Abstract

First, for a process , we introduce a new concept, called the weak -pullback exponential attractor, which is a family of sets , for any , satisfying the following: (i) is compact, (ii) is positively invariant, that is, , and (iii) there exist such that ; that is, pullback exponential attracts . Then we give a method to obtain the existence of weak -pullback exponential attractors for a process. As an application, we obtain the existence of weak -pullback exponential attractor for reaction diffusion equation in with exponential growth of the external force.

1. Introduction

Pullback attractor is a suitable concept to describe the long time behavior of infinite dimensional nonautonomous dynamical systems or process generated by nonautonomous partial differential equations. There are many references concerned with the existence of pullback attractors for nonautonomous PDEs (see [15]). In [2], Caraballo introduced the notion of -pullback attractor for nonautonomous dynamical systems and gave a general method to prove the existence of -pullback attractor. However, pullback attractors or -pullback attractors attract any bounded set of phase space, but the attraction to it may be arbitrarily slow. In order to describe the attracting speed, the concept of pullback exponential attractor is put forward (see [6]), which is a positively invariant family of compact subsets with finite fractal dimension (see [7, 8]) and exponentially attracts each bounded subset. In [6], a new method is given to prove the existence of pullback exponential attractor and it is applied to reaction diffusion equation when the external force is normal; in [9], the same result is obtained when the nonlinear term satisfies . In fact, these conditions are relatively strict; for general conditions, we can not get the result.

Motivated by these problems and some ideas in [36], we introduce a new attractor, called the weak -pullback exponential attractors that is for a process , for any , there exists a family of sets satisfying the following:(i) is compact.(ii) is positively invariant; that is, .(iii), there exist such that , that is, pullback exponential attracts for all .

Compared with the pullback exponential attractor, the fractal dimension of the weak -pullback exponential attractor is not necessarily uniformly bounded or even unbounded, and the positively invariant only holds for any , compared with the -pullback attractor, which pullback attracts bounded set with exponential speed and contains -pullback attractor.

The paper is organized as follows. In Section 2, we recall some basic concepts about pullback attractor. In Section 3, we construct a weak -pullback exponential attractor for nonautonomous dynamical systems and we provided a method to verify the existence of weak -pullback exponential attractor. In Section 4, we apply our result to prove the existence of weak -pullback exponential attractor for nonautonomous reaction diffusion system in with exponential growth of the external force.

2. Preliminaries

Let be a complete metric space; let be the set of all bounded subsets of ; is a nonempty class of parameterised sets or ; and a two-parameter family of mappings act on , that is, , .

Definition 1. A two-parameter family of mappings is said to be a process in , if(1), ,(2) is the identity operator, .The pair is generally referred to as a nonautonomous dynamical system, and () is called a nonautonomous discrete dynamical system generated by . If is continuous in , we say that the process is continuous process; if as , we say that the process is the norm-to-weak continuous process. Obviously, continuous process is also a norm-to-weak continuous process.

Definition 2. A family of sets is called -pullback bounded absorbing sets for the process if, for any and any bounded sets , there exists such that for all .

Definition 3. The family is said to be a -pullback attractor for if the following hold: (1) is compact for all ;(2) is invariant; that is,   ;(3) is -pullback attracting; that is, , , and ;(4)if is another family of closed attracting sets, then   .Here denotes the nonsymmetric Hausdorff distance between sets in ; that is, .

Definition 4. The Kuratowski measure of noncompactness of is defined by

The following summarizes some of the basic properties of the measure of noncompactness.

Lemma 5 (see [10]). Let . Then if, and only if, is compact;; for ;;if are nonempty closed sets in such that as , then is nonempty and compact.In addition, let be an infinite dimensional Banach space with a decomposition and let , be projectors with . Then , where is a ball of radius ; for any bounded subset of for which the diameter of is less than .

Definition 6 (see [35]). A process is called -pullback -limit compact for if, for any , there exists a such that .

Lemma 7 (see [35]). Assume that the process is -pullback -limit compact for ; then, for any sequence , as and for any sequence , there exists a convergence subsequence of whose limit lies in ; here is defined by

Theorem 8 (see [3, 5]). Let be a continuous or norm-to-weak continuous process and is -pullback -limit compact; let be a family of -pullback bounded absorbing sets for the process. Then the process has a -pullback attractor , and

For a discrete process , the above conclusions also hold true.

3. The Existence of Weak -Pullback Exponential Attractor

Let be a Banach space; denotes the norm of , is a nonempty class of parameterised sets or , and is a continuous process on .

Now, we give our main theorems which describe the relationship between the measure of noncompactness and the weak -pullback exponential attractor.

Theorem 9. Assume that is positively invariant -pullback bounded absorbing sets of ; that is, for any , , there exists , such that for any , and for any ; then the following are equivalent: The measure of noncompactness -pullback decays exponentially for the discrete process ; that is, there exist such thatThe process has a weak -pullback exponential attractor; that is, there exists a family of sets satisfying the following: is compact; is positively invariant; that is, ; attracts exponentially in a -pullback sense; more precisely,

Proof. Since the measure of noncompactness -pullback decays exponentially for , from Definition 6, we find that is -pullback -limit compact. By Theorem 8, we get that is a -pullback attractor of . Using of Lemma 5, we find that and by the definition of the measure of noncompactness, for any , there exist finite points such that Letting and , we getConsequently, for all , the family is positively invariant.
Let ; we claim that satisfies (II).
(Compactness) for any sequence , there exist and such that . By (I), we get that the process is pullback -limit compact; we deduce from Lemma 7 that has subsequence convergent in . We get that is compact.
(Positively invariant) since , , we get (Exponential attracting) for any , there exists , such that Since is positively invariant, we getso we obtain Since , we get for any .
() By the definition of , we get and, for any , we have Therefore, for any , there exists , such that We get Since is a compact set, we get that there exist such that Therefore, for any , there exists such that We get and, by Definition 4, we obtain and, by of Lemma 5, we get which say that the measure of noncompactness -pullback decays exponentially.

Theorem 10. Assume that is positively invariant -pullback bounded absorbing sets of ; that is, for any , , there exists , such that for any , and for any , and there exists a continuous function that satisfies for any , ; then the following are equivalent:The measure of noncompactness -pullback decays exponentially for the process ; that is, there exist such thatThe process has a weak -pullback attractor; that is, there exists a family of sets satisfying the following: is compact; is positively invariant; that is, ; attracts exponentially in -pullback sense; more precisely,

Proof. () By Theorem 9, we know that the discrete process generated by has a weak -pullback exponential attractor , that is, is compact and positively invariant and -pullback exponentially attracts . We set , , for all . As proof of Theorem 9, it is easy to prove that is compact and positively invariant. Next, we will prove that attracts exponentially in -pullback sense.
For any , there exists such that for any . For discrete process , by Theorem 9, there exist such that For any , there exist such that , ; thereforeWe obtain that attracts exponentially in a -pullback sense.
() The proof is the same as that of Theorem 9, so we omit it.

We now present a method to verify that the measure of noncompactness -pullback decays exponentially for the process .

Let be a uniformly convex Banach space; that is, for all , there exists such that, given , , , ; then . Requiring a space to be uniformly convex is not a severe restriction in application, since this property is satisfied by all Hilbert spaces, the space with , and most Sobolev spaces with .

Definition 11 (enhanced flattening property). Let be a uniformly convex Banach space; for a family of bounded sets , there exist , and for any finite dimension subspace of , such that(i) is bounded;(ii), ,for all . Here denote the norm in and is real-valued function satisfying

Theorem 12. Assume that the process satisfies the enhanced flattening property; then the measure of noncompactness -pullback decays exponentially for .

Proof. For any , from and of Lemma 5, and the enhanced flattening property, we get Since , for , there exists , for any ; we haveHence, ; that is, the measure of noncompactness of -pullback decays exponentially.
Let be the set of all functions such that for some , , and denote by the class of all families such that for some , denote the closed ball in with radius .

Theorem 13. Assume that the process satisfies for some , , and and for any ; then the process has a family of positively invariant -pullback bounded absorbing sets ; that is, for any , there exists such that for any and .

Proof. Let us define For every , there exists such that Obviously, is a family of -pullback bounded absorbing sets. Moreover, there exists such that Note that these can not hold for any . Let We know that and is also a family of -pullback bounded absorbing sets. We also have

By Theorems 1013, we get the following theorem.

Theorem 14. Let be a uniformly convex Banach space; is a process on , and the process satisfies the following: for some , , and and any ., , for all . Here is the dimension of subspace of , and is real-valued function that satisfies, for any , ; then the process has a weak -pullback exponential attractor; that is, for any , there exists a family of sets satisfying the following: is compact. is positively invariant; that is, . attracts exponentially in a -pullback sense; more precisely,

4. Application to Nonautonomous Reaction Diffusion Equation

As an application of Theorem 14, we prove the existence of the weak -pullback exponential attractor in for the process generated by the solution of the following nonautonomous reaction diffusion equation: where , , is a bounded open subset of , and there exist ,  ,  , such thatfor all .

We set , naming the first eigenvalue of , and denote by scalar product and norm ; let and denote the scalar product and norm of and for all . Moreover, we suppose for any that there exist and such that

For this initial boundary value problem, we know from [7, 8] that, for any ,  , there exists a unique solution .

Thanks to the existence theorem, the initial boundary value problem is equivalent to a process defined bywhere is the solution of (38)–(40) with as initial data at time .

Theorem 15 (see [3]). Assume that and satisfy (39)–(41) and is a weak solution associated with (38). Then the following inequality holds for :

Theorem 16. Assume that and satisfy (39)–(41), where , (). Then the process defined by (42) has a weak -pullback exponential attractor in .

Next, we will prove that the process defined by (42) satisfy (I)–(III) of Theorem 14.

Proof. By (42), for , and, for ,Therefore, for any , we have Using the same proof, we can get By (43) and using (46) and (47) we find that there exists , for any ; we have By Theorem 13, for any fixed , the process generated by (38) is a family of positively invariant -pullback bounded absorbing sets and for any , .
Let be the set of all functions such that and denote by the class of all families such that for some , denotes the closed ball in with radius .
Since is a continuous compact operator in , by the classical spectral theorem, there exist a sequence , and a family of elements of which are orthogonal in such thatLet in and is an orthogonal projector. For any we writeWe set and to be solutions associated with (38) with initial data . Let ; by (38) we get Taking inner product of (52) with in , we haveTaking into account (40) and Hölder inequality, it is immediate to see that Since , , using Sobolev embedding theorem, we obtain Since , we get From (53)–(56), we have Therefore We obtain For any we write