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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 719063, 13 pages
http://dx.doi.org/10.1155/2013/719063
Research Article

Pullback -Attractor of Nonautonomous Three-Component Reversible Gray-Scott System on Unbounded Domains

College of Science, Guilin University of Technology, Guilin 541004, China

Received 22 April 2013; Accepted 29 July 2013

Academic Editor: Shaoyong Lai

Copyright © 2013 Anhui Gu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The long time behavior of solutions of the nonautonomous three-components reversible Gray-Scott system defined on the entire space is studied when the external forcing terms are unbounded in a phase space. The existence of a pullback global attractor for the equation is established in and , respectively. The pullback asymptotic compactness of solutions is proved by using uniform estimates on the tails of solutions on unbounded domains.

1. Introduction

In this paper, we consider the dynamical behavior of the nonautonomous three-components reversible Gray-Scott system on with initial data where ; all the parameters are arbitrarily given positive constants; for and is an external forcing term which is locally square integrable in time. That is, , .

Historically, when , , and the external forces , , system (1) reduces to the two-component Gray-Scott system which signified one of the Brussell school led by the renowned physical chemist and Nobel Prize laureate (1977), Ilya Prigogine, which originated from describing an isothermal, cubic autocatalytic, continuously fed, and diffusive reactions of two chemicals (see, e.g., [1, 2]). The three-component reversible Gray-Scott model was firstly introduced by Mahara et al., which is based on the scheme of two reversible chemical or biochemical reactions [3]. Then in [4], You takes some nondimensional transformations, and the three-component reversible system is reduced to the system (1) without external forces. In [4], You considers the existence of global attractor for the system with Neumann boundary condition on a bounded domain of space dimension by the method of the rescaling and grouping estimation. For more recent dynamical behaviors of the nonautonomous three-component reversible Gray-Scott system, we can refer to [5, 6] for the existence of random attractors of the stochastic cases on bounded and unbounded domains and [7] for the existence of uniform attractor of the deterministic case on a bounded domain.

As pointed in [8], we can discuss the same or similar coupled reaction-diffusion systems on a higher dimensional domain with the space dimension and on an unbounded domain, to work with various different phase spaces. Here, we intend to investigate the dynamical behavior of the nonautonomous three-component reversible Gray-Scott system on unbounded domains. It is worth mentioning that the Sobolev embeddings are not compact on domains of infinite volume. This introduces a major obstacle for proving the existence of attractors for PDEs on unbounded domains. For some deterministic equations, the difficulty caused by the unboundedness of domains can be overcome by the energy equation approach which is developed by Ball in [9] and used by many authors (see, e.g., [10, 11]). In this paper, we will use the uniform estimates on the tails of solutions to circumvent the difficulty caused by the unboundedness of the domain. This idea was developed in [12] to prove the asymptotic compactness of solutions for autonomous parabolic equations on and later extended to stochastic equations in, for example, [6, 1315]. Here, we will use the method of tail estimates to investigate the asymptotic behavior of system (1) with initial data (2). We first establish the pullback asymptotic compactness of solutions of system (1) and prove the existence of a pullback global attractor in . Then we extend this result and show the existence of a pullback global attractor in .

The paper is organized as follows. In the next section, we recall the fundamental concepts and results for pullback attractors for nonautonomous dynamical systems. In Section 3, we define a cocycle for the nonautonomous three-component reversible Gray-Scott system on . Section 4 is devoted to deriving the uniform estimates of solutions for large space and time variables. In the last section, we prove the existence of a pullback global attractor for the equation in and .

The following notations will be used throughout the paper. We denote by and the norm and inner product in or and use , to denote the norm in or , or ; the dual of ; . The letters is a generic positive constant which may change its value from line to line or even in the same line.

2. Preliminaries

In this section, we recall some basic concepts related to pullback attractors for nonautonomous dynamical systems. It is worth noticing that these concepts are quite similar to those of random attractors for stochastic systems. We can refer to [1619] for more details.

Let be a nonempty set and a metric space with distance .

Definition 1. A family of mappings from to itself is called a family of shift operators on if satisfies the group properties:(i), for all ,(ii), for all and .

Definition 2. Let be a family of shift operators on . Then a continuous -cocycle on is a mapping which satisfies, for all and , the following:(i) is the identity on ,(ii),(iii) is continuous.

Hereafter, we always assume that is a continuous -cocycle on and is a collection of families of subsets of :

Definition 3. Let be a collection of families of subsets of . Then is called inclusion closed if and with for all imply that .

Definition 4. Let be a collection of families of subsets of , and is called a pullback absorbing set for in if, for every and , there exists such that

Definition 5. Let be a collection of families of subsets of . Then is said to be -pullback asymptotically compact in if, for every , has a convergent subsequence in whenever and with .

Definition 6. Let be a collection of families of subsets of and . Then is called a pullback global attractor for if the following conditions are satisfied, for every :(i) is compact,(ii) is invariant, that is, (iii) attracts every set in , that is, for every , where is the Hausdorff semimetric given by for any and .

Theorem 7. Let be an inclusion-closed collection of families of subsets of and a continuous -cocycle on . Suppose that is a closed absorbing set for in and is -pullback asymptotically compact in . Then has a unique -pullback global attractor which is given by

3. Cocycle Related to Three-Component Reversible Gray-Scott System

In this section, we constructed a -cocycle for the nonautonomous three-component reversible Gray-Scott system defined on . For every and , system (1) with initial data (2) can be rewritten as with initial condition where and here denotes the transposition.

As in the case of bounded domains, by conducting a priori estimate on the Galerkin approximate solutions of system (9)-(10) similar to the autonomous system studied in [7], we can prove that if , then problem (9)-(10) is well defined in . That is, for any and , (9) possesses a unique solution satisfying which continuously depends on the initial data . To construct a cocycle for problem (9)-(10), we denote by and define a shift operator on for every by Let be a mapping from to given by where , , , and is the solution of problem (9)-(10). By the uniqueness of solutions, we find that, for every , and , Then it follows that is a continuous -cocycle on . The purpose of this paper is to study the existence of pullback attractors for in an appropriate phase space.

Let be a subset of and denote Let be a family of subsets of . That is, for every and satisfying where is a positive number given in (1). Hereafter, we use to denote the collection of all families of subsets of satisfying (17), that is, Throughout this paper, we assume the following conditions for the external term:

It is useful to note that condition (21) implies that, for every and , there is such that As we will see later, inequality (22) is crucial for deriving uniform estimates on the tails of solutions, and these estimates are necessary for proving the asymptotic compactness of solutions.

4. Uniform Estimates of Solutions

In this section, we derive uniform estimates of solutions of problem (9)-(10) defined on when . We start with the estimates in .

Lemma 8. Suppose that (19) holds. Then for every and , there exists such that for all , where and depends on , , , and .

Proof. Define then the system (1) becomes Taking the inner products , , and and then suming up the resulting equalities, we get Setting then (28) becomes Multiplying (30) by and then integrating the resulting inequality on with , we find that Notice that and . We find that for every , there exists such that for all , By (31) and (32), we get that, for all , which completes the proof.

The following lemma is useful for deriving uniform estimates of solutions in .

Lemma 9. Suppose that (19) holds. Then for every and , there exists such that for all , where and relies on , , , and .

Proof. By (30) we find that Let and . Multiplying the above by and integrating over , we get Therefore, there exists such that for all and , Integrate the above with respect to on to obtain On the other hand, for , (37) implies that Multiplying (30) by and then integrating over , by (38) we get that, for all , which along with (39) completes the proof.

Note that for any . So as an immediate consequence of Lemma 9 we have the following estimates.

Corollary 10. Suppose that (19) holds. Then for every and , there exists such that for all , where and depends on , , , and .

Before the derivation of uniform estimates of solutions in , we firstly give two propositions which will frequently be used in the next results.

Proposition 11. Suppose that (20) holds. Then for every and , there exists such that for all , where and depends on , , , and .

Proof. Let , then (25)–(27) become Taking the inner products , and and then suming up the resulting equalities, we get By Young's inequality, From (46) then (45) yields Denote then from (47) implies that Multiplying (49) by and then integrating the resulting inequality on with , we find that
Notice that and . We find that, for every , there exists such that for all , By (50) and (51), we get that, for all , which completes the proof.

Similarly, we have the following.

Proposition 12. Suppose that (20) holds. Then for every and , there exists such that for all , where and depends on , , , and .

Proof. The proof is similar to Proposition 11 except for few trivial details, and thus we omit it here.

Lemma 13. Suppose that (19) and (20) hold. Then for every and , there exists such that for all , where and depends on , , , , and .

Proof. Taking the inner product of the first equation of system (1) with , the second equation with , and the third equation with , respectively, in and then replacing by , we obtain where (55) and (56) are partly due to the Young inequality . Denote , , , and with , , , and , respectively. Adding the three inequalities (55)–(57) together, we have Let Then (58) yields That is, Let and . By integrating (61) over , we get Now integrating the above with respect to s on , we find which along with Corollary 10 and Proposition 11 implies that there exists such that for all , Similarly, first integrating (61) with respect to on and then integrating with respect to on , by using Corollary 10 and Proposition 11, we can get that for all , Now integrating (60) over , we obtain which along with (65) shows that, for all , Then Lemma 13 follows from (64) and (67) which completes the proof.

Lemma 14. Suppose that (19) and (20) hold, and let Then for every and , there exists such that for all , where and depends on , , , , , and (a positive constant in the Gagliardo-Nirenberg inequality).

Proof. Let , ,  and and differentiate system (1) with respect to to get that Taking the inner products , , and in and then putting the three equalities together, we have That is, Due to the Hölder inequality and Gagliardo-Nirenberg inequality, where . Then (73) implies that By the Gronwall lemma, letting and and integrating on , by we get Now integrating (75) with respect to on , we find which along with Lemma 13 and Proposition 12 shows that there exists such that for all , which completes the proof.

We now establish uniform estimates on the tails of solutions when . We show that the tails of solutions are uniformly small for large space and time variables. These uniform estimates are crucial for proving the pullback asymptotic compactness of the cocycle .

Lemma 15. Suppose that (19) and (20) hold. Then for every , , and , there exist and such that for all and , where , depends on and , and depends on , , and .

Proof. Choose a smooth cut-off function satisfying for , for , and for . Suppose that there exists a constant such that for .
Taking the inner product of (25), (26), and (27) with , , and in , respectively, we get Add up the three equalities. Then we have That is,