On Global Attractors for a Class of Reaction-Diffusion Equations on Unbounded Domains with Some Strongly Nonlinear Weighted Term

Zhang, Jin; Zhong, Chengkui

doi:https://doi.org/10.1155/2015/251614

Discrete Dynamics in Nature and Society

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Research Article | Open Access

Volume 2015 | Article ID 251614 | https://doi.org/10.1155/2015/251614

On Global Attractors for a Class of Reaction-Diffusion Equations on Unbounded Domains with Some Strongly Nonlinear Weighted Term

Jin Zhang¹and Chengkui Zhong²

Academic Editor: Seenith Sivasundaram

Received04 Nov 2014

Revised06 Apr 2015

Accepted03 Jun 2015

Published14 Jun 2015

Abstract

We consider the existence and properties of the global attractor for a class of reaction-diffusion equation , in ; , in . Under some suitable assumptions, we first prove that the problem has a global attractor in . Then, by using the -index theory, we verify that is an infinite dimensional set and it contains infinite distinct pairs of equilibrium points.

1. Introduction

In this paper, we are mainly concerned with the long-time behaviour of solutions for the following reaction-diffusion equation:where and satisfies(e.g., with ). The function is continuous and satisfies the following assumptions: is odd; that is, , for all ; there exists a constant , , such that there exists a positive constant such that ; there exists a positive constant such that , where

As we know, the basic problems to consider the long-time behaviour of solutions for the above equation are to prove the existence of global attractors for the semigroup of solutions and discuss some properties of the global attractors, such as the dimension property and the existence of multiple equilibrium points.

Those problems for the equations in bounded domains have been studied extensively by many authors and have been rather well understood; see, for example, [1–7]. However, the solution for the equation is different in unbounded domain. The main difference is the fact that, in contrast to the case of bounded domains, the global attractors for the reaction-diffusion equations in unbounded domains admit finite dimension under some specific assumptions and infinite dimension under general assumptions.

For the kind of equation in unbounded domains. In pioneering work [8], the authors used weighted spaces instead of the usual spaces to prove the existence of the global attractors; further details can be found in [9–11]. In [12–15], the authors have developed some new ideas and methods to deal with more general cases in unbounded domains, including uniformly local Sobolev spaces, locally compact attractor, and the so-called entropy theory, and have obtained the existence of the locally compact global attractors for the semigroups associated with the equations. Under some structural assumptions on the term (i.e., or ), the authors in [16, 17] prove the existence of global attractor for the equation in unbounded domain in usual space .

On the other hand, we have noticed that the -index is a powerful method to find multiple critical points of some even functional. The authors in [18] used the -index to obtain the existence of infinite dimensional global attractor for a class of -Laplacian equation in bounded domain, for which is necessary. Additional information about other attractor problems can be found in [19–23].

Motivated by the above papers, in this paper, we are interested in finding a semigroup associated with a reaction-diffusion equation in unbounded domain, such that the semigroup has a global attractor in the usual space; furthermore the dimension of the global attractor is infinite.

The main results of this paper can be stated as follows.

Theorem 1. Assuming that , , and satisfies condition (2) and the nonlinear term satisfies , then reaction-diffusion equation (1) has a global attractor in .

Theorem 2. Assume that , , and satisfies condition (2) and the nonlinear term satisfies . Let be the global attractor of (1). Then, for any , there exists a neighborhood of the origin, such that , where denotes the -index of the set .

We recall that, from [24], any compact set , with fractal dimension , can be mapped into spaces by a linear odd Hölder continuous one-to-one projector. Thus, we obtained the following corollary.

Corollary 3. Under the assumptions of Theorem 2, the fractal dimension of the global attractor is infinite.

Theorem 4. Assume that , , and satisfies condition (2) and the nonlinear term satisfies . Let be the global attractor of (1). Then, contains infinite distinct pairs of equilibrium points.

The proofs of the above theorems are, respectively, given in Sections 3 and 4. Some preliminaries and associate lemmas can be found in Section 2.

2. Some Preliminaries

Initially, backgrounds about global attractors and -index theory are reviewed. Proofs are then given for the lemmas and the existence of solution for (1).

In this paper, we define the following space: with the corresponding norm where

Let be a Banach space, and define closed, as the class of closed symmetric subsets of . Based on this, the formal definition of -index can be given.

Definition 5 (see [25]). Let , . The -index or Krasnoselskii genus of is defined by In particular, if , , then define .

The properties of -index are provided in the following lemma.

Lemma 6 (see [25]). Let be an odd map and . Then the -index on satisfies the following properties: , ; ; ; , for all , and is odd and continuous; If is compact and , then and there exists a neighborhood of in such that and ; For any bounded symmetric neighborhood of the origin in there holds .

Applying the index theory to an even functional on some Banach space , we can obtain a sequence of minimax values. Moreover, if satisfies the condition, the sequence of minimax values must be the critical values of the functional .

Definition 7 (see [25]). Let be a Banach space, , and . The functional is said to satisfy the condition if any sequence such that has a convergent subsequence.

Lemma 8 (see [26]). Suppose is a Banach space and suppose , for all . satisfies the following conditions: there exists a subspace with and , such that where ; there exists a closed subspace with , such that satisfies the condition.Then if , the functional possesses at least pairs of critical points.

Following the proof in [8], we will prove the existence of a unique weak solution of (1) for any initial data .

Firstly, we consider the problem in the bounded domain. We denote and the function with , , satisfying

It is well known that (see, e.g., [5, 6])has a unique solution , where . And for every it satisfies

The following lemmas give some estimates for solution of the bounded problem (14).

Lemma 9. Let , , satisfy condition (2), and satisfies ; let be a solution of problem (14). Then, for any , the following estimates hold:where the constants , depend on data , , and but are independent of .

Proof. Firstly, for any , utilizing Hölder inequality, we haveThen utilizing Young’s inequality, we havewhere Multiplying (14) by and integrating over , we haveand it follows from (19) and that Integrating between and yields It follows that which implies second estimate (17).
On the other hand, it follows from (21) and that Referring to Gronwall’s inequality, first estimate (16) can be easily obtained.

Lemma 10. Let , , and satisfy condition (2), and satisfies ; let be a solution of problem (14). Then, for any , the following estimate holds:where and the constant depends on data , , and but independent of .

Proof. For any , we haveBy we find that so we get Applying the Hölder inequality to each term, it follows that Substituting into inequality (27), there exists a constant , such that Then, referring to Lemma 9, the estimate yields the conclusion.

It is worth noting that both estimates in Lemmas 9 and 10 are independent of , so we let , providing the existence and uniqueness of the solution of problem (1). Before giving the proof of the existence theorem, we first state the following two lemmas.

Lemma 11 (see [5]). Let be Banach spaces, with V reflexive. Suppose that is a sequence uniformly bounded in and is uniformly bounded in , for . Then there is a subsequence that converges strongly in .

Lemma 12 (see [27]). Let , , and be the standard scalar product in . Then, there exists a constant such that

Theorem 13. Let , , satisfy condition (2), and satisfies , then for any and , there exists a unique weak solution of (1) which satisfies Furthermore, is continuous on .

Proof. We choose such that as . Denote that are the solutions of boundary problem (14) in the domain .
Now, we extend the functions from into . For each , define the function as zero for and multiply by , where is defined by (13). For simplicity, we denote by the extended functions and . Since we obtain that Lemmas 9 and 10 are still valid. It follows that Hence, taking a subsequence of if necessary there exists such that Similarly to the proof in [5, 6], we can obtain In addition, referring to Lemma 11, we have Therefore, for any , Thus, is the weak solution of (1).
In the following, we will prove uniqueness of solution and the continuous dependence. Let , be any two solutions of (1) with initial data , ; setting , we have with initial data . Multiplying by and integrating on , we obtain and it follows from Lemma 12 that and by condition we find that Then note that ; we have and integrating this giveswhich implies uniqueness if and the continuous dependence on initial data.

The following theorem shows the existence of global attractors when an absorbing set exists.

Lemma 14 (see [5, 6]). If a continuous semigroup has a compact absorbing set , then there exists a global attractor , where is the -limit set of the set .

3. The Existence of a Global Attractor

In this section, we will prove Theorem 1. Before the proof, we first give the following lemma.

Lemma 15. Assuming that , , and satisfies assumption (2), then is compactly embedded in .

Proof. Assume that is a bounded sequence in . Then there exists a constant , such that so it has a subsequence satisfying For arbitrary , choose the constant sufficiently large, such thatNote that in and due to the boundedness of the domain , the Sobolev embedding theorem can be used, yielding Then there exists sufficiently large such that, for all , we haveand it follows that Utilizing the Hölder inequality and inequality (49), we have which implies This completes the proof of Lemma 15.

Proof of Theorem 1. In order to prove that (1) has a global attractor, referring to Lemma 14, it is sufficient to show the existence of a compact absorbing set in .
Let be the solution of (1); multiplying the first equation of (1) by and integrating on , it follows thatSince and , we have Similar to estimate (19), we haveand thus Note that yielded and then Gronwall’s inequality can be applied, yieldingwhere . Now, combining estimates (55) and (57), we haveIntegrating between and , it follows that When , it follows from (60) thatNow, multiplying the first equation of (1) by and integrating on , we obtainso it follows from (61) and (64) that Integrating between and (), it holds thatThen integrating the equation with respect to between and again, we have when , so it follows from (63) and that which implies that there exists a constant , such that Finally, referring to Lemma 15, is compactly embedded in , and we obtain a compact absorbing set in which concludes the proof of Theorem 1.

4. The Dimension of the Global Attractor and the Equilibrium Points

Next we will estimate the -index of the global attractor obtained in Theorem 1. Before the formal proof of Theorem 2, we first consider the energy function

It is well known that functional (70) has an infinite dimensional negative subspace of ; that is, there exists linearly independent nonzero functions , satisfying

Let be a subspace of with , where are orthogonal in both and .

Now, we give the proof of Theorem 2.

Proof of Theorem 2. For arbitrary , we first prove that there exists a set with and a neighborhood of the origin, such that It follows from (64) that that is, for any , the function is nonincreasing. For arbitrary and , we have Denoting , then is compact in ; thus, there exists , such that for all Referring to Lemma 6 , for every constant , we havewhere . Thus, for , it follows that Recalling and condition , when is sufficiently small, we have In addition, since and is nonincreasing, then . Since is closed and compact, there exists open neighborhood of 0, such that Let ; we have completed the proof of the first step.
Next, we only need to prove . Referring to Lemma 6 , there exists , which is a neighborhood of , satisfying In addition, referring to the definition of -limit set , there exists such that Thus, It is obvious that is odd since . Then, referring to Lemma 6 , we have , and then . The proof is complete.

At last, we want to investigate existence of the multiple equilibrium points of the equation, that is, solutions of the following elliptic equation:

We consider the critical values of the energy functional defined by (70). In order to obtain infinite critical values by Lemma 8, we verify that the functional is bounded from below and satisfies the condition.

Lemma 16. The functional defined by (70) is bounded from below.

Proof. The functional isBy estimate (57), it is easy to verify that the functional is bounded from below.

Lemma 17. Let be a functional defined by (70) and be a constant, then any sequence such that contains a convergent subsequence.

Proof. Since , we obtain Similarly to estimate (19), we have and combining the above estimates and , we have It follows that is bounded.
Going if necessary to a subsequence, we assume that By Lemma 15, we have in . Observe that Since and in , we can obtain that Thus we have proved that , .

Now, we prove that the global attractor contains infinite distinct pairs of equilibrium points.

Proof of Theorem 4. By the proof of Theorem 2, we obtain that, for arbitrary , there exists a subspace with and , such that The above two lemmas show that also satisfies conditions and of Lemma 8; thus we obtain infinite pairs of critical points, which implies the conclusion.

Remark 18. In this paper, we suppose that the nonlinear term is continuous. If is a weak continuous function in space or is a Carathéodory mapping of in , all conclusions in this paper are still valid.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to express their sincere thanks to the anonymous referees for their valuable comments and suggestions which led to an important improvement of their original paper. This work was partly supported by NSFC Grant (no. 11031003).

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Copyright © 2015 Jin Zhang and Chengkui Zhong. 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.

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