (5.17) that

H"(A; © b) 6µ

1 + log2

R

"

¶log2 P

+ H"=(8L1)

µH(g)j[0;T (1+ log2(R="))]£ « ;

C

µ·0; T

µ1 + log2

R

"

¶¸; L2

b( « )

¶¶: (5.19)

In order to complete the proof of the theorem, it remains to verify that

log2 P 6 C vol(« \ BReff + C1

0 ): (5.20)

Indeed, according to our choice of space © e¬ and, due to (5.9), there exists a constantC1 = C1(L1; ) such that

log2 P 6 H1=(16L1)(B(1; 0; W 2¡ ¯ 1;2b ( « \ BReff + C1

0 ); © b( « \ BReff + C1

0 ))): (5.21)

It is well known (see, for example, [37,38]) that the right-hand side of (5.21) canbe estimated as follows:

H1=(16L1)(B(1; 0; W 2¡ ¯ 1;2b ( « \ BReff + C1

0 ); © b( « \ BReff + C1

0 )))

6 C2 vol(« \ BReff + C1

0 ); (5.22)

where the constant C2 is independent of Re¬ . Theorem 5.7 is proved.

Remark 5.9. We note that, in the autonomous case g = g(x), equation (5.6)implies the following estimate for the fractal dimension of A:

dimF (A; © b) 6 C vol(« \ BReff + C1

0 ): (5.23)

We also recall that, for bounded domains « , we have the estimate

dimF (A; © b) 6 C vol(« ): (5.24)

6. Exponential attractors

The main aim of this section is to construct an (in nite-dimensional) exponentialattractor for the non-autonomous problem (1.1). For the reader’s convenience, westart our considerations by giving the standard de nition of exponential attractorsfor semigroups (see [14]).


Definition 6.1. Let Sh : © b ! © b, h > 0, be a semigroup. Then, a set M is anexponential attractor for Sh if the following hold.

(1) M is compact in © b.

(2) M is semi-invariant with respect to Sh, i.e. ShM » M 8h > 0.

(3) M attracts the bounded subsets of © b exponentially, i.e. there exists ® > 0such that, for every bounded subset B » © b, there exists a constant C = C(B)such that

dist© b

(ShB; M) 6 Ce¡ ¸ h; ¸ > 0; h > 0: (6.1)

(4) The set M has nite fractal dimension, i.e.

dimF (M; © b) < 1: (6.2)

For non-autonomous equations, a de nition of an exponential attractor can beobtained from de nition 6.1 by using the skew-product technique and by projectingthe exponential attractor for the extended semigroup onto the rst component(see [20,21]). We give this de nition in the case of equation (4.1).

Definition 6.2. Let U¹ (t; ½ ) : © b ! © b be the solving operators for (4.1). Then aset M is an exponential attractor for this equation if the following hold.

(1) M is compact in © b.

(2) M attracts exponentially the trajectories of the family of equations (4.1),i.e. for every bounded subset B » © b, there exists C = C(B) such that

sup¹ 2 H (g)

dist© b

(U ¹ (t; 0)B; A) 6 Ce¡ ¸ t; ¸ > 0: (6.3)

(3) For every u0 2 M, there exists ¹ 2 H(g) such that U¹ (t; 0)u0 2 M, for everyt > 0.

(4) The set M has nite fractal dimension: dimF (M; © b) < 1.

As in the autonomous case, it follows immediately from the de nition that theuniform attractor is a subset of any exponential attractor,

A » M: (6.4)

We note that, although this de nition of non-autonomous exponential attractorsis well adapted to the study of non-autonomous equations in bounded domains «with periodic or quasi-periodic external forces (see [15, 20]), it is not convenientfor more general translation-compact time dependencies and/or for unboundeddomains. Indeed, as already mentioned, there is no reason to expect the uniformattractor to have nite dimension in general and condition (4) of de nition 6.2,together with the embedding (6.4), implies that dimF A < 1. Consequently, thereis also no reason to expect the existence of an exponential attractor in the sense ofde nition 6.2 in general.


Thus condition (4) of de nition 6.2 should be modi ed. We note, however, thatthis condition cannot be dropped completely. Indeed, in that case, every compact(semi-invariant) absorbing set of (3.1) would be an exponential attractor, whichdoes not make sense.

We note that, although an exponential attractor must have in nite dimension(if the same is true for the uniform attractor), it is reasonable to construct it as`small’ as possible. Using Kolmogorov’s entropy in order to compare the `size’ ofin nite-dimensional sets, we come to the following problem: construct an expo-nential attractor M whose entropy H"(M) has in some sense the same type ofasymptotics, as " ! 0, as the entropy H"(A) of the uniform attractor,

H"(M; V0) ¹ H"(A; V0): (6.5)

Having estimate (5.6) for the entropy of the right-hand side of (6.5), it looks reason-able to give the following de nition of (in nite-dimensional) exponential attractorsfor equation (3.1).

Definition 6.3. A set M is an (in nite-dimensional) exponential attractor forequation (3.1) if conditions (1){(3) of de nition 6.2 are satis ed and if, in addition,

H"(M; © b) 6 C1 vol(« \ BReff + C 0

0 ) log2

1

"+ H"=L1

(H(g)j[0;K1 log2 1="]£ « ; C([0; K1 log2 1="]; L2b( « ))); (6.6)

for appropriate constants C1, C 0, L1 and K1 depending only on kgk § , a, ¶ 0 and f .

Remark 6.4. We note that, if the external force g is in some proper sense nitedimensional (e.g. if it is quasi-periodic (see also the examples in x 7 below)), thenthe second term in the right-hand side of (6.6) has the asymptotics L00 log2 1=" and,consequently, equation (6.6) implies that M is nite dimensional (that is the reasonwhy we put into parentheses the words `in nite dimensional’ in de nition 6.3). Inthis situation, de nition 6.3 for exponential attractors coincides with the standardde nition (i.e. de nition 6.2).

The main result of this section is the following theorem.

Theorem 6.5. Let the assumptions of theorem 4.1 hold. Then equation (3.1) pos-sesses an (in¯nite-dimensional) exponential attractor M in the sense of de¯ni-tion 6.3.

Proof. We rst note that it is su¯ cient to verify the exponential attraction for anabsorbing set for equation (3.1) (indeed, the image of every bounded subset of © b

enters the absorbing set after a nite time).Let us construct an absorbing set B » © b £ H(g) for the extended semigroup

St : © b £ H(g) ! © b £ H(g) via the expression

B := V R £ H(g); (6.7)

where VR := VReff is the same as in theorem 4.4. We now x the time T = T (kgk § )exactly as in the proof of theorem 5.7 (in order to satisfy (5.7) and (5.8)). Then,obviously,

ST B » B: (6.8)


As usual, we rst construct an exponential attractor for the discrete semigroupS(n) := SnT , n 2 N. Then we extend this result to the continuous case.

We will use the notations of theorem 5.7. Namely, for every k 2 N, we consider theR=(K12k + 3)-net G(k) := f ¹ k

1 ; : : : ; ¹ kN g in the hull H(g)j[0;(k + 1)T ] of the right-hand

side, restricted to the time-interval [0; (k + 1)T ]. Then the number N = Nk satis esestimate (5.80). For every j 2 [1; : : : ; N (k)] and l 6 k, we x the sets U j

l (k) » V R

constructed in the proof of theorem 5.7 (except that, in contrast to x 5, we nowindicate explicitly the dependence of sets G := G(k) and U j

l := U jl (k) on k). We

also recall that# U j

l (k) 6 P l; l 6 k; j 2 [1; : : : ; N (k)]; (6.9)

where the constant P is de ned in (5.9).Then it can be proved (repeating exactly the proof of lemma 5.8) that the system

of R=(2k)-balls for the © b-metric centred at the points of U k(k) :=SN(k)

j = 1 U jk(k)

covers the set ¦ 1SkB. We reformulate this property in the following equivalentway:

dist© b

( ¦ 1(SkB); U k(k)) 6 R

2k: (6.10)

Roughly speaking, the main idea of our construction of exponential attractor is totake this attractor as the union [

Sk 2 N U k(k)]© b . Then (6.10) gives the exponential

attraction, whereas (6.9) gives the proper entropy estimate. However, in order tosatisfy the other conditions of de nition 6.3, we have to be a little more accurate.

Let us rst de ne the lifting of the sets U jk(k) (which will be denoted below by

U j(k) in order to simplify the notations) to the extended phase space © b £ H(g).To this end, for every v 2 U j(k), we take a point (uv ; ¹ v) 2 B such that

kv ¡ U¹ v (k; 0)uvkV06 R

2k; k ¹ j ¡ ¹ vkC([0;(k + 1)T ];L2

b) 6 R

(2k + 3K1): (6.11)

(It is clear from the proof of lemma 5.8 that the points v 2 U j(k) for which such apoint does not exist can be dropped out of U j

k(k), thus preserving (6.10).)We set Uj(k) := fSk(uv ; ¹ v) 2 © b £ H(g) : v 2 U j(k)g and U(k) :=

SNj = 1Uj(k).

Then (due to (6.9){(6.11))

dist© b

( ¦ 1(SkB); ¦ 1U(k)) 6 R

2k¡1; U(k) » SkB; #U(k) 6 N (k)P k: (6.12)

We are now in a position to construct the discrete exponential attractors M d andM d := ¦ 1M d . To this end, we rst de ne a sequence of sets E(k) by induction,

E(0) := U(0); E(k + 1) := S1E(k) [ U(k + 1); (6.13)

and we then de ne the exponential attractor M d as follows:

M d :=

· [

k 2 N

E(k)

¸

© b£H (g)

: (6.14)

Indeed, the semi-invariance (S1M d » M d ) is an immediate consequence of (6.13)and (6.14). Furthermore, the exponential attraction follows from the rst formulaof (6.12). Thus there only remains to verify the entropy estimate. We recall that the


entropy of a set coincides with that of its closure. Consequently, we will estimatethe entropy of M d

1 :=S

k 2 N ¦ 1E(k).Let us x an arbitrary ", R > " > 0, and compute k = k(") from the inequality

R

2k< " <

R

2k¡1: (6.15)

We then split M d1 as follows:

M d1 =

µ[

l6k

¦ 1E(l)

¶[

µ[

l>k

¦ 1E(k)

¶: (6.16)

We note that, according to our construction, E(k) » SkB. Consequently (since Bis semi-invariant), the second set in decomposition (6.16) is a subset of Sk + 1B. Werecall that (due to (6.12)) the system of R=2k(< ")-balls centred at the points of¦ 1U(k + 1) covers ¦ 1(Sk + 1B) and, consequently, covers the second set in (6.16).Thus the minimal number N"(M d ; © b) of "-balls which cover the attractor satis es

N"(Md ; © b) 6X

l6k

#E(k) + #U(k + 1) 6X

l6k + 1

#E(k): (6.17)

It follows immediately from the inductive de nition of the sets E(l) that

#E(l) 6X

m6 l

#U(m) 6 l#U(l) 6 (k + 1)#U(k + 1) 6 (k + 1)N (k + 1)P k: (6.18)

Inserting this estimate into (6.17) and applying the log2, it follows that

H"(M d ; © b) 6 2 log2(k + 1) + log2 N (k + 1) + (k + 1) log2 P: (6.19)

Expressing k = k(") from (6.15) and inserting this expression into (6.19), we obtain(as in the end of the proof of theorem 5.7) estimate (6.6). Thus the set M d isindeed a discrete exponential attractor. We also note that the upper bound (5.6)for the entropy of the global attractor di¬ers from the estimate for the exponentialattractor M d by the term log2(k(") + 1), which is proportional to the logarithm ofthe logarithm of ".

To complete the proof of the theorem, it remains to extend M d to a continuousexponential attractor.

Lemma 6.6. We setM = Mc :=

[

t 2 [0;T ]

StM d : (6.20)

Then M := ¦ 1M is an (in¯nite-dimensional) exponential attractor for (3.1).

Proof. Let us verify the conditions of de nition 6.3. The semi-invariance followsimmediately from the discrete semi-invariance of M d and from (6.20).

Let us now verify the exponential attraction. To this end, we x an arbitrary(v; ¹ ) 2 B with corresponding trajectory u(t) := U¹ (t; 0)v and we consider the timet = kT + s, where k 2 N and 0 6 t < T . Then, according to the construction of thesets U(k), there exists (v ¤ ; ¹ ¤ ) 2 B such that Sk(v ¤ ; ¹ ¤ ) 2 U(k) and

ku(k) ¡ U¹ (k; 0)v ¤ k © b6 R

2k¡1; k¹ ¡ ¹ ¤ kC([0;(k + 1)T ];L2

b( « )) 6 R

2k + 2K1: (6.21)


The second inequality follows from the second inequality in (6.11) and from thefact that G(k) = f ¹ j ; j = 1; : : : ; Ng is an R=2k + 2K1-net. In particular, we notethat this inequality implies that

kTk ¹ ¡ Tk ¹ ¤ kC([0;T ];L2b( « )) 6 R

2k + 2K1: (6.22)

(That was the reason why we have considered the time-interval [0; (k +1)T ] insteadof [0; kT ] from the very beginning.)

We note that, by de nition of M, the point u ¤ (t) := UTk ¹ ¤ (kT + s; kT )U ¹ (kT; 0)v ¤

belongs to M. Moreover, estimate (3.23) implies that

ku(t) ¡ u ¤ (t)k© b 6 L00(ku(kT ) ¡ u ¤ (kT )k © b + kTk ¹ ¡ Tk ¹ ¤ kC([0;T ];L2b( « ))): (6.23)

(We note that the constant L00 > 1 is chosen such that (6.23) holds uniformly withrespect to u(k); u ¤ (k) 2 VR and 0 < t ¡ kT < T . It is possible to do so thanks totheorem 3.9.)

Inserting estimates (6.21) and (6.22) into the right-hand side of (6.23), we have

ku(t) ¡ u ¤ (t)k© b6 L12¡k 6 L121¡t=T : (6.24)

Since u¤ (t) 2 M, we obtain the attraction property.So it only remains to verify the entropy estimate. To this end, it is convenient to

introduce the operator

S : © b £ H(g) ! C([0; T ]; © b); S(v; ¹ ) := U¹ (t; 0)v; (6.25)

and to consider the set M := SM » C([0; T ]; © b). (Roughly speaking, we replaceevery point of M by the piece of the corresponding trajectory of length T .) We alsoneed the following proposition.

Proposition 6.7. The L00R=2k¡2-balls for the topology of C([0; T ]; © b) centred atthe points of S(U(k)) cover S(SkB).

Proof. The proof of this proposition is similar to our proof of the attraction prop-erty. We take an arbitrary point (v; ¹ ) 2 B and we set u(t) := U ¹ (t; 0)v. Then thereexists a point (v ¤ ; ¹ ¤ ) 2 B, u ¤ (t) = U¹ ¤ (t; 0)v ¤ , such that (u ¤ (k); Tk ¹ ¤ ) 2 U(k) and

ku(kT ) ¡ u ¤ (kT )k © b6 R

2k¡1; k ¹ ¡ ¹ ¤ kC([0;(k + 1)T ];L2

b) 6 R

2k + 1: (6.26)

Thus, according to estimate (6.23),

ku(s + kT ) ¡ u ¤ (s + kT )k © b6 L00

µR

2k¡1+

R

2k + 1

¶6 L00 R

2k¡2; 0 < s < T; (6.27)

and proposition 6.7 is proved.

Let us now x ", R > " > 0, and let us construct the "-covering of M. As in thediscrete case, we compute k = k(") from the inequality

L00 R

2k< " < L00 R

2k¡1: (6.28)


We then decompose the set M as follows:

M =[

l6k

S(E(l)) [ Sµ[

l>k

E(l)

¶:= M1 [ M2: (6.29)

We note that, as in the discrete case, M2 » S(Sk + 1B) and, due to proposition 6.7,it can be covered by the "-balls centred at the points of S(U(k + 1)) » S(E(k + 1)).Thus we have proved that the system of "-balls centred at the points of S(E(l)),l = 1; : : : ; k+1, covers M. Consequently (cf. (6.19)), the entropy H" of this coveringsatis es

H"(M; C([0; T ]; © b)) 6 H" 6 (k + 1) log2 P + log2 N (k) + 2 log2(k + 1): (6.30)

Computing k = k(") from (6.28) and inserting this value into (6.30), we obtain anentropy estimate of the form (6.6) for the set M.

We are now in a position to estimate the entropy of M and to complete the proofof the lemma. To this end, we introduce the projector

PR : C([0; T ]; © b) ! 2© b ; Pr(v) := fv(t) : t 2 [0; T ]g:

Obviously, M = PR(M) and, consequently,

dist© b

µM;

[

l6k + 1

PR(S(E(l)))

¶6 "; (6.31)

where k = k(") is de ned by (6.25). Thus it remains to construct the covering ofthe sets O"(PR(S(E(l)))), where O" denotes the "-neighbourhood in © b. We recallthat the sets E(l) contain a nite number of points. Therefore, it is su¯ cient toknow how to cover the sets

O"(S(v; ¹ )) = fw 2 © b : 9t 2 [0; T ]; kv(t) ¡ wk © b 6 "; v(t) := U ¹ (t; 0)vg; (6.32)

for every (v; ¹ ) 2 E(l). In order to construct such a covering, we note that, accordingto our construction, M » S1B (since we take a union in (6.14) not from k = 0, butfrom k = 1). Then, according to corollary 3.7, M is bounded in W 2¡ ¯ 0 ;2

b ( « ) and,consequently (due to corollary 3.8), every trajectory u(t) := U¹ (t; 0)v starting froman arbitrary point (v; ¹ ) 2 M is uniformly H�older continuous in © b, i.e.

ku(t + s) ¡ u(t)kV06 Cs® ; t > 0; 0 < s < T: (6.33)

In particular, equation (6.33) holds uniformly with respect to (v; ¹ ) 2 E(l), l 2 N.Let us x s0 = s0(") = ("=C)1=® and let us consider the following discrete subset

of (6.32),

L(v;¹ ) := fv(ns0) : n = 0; 1; : : : ; [T=s0]g: (6.34)

Then, obviously, the 2"-balls centred at the points of L(v;¹ ) cover (6.32). Moreover,

#L(v;¹ ) = 1 +

·T

s0

¸6

µC

"

¶1=®

; (6.35)


and, consequently, the system of 2"-balls centred at the points of L(v;¹ ), (v; ¹ ) 2E(l), l 6 k + 1, covers M. The entropy of this covering can be estimated byH" + (1=® ) log2 C=". Thus

H2"(M; © b) 6 (k + 1) log2 P + log2 N (k) + 2 log2(k + 1) +1

®log2

C

": (6.36)

Computing k(") from inequality (6.28) and inserting this expression into (6.36), weobtain estimate (6.6). Lemma 6.6, and thus theorem 6.5, is proved.

7. Examples and concluding remarks

In this section, we derive several corollaries of the results obtained above whichshow, in particular, that estimates (5.6) and (6.6) are in a sense sharp. We restrictourselves to the case where the attractors A and M are nite dimensional only(examples for the in nite-dimensional case are given in [20]). We start our consid-erations with the autonomous case

g = g(x) 2 L2b( « ): (7.1)

Then estimates (5.6) and (6.6) imply that

dimF (A; © b) 6 dimF (M; © b) 6 C vol(« \ BReff + C 0

0 ); (7.2)

where Re¬ is de ned in theorem 4.4 and C and C 0 depend only on kgkL2b. So, in

that case, we have a nite-dimensional exponential attractor in complete agreementwith the standard de nition. Moreover, in the particular case « = R3, (7.2) reads

dimF (A; © b) 6 dimF (M; © b) 6 C1(Re¬ + 1)3; (7.3)

where C1 depends only on kgkL2b

and is independent of Re¬ .Let us now verify the sharpness of estimate (7.3) with respect to the parameter

Re¬ ! 1. To this end, we consider the following particular scalar (k = 1) case ofequation (3.1),

@tu = ¢ xu ¡ f(u) ¡ ¶ 0u + gR(x); (7.4)

where f (u) is an interaction function satisfying the following conditions.

Assumptions 7.1.

(1) f 2 C2(R; R).

(2) f (u) ¢ u > 0.

(3) 9 · 0 2 R; f 0( · 0) 6 ¡ ¶ 0 ¡ 1.

For instance, we can consider a Chafee{Infante equation

@tu = ¢ xu ¡ f(u) ¡ ¶ 0u;

where f (u) = au3 + bu2 + cu is such that assumption 7.1 (2) is satis ed and f is notmonotonic (for instance, for ¶ 0 = 1

3 , f (u) = u3 ¡ 7u2+14u satis es assumptions 7.1).For a given R > 1, we construct an equilibrium point uR(x) of equation (7.4) via

uR(x) := · 0 À

µjxj2R

¶; (7.5)


where À (z) 2 C 1 (R) is such that 0 6 À (z) 6 1, À (z) = 1 for jzj 6 1 and À (z) = 0for jzj > 2. Having the functions uR(x), we nally de ne a family of external forcesgR(x) such that uR(x) is the equilibrium point of equation (7.4), namely,

gR(x) := f (uR(x)) + ¶ 0uR(x) ¡ ¢ xuR(x): (7.6)

Then kgRkL1 6 C , uniformly with respect to R > 1. Moreover, supp gR » BR0 and,

consequently, we have the following estimate for the e¬ective radius Re¬ = Re¬ (gR):

C1R 6 Re¬ (gR) 6 C2R: (7.7)

In order to obtain a lower bound on the dimension of the global attractor A, weuse the well-known fact that this dimension is greater than the unstable index ofany equilibrium point of equation (7.4),

dimF (A; © b) > Ind +uR

; (7.8)

(see, for example, [4] or [31]). On the other hand, it is not di¯ cult to verify, usingthe min{max principle, that

Ind +uR

> Ind + ( ¢ x + Id; BR=20 ) > C3R3 (7.9)

(see [4] for details). Thus we have proved the following result.

Proposition 7.2. Let the above assumptions hold. Then the following estimatesare satis¯ed for the global and exponential attractors of (7.4),

C0R3 6 dimF (A; © b) 6 dimF (M; © b) 6 C 00R3; (7.10)

where C0 and C 00 are independent of R > 1.

Thus (7.10) shows that (5.6) and (6.6) are sharp as Re¬ ! 1.For our next applications, it will be useful to have an exponential attractor not

only in the phase space © b, but also in the space L 1 ( « ).

Proposition 7.3. Let the assumptions of theorem 4.1 hold and, in addition, letequation (7.1) be satis¯ed. Then there exists an exponential attractor M 2 © b

of (3.1) such that the following hold.

(1) There exists a positive constant ® > 0 and a monotonic function Q, whichare independent of Re¬ , such that, for every bounded set B » L 1 ( « ),

distL1 ( « )

(StB; M) 6 Q(kBkL1 ( « ))e¡ ® t: (7.11)

(2) The fractal dimension of M satis¯es the estimate

dimF (A; L 1 ( « )) 6 C vol(« \ BReff + C 0

0 ); (7.12)

where C and C 0 are also independent of Re¬ .


Proof. The assertion of proposition 7.3 for the phase space © b is proved in theo-rem 6.5. In order to extend this result to L 1 ( « ), it remains to note that, as provedin [16], equation (3.1) possesses an L 1 ( « ) ! © b smoothing property of the form

ku(1)k © b6 Q(ku(0)kL1 ) + Q(kgkL2

b); (7.13)

where the function Q is independent of Re¬ , which nishes the proof of proposi-tion 7.3.

In our next application, we consider the following reaction{di¬usion system oftype (3.1), with a small di¬usion parameter ¸ ½ 1, in the domain « ,

@tu = ¸ ¢ xu ¡ ¶ 0u ¡ f (u) + g(x); (7.14)

where the nonlinearity is independent of rxu.

Proposition 7.4. Let the function f satisfy

f 2 C2(Rk; Rk) and f (u):u > 0; (7.15)

and, in addition, let the external forces satisfy

g 2 _L 1 ( « ): (7.16)

Then equation (7.14) possesses an exponential attractor M ¸ , for every ¸ > 0, inL 1 ( « ). Moreover, these attractors satisfy (7.11) uniformly with respect to ¸ > 0and their fractal dimension can be estimated as follows,

dimF (A ¸ ; L 1 ( « )) 6 dimF (M ¸ ; L 1 ( « )) 6 C¸ ¡3=2; (7.17)

where the constant C is independent of ¸ > 0.

Proof. Indeed, the form of equation (7.14) is preserved under the rescaling x !x0 ¸ 1=2 (which obviously also preserves the L 1 -norm). We should then only replacethe domain « by « 0 = ¸ ¡1=2 « , the di¬usion coe¯ cient ¸ by 1 and the externalforces g(x) by g ¸ (x0) := g(x0 ¸ 1=2). We also note that the L2

b -norms of g ¸ are uni-formly bounded as ¸ ! 0 (to this end, we need the assumption g 2 L 1 ( « )). Thus,constructing an exponential attractor for the rescaled equation, returning then tothe initial problem and using the invariance of the L 1 -norm, we construct expo-nential attractors M ¸ enjoying property (7.11) uniformly with respect to ¸ andsatisfying the following estimates,

dimF (M ¸ ; L 1 ( « )) 6 C vol( ¡1=2 « \ BReff (g )+ C 0) 6 C1(Re¬ (g ¸ ) + 1)3; (7.18)

where the constant C1 is independent of ¸ . So it remains to estimate the quantityRe¬ (g ¸ ) with respect to ¸ . To this end, we note that the assumption g 2 _L 1 ( « ),the obvious formula

R0;2(H(g); z) 6 CR

0; 1 (H(g); z)

:= C sup¹ 2 H(g)

supx 2 «

fexp( ¡ dist(x; R3 n Bz0 ))j¹ (0; x)jg;


where C is independent of z and g, the equality

R0; 1 (H(g ¸ ); z) = R

0; 1 (H(g); z ¸ 1=2)

and the de nition of Re¬ (see theorem 3.9) immediately imply the estimate

Re¬ (g ¸ ) 6 Cg ¸ ¡1=2: (7.180)

Inserting (7.180) into (7.18), we nally obtain (7.17) and proposition 7.4 is proved.

Remark 7.5. It is well known (see, for example, [4,23,31]) that estimate (7.17) issharp with respect to the parameter ¸ .

Remark 7.6. We note that we have rigorously proved estimate (7.17) only forscalar di¬usion matrices a. Nevertheless, we have used this assumption only in orderto derive the dissipative estimate (3.4) for the L 1 -norm (applying the maximumprinciple). Consequently, if this estimate is a priori known, then we can extend theresult of proposition 7.4 to larger classes of reaction{di¬usion systems. In particular,instead of (7.14), we can consider the system

@tu = a¸ ¢ xu ¡ ¶ 0u ¡ f (u) + g(x); (7.19)

with an arbitrary di¬usion matrix a satisfying a + a ¤ > 0 and a nonlinearity fsatisfying the assumptions

f 2 C2(Rk; Rk); f 0(u) > ¡ K; f (u)¢u > 0; jf (u)j 6 C(1+jujp); (7.20)

where p is arbitrary. The dissipative estimate for the L 1 -norm, of the form (3.4)(for ¸ = 1 after rescaling), has been obtained in [35]. Thus equations (7.19) possessexponential attractors M ¸ which satisfy (7.11) uniformly with respect to ¸ andtheir fractal dimension satis es the sharp estimate (7.17).

Let us now consider the case of quasi-periodic external forces g(t). More precisely,we assume that the function g has the following structure,

g(t) := G(T !t ¿ 0); G 2 C1(Tm; _L2

b( « )); ¿ 0 2 Tm; (7.21)

where Tm is the m-dimensional torus, ! = (!1; : : : ; !m) is a rationally independentfrequency vector and T !

t ¿ 0 := ( ¿ 0 + !t) (mod (2 º )m) is a standard linear ®ow onthe torus. In that case, the hull H(g) possesses the following description,

H(g) = fG(T !t ¿ ); ¿ 2 Tmg; (7.22)

and, consequently, using the obvious fact that the standard linear ®ow preservesthe metric on the torus and the smoothness of G (G 2 C1), we have

H"=L1

µH(g); L 1

µ·0; K1 ln

1

"

¸; L2

b( « )

¶¶6 H"=L1

(H(g); C(R; L2b( « )))

6 H"=(CGL1)(Tm; Rm)

6 (m + ··o(1)) ln1

"; (7.23)


where CG only depends on kGkC1(R;L2b( « )). Thus theorem 6.5 and (7.23) imply the

following result.

Proposition 7.7. Let the assumptions of theorem 4.1 hold and, in addition, let theexternal forces g have structure (7.21) (i.e. let g be quasi-periodic with m indepen-dent frequencies). Then there exists ¯nite-dimensional uniform A and exponentialM attractors of equation (4.1), whose fractal dimensions satisfy

dimF (A; © b) 6 dimF (M; © b) 6 C vol(« \ BReff + C 0

0 ) + m; (7.24)

where the ¯rst term in the right-hand side of (7.24) is the same as in the autonomouscase (see (7.2)).

As in the autonomous case, it is not di¯ cult to verify the sharpness of esti-mate (7.24).

Proposition 7.8. Under the assumptions of proposition 7.7, there exists a familyof equations of the form (4.1) in « = R3 such that

C1R3e¬ + m 6 dimF (A; © b) 6 dimF (M; © b) 6 C2R3

e¬ + m; (7.25)

where Ci, i = 1; 2, are independent of Re¬ and m.

Proof. Let us consider equation (7.4) in « = R3 under assumptions 7.1 on thenonlinearity and with the external forces gR(x) de ned in (7.6). Then, without lossof generality, we may assume that the equilibrium uR(x) is hyperbolic (it is wellknown (see, for example, [4,37]) that the hyperbolicity of an equilibrium is a genericproperty, so, if it is violated for the initial gR, one can nd a new ~gR arbitrarilyclose to gR for which this assumption is satis ed). Thus we have

© b = © ¡b + © +

b ; (7.26)

where the subspaces © +b := ¦ + © b and © ¡

b := ¦ ¡ © b correspond to the stable andunstable parts of the linear operator ¢ x ¡ ¶ 0 ¡ f 0(uR). Moreover, it is well knownthat the unstable subspace is nite dimensional,

dim © +b < 1; (7.27)

and there exists the nite-dimensional local unstable manifold M + (uR), de ned by

M + (uR) :=n

u0 2 © b : ku0 ¡ uRk © b6 "; 9u 2 Cb(R¡; © b)

such that u(t) solves (7.4), u(0) = u0 and limt! ¡1

u(t) = uR

o;

(7.28)

where " > 0 is small enough (see, for example, [3,37] for details). This manifold isC1-di¬eomorphic to © +

b , i.e. there exists

M 2 C1(B("; 0; © +b ); © ¡

b ); M (0) = M 0(0) = 0; (7.29)

such that

M + (uR) = fv 2 © B : 9u +0 2 B("; 0; © +

b ); v = uR + u +0 + M (u +

0 )g:


Let us now construct the small non-autonomous perturbation of equation (7.4). Tothis end, we x an arbitrary quasi-periodic function w(t; ¿ 0) satisfying

w(t; ¿ 0) := W (T !t ¿ 0); W 2 C2(Tm; _C2(R3)); (7.30)

de ne the non-autonomous perturbation of the external forces gR(x), for everysmall ¯ > 0, by

gR;m;¿ 0 (t) := gR + ¯ g1(t; ¯ ; ¿ 0); (7.31)

where

g1(t; ¯ ; ¿ 0) := @tw(t; ¿ 0) ¡ a¢ xw(t; ¿ 0) +1

¯[f (uR + ¯ w(t; ¿ 0)) ¡ f (uR)]; (7.32)

and consider the following equation:

@tu = ¢ xu ¡ f (u) + gR;m;¿ 0 (t): (7.33)

We note that the external forces in (7.33) are de ned such that the function~u(t) := uR + ¯ w(t; ¿ 0) is a solution of (7.33).

Then, according to the standard theory of non-autonomous perturbations ofunstable manifolds (see [17,28]), for every ¿ 0 2 Tm, the set

M +¯ (uR; ¿ 0) := fu0 2 © b : 9u 2 Cb(R¡; © b) such that u(t) solves (7.33),

u(0) = u0 and ku(t) ¡ uRk © b6 " 8t 6 0g (7.34)

is a C1-submanifold of © b that is di¬eomorphic to © +b if ¯ > 0 is small enough.

Moreover, these manifolds tend to (7.28) as ¯ ! 0; more precisely, there exists

N 2 C1([0; ¯ 0] £ B("; 0; © +b ) £ Tm; © b); (7.35)

such that

M +¯ (uR; ¿ 0) = fv 2 © b; 9u +

0 2 B("; 0; © +b );

v = V ( ¯ ; u +0 ; ¿ 0) := uR + u +

0 + M (u +0 ) + ¯ N ( ¯ ; u +

0 ; ¿ 0)g:

We now recall that, according to the construction of the uniform attractor A ofproblem (7.33), [

¿ 0 2 Tm

M +¯ (uR; ¿ 0) » A: (7.36)

Consequently, there only remains to nd an m + dim © +b -dimensional submanifold

of © b which is contained in the left-hand side of (7.36). Thanks to the implicitfunction theorem, such a submanifold exists if

rankfDu+0

V ( ¯ ; 0; ¿ 0); D ¿ 0 V ( ¯ ; 0; ¿ 0)g

= rank

µE µ ¯ Du

+0

N ( ¯ ; 0; ¿ 0)

¯ ¦ + D ¿ 0 N ( ¯ ; 0; ¿ 0) ¯ ¦ ¡D ¿ 0 N ( ¯ ; 0; ¿ 0)

¶

= µ + m; (7.37)

at least for one point ¿ 0 2 Tm, where µ := Ind +uR

and Eµ is the identity matrix.


It is not di¯ cult to verify that condition (7.37) is satis ed for su¯ ciently small¯ > 0 if

rankf¦ ¡D ¿ 0 N (0; 0; ¿ 0)g = m: (7.38)

It remains to note that the derivative in (7.38) can be found as a unique boundedsolution of the variation equation associated with (7.33). More precisely, let ® 2 Rm

and let v ®¡(t) be the unique bounded solution of

@tv®¡ = ¦ ¡( ¢ xv ®

¡ ¡ f 0(uR)v ®¡) + ¦ ¡D ¿ 0 g1(t; 0; ¿ 0) ® ; t 6 0: (7.39)

Then D ¿ 0 N (0; 0; ¿ 0) ® = v ®¡(0). On the other hand, equation (7.32) implies that

D ¿ 0 g1(t; 0; ¿ 0) = @tD ¿ 0 w(t; ¿ 0) ¡ a¢ xD ¿ 0 w(t; ¿ 0) + f 0(uR)D ¿ 0 w(t; ¿ 0): (7.40)

Comparing (7.39) and (7.40), we conclude that

¦ ¡D ¿ 0 N (0; 0; ¿ 0) = ¦ ¡D ¿ 0 W ( ¿ 0): (7.41)

Therefore, condition (7.38) is obviously satis ed if the non-autonomous perturba-tion satis es the additional generic assumption

rankf ¦ ¡D ¿ 0 W ( ¿ 0)g = m for some ¿ 0 2 Tm: (7.42)

Thus, under assumption (7.42), the uniform attractor A of equation (7.33) containsa submanifold that is di¬eomorphic to RIn d +

uR+ m, if ¯ > 0 is small enough, and,

consequently,

dimF (A; © b) > Ind +uR

+m: (7.43)

Combining now (7.9), (7.43) and proposition 7.7, we nish the proof of proposi-tion 7.8. Thus (7.24) is indeed sharp with respect to Re¬ and m.

To conclude, let us formulate the non-autonomous analogue of proposition 7.4.

Proposition 7.9. Let the function f = f (u) satisfy (7.15) and, in addition, letthe external forces g be quasi-periodic with m independent frequencies,

g(t) := G(T !t ¿ 0); G 2 C1(Tm; _L 1 ( « )): (7.44)

Then there exists a family M ;m of exponential attractors for equations (7.14)(with g replaced by g(t)) such that the following hold.

(1) There exists a positive constant ® > 0 and a monotonic function Q, which areindependent of ¸ and m (but which obviously depend on kGkC1 ), such that,for every bounded set B » L 1 ( « ),

distL1 ( « )

(Ug( ½ + t; ½ )B; M ;m) 6 Q(kBkL1 ( « ))e¡ ® t: (7.45)

(2) The fractal dimension of M ¸ ;m satis¯es the inequalities

dimF (A ¸ ;m; L 1 ( « )) 6 dimF (M ¸ ;m; L 1 ( « )) 6 C1 ¸ ¡3=2 + m; (7.46)

where C is independent of ¸ and m.


The proof of proposition 7.9 is based on a rescaling argument and is very similarto that of proposition 7.4, so we omit it here.

Remark 7.10. Applying the rescaling arguments to proposition 7.8, we obtainexamples of non-autonomous equations (7.14) for which

C1 ¸ ¡3=2 + m 6 dimF (A ¸ ;m; L 1 ( « )) 6 dimF (M ¸ ;m; L 1 ( « )) 6 C2 ¸ ¡3=2 + m;(7.47)

where C1 and C2 are independent of ¸ and m. We also note that estimates of theform (7.47) for uniform attractors in bounded domains « are obtained in [11].

Remark 7.11. To conclude, we note that, although we have considered in thispaper only the case of three-dimensional domains « » R3, the main results ofthe paper remain true (after minor changes) for an arbitrary space dimension n.Obviously, in that case, instead of assumption (3.2), we should require that

g 2 Cb(R; Lqb( « )); with q > 1

2 n; (7.48)

and is translation-compact in the local topology of this space, and consider prob-lem (1.1) in the phase space

© qb := W 2¡ ¯ ;q

b ( « )

(see also [18,38]).

Acknowledgments

This research was partly supported by INTAS project no. 00-899 and CRDF grantno. 10545.

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(Issued 7 May 2004)

Global and exponential attractors for nonlinear reaction–diffusion systems in unbounded domains

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