Discrete Dynamics in Nature and Society

Volume 2012 (2012), Article ID 672762, 16 pages

doi:10.1155/2012/672762

Research Article

Global Attractors in for Nonclassical Diffusion Equations

Qiao-zhen Ma, Yong-feng Liu, and Fang-hong Zhang

College of Mathematics and Information Science, Northwest Normal University, Gansu, Lanzhou 730070, China

Received 14 May 2012; Accepted 22 October 2012

Academic Editor: Chuanxi Qian

Copyright © 2012 Qiao-zhen Ma et al. 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.

We study the existence of global attractors for nonclassical diffusion equations in . The nonlinearity satisfies the arbitrary order polynomial growth conditions.

1. Introduction

In this paper, we investigate the long-time behavior of the solutions for the following nonclassical diffusion equations:

with the initial data

where , and the nonlinearity satisfies

 () , and , () , and , and () ,

where , , , , , and are all positive constants. Moreover, without loss of generality, we also assume .

In 1980, Aifantis in [1–3] pointed out that the classical reaction-diffusion equation

does not contain each aspect of the reaction-diffusion problem, and it neglects viscidity, elasticity, and pressure of medium in the process of solid diffusion and so forth. Furthermore, Aifantis found out that the energy constitutional equation revealing the diffusion process is different along with the different property of the diffusion solid. For example, the energy constitutional equation is different, when conductive medium has pressure and viscoelasticity or not. He constructed the mathematical model by some concrete examples, which contains viscidity, elasticity, and pressure of medium, that is the following nonclassical diffusion equation:

This equation is a special form of the nonclassical diffusion equation used in fluid mechanics, solid mechanics, and heat conduction theory (see [1–4]). Recently, Aifantis presented a new model about this problem and scrutinized the concrete process of constructing model; the reader can refer to [5] for details.

The longtime behavior of (1.1) acting on a bounded domain has been extensively studied by several authors in [6–13] and references therein. In [12] the existence of a global attractor for the autonomous case has been shown provided that the nonlinearity is critical and . Furthermore, for the non-autonomous, the existence of a uniform attractor and exponential attractors has been scrutinized when the time-dependent forcing term only satisfies the translation bounded domain instead of translation compact, namely, . A similar problem was discussed in [13] by virtue of the standard method based on the so-called squeezing property. To our best knowledge, the dynamics of (1.1) acting on an unbounded domain has not been considered by predecessors.

As we know, if we want to prove the existence of global attractors, the key point is to obtain the compactness of the semigroup in some sense. For bounded domains, the compactness is obtained by a priori estimates and compactness of Sobolev embeddings. This method does not apply to unbounded domains since the embeddings are no longer compact. To overcome the difficulty of the noncompact embedding, in [14], using the idea of Ball [15], the author proved that the solutions are uniformly small for large space and time variables and then showed that the weak asymptotic compactness is equivalent to the strong asymptotic compactness in certain circumstances. In [16], the authors provided new a priori estimates for the existence of global attractors in unbounded domains and then applied this approach to a nonlinear reaction-diffusion equation with a nonlinearity having a polynomial growth for arbitrary order and with distribution derivatives in homogeneous term. More recently, in [17] the authors achieved the existence of global attractors for reaction-diffusion equations in , by using the methods presented in [18]. Our purpose in this paper is to study the existence of global attractors of (1.1) on the unbounded domains , and we borrow the idea of [17, 18]. Our main result is Theorem 4.6.

This paper is organized as follows. In Section 2, we recall some basic definitions and related theorems that will be used later. In Section 3, we prove the existence of weak solution for nonclassical diffusion equations in . The main result is stated and proved in Section 4.

2. Preliminaries

In this section, we iterate some notations and abstract results.

Definition 2.1 (see [18]). Let be a metric space, and let be bounded subsets of . The Kuratowski measure of noncompactness of defined by

Definition 2.2 (see [18]). Let be a Banach space, and let be a family of operators on . We say that is a continuous semigroup ( semigroup) (or norm-to-weak continuous semigroup) on , if satisfies

(i) (the identity),(ii),  for all ,(iii), if in (or , if in ).

Definition 2.3 (see [18]). A semigroup (or norm-to-weak continuous semigroup) in a complete metric space is called -limit compact if for every bounded subset of and for every , there is a , such that

Condition C (see [18]). For any bounded set of a Banach space , there exists a and a finite dimensional subspace of such that is bounded and

where is a bounded projector.

Lemma 2.4 (see [18]). Let be a Banach space, and let be a semigroup (or norm-to-weak continuous semigroup) in .

(1)If Condition C holds, the is -limit compact.(2)Let be a uniformly convex Banach space. Then is -limit compact if and only if Condition C holds.

Lemma 2.5 (see [18]). Let be a Banach space, and let be a semigroup (or norm-to-weak continuous semigroup) in .

(1)If Condition C holds, the is -limit compact;(2)Let be a uniformly convex Banach space. Then is -limit compact if and only if Condition C holds.

Theorem 2.6 (see [18]). Let be a Banach space. Then the semigroup (or norm-to-weak continuous semigroup) has a global attractor in if and only if

(1)there is a bounded absorbing set .(2) is -limit compact.

Lemma 2.7 (see [19]). Let be an absolutely continuous positive function on , which satisfies for some the differential inequality

for almost every , where and are functions on such that

for some and , and

for some . Then

for some and

Lemma 2.8 (see [20]). Let be Banach spaces, with reflexive. Suppose that is a sequence that is uniformly bounded in , and is uniformly bounded in , for some . Then there is a subsequence that converges strongly in .

3. Unique Weak Solution

Theorem 3.1. Assume , , and are satisfied. Then for any and , there is a unique solution u of (1.1)-(1.2) such that

Moreover, the solution continuously depends on the initial data.

Proof. We decompose our proof into three steps for clarity.

Step  1. For any , we consider the existence of the weak solution for the following problem in :

Choose a smooth function with

Since is a bounded domain, so the existence and uniqueness of solutions can be obtained by the standard Faedo-Galerkin methods; see [6, 8, 11, 16]; we have the unique weak solution

Step  2. According to Step 1, we denote ; then satisfies

For the mathematical setting of the problem, we denote , , , .

Multiplying (3.5) by in , using , and , we have

By the Poincaré inequality, for some , we conclude that

Hence, it follows that

We get the following estimate:

Similar to (3.9), using , , and , we get

where is independent of .

and yield

Choose such that ; then . Noting that , then , and we have the embedding . According to (3.12) and (3.13), we get

where is independent of .

Thanks to (3.14), is bounded in , and is bounded in .

For ,

where is independent of . We can obtain that is bounded in .

Since ,

Therefore, there exists , such that , , , and are continuous embedding to .

According to (3.5) and (3.14)–(3.16), we obtain

So has a subsequent (we also denote ) weak* convergence to in and ; has a subsequent (we also denote ) weak* convergence to . Let outside of ; we can extend to .

As introduced in [6, 20], is dense in the dual space of , , , and , so we can choose for all as a test function such that

Since for all , there exists bounded domain such that , . It follows that is uniformly bounded in , and . Since , according to Lemma 2.8, there is a subsequence (we also denote ) that converges strongly to in .

Using the continuity of and , we have

In line with (3.18) and (3.19), and let , we geting for all :

Thus, is the weak solution of (3.2) and satisfies

Step  3 (uniqueness and continuous dependence). Let , be in , and setting , we see that satisfies

Taking the inner product with of (3.22), using , , and , we obtain

By the Gronwall Lemma, we get

This is uniqueness and is continuous dependence on initial conditions.

Thanks to Theorem 3.1, and leting , is a semigroup.

4. Global Attractor in

Lemma 4.1. Assume , , and are satisfied. There is a positive constant such that for any bounded subset , there exists such that

Proof. Multiplying (1.1) by in , using , and , we have

By virtue of the Poincaré inequality, for some , there holds

Furthermore,

By the Gronwall Lemma, we get

We completed the proof.

According to Lemma 4.1, we know that

is a compact absorbing set of a semigroup of operators generalized by (1.1)-(1.2), , , and .

Lemma 4.2. Assume , , and hold. Then for any and , there are some and such that

whenever and .

Proof. Choose a smooth function with

where ,  , and there is a constant such that .

Multiplying (1.1) with and integrating on , we obtain

Next we deal with the right hand side of (4.9) one by one:

According to the condition and the bounded absorbing set in for , it follows that

where is independent of . For any given, let

Hence, combining (4.10) with (4.11), when , we conclude that

Using and , it yields

Since , there exist , such that

Then

From the assumption , provide , such that

Thus combining (4.9), (4.13), (4.16), and (4.17), we finally obtain

Furthermore, there holds

According to Lemma 2.7, we obtain

Thus, we get

provided and , we complete the proof.

Lemma 4.3. Assume , , and hold. There is a positive constant such that for any bounded subset , there exists such that

Proof. Multiplying (1.1) by in , we find

Using , , and , we have the following estimates:

Together with (4.6) and (4.19)–(4.21), by the Poincaré inequality, for some , this yields

By the Gronwall Lemma, we get

We complete the proof.

Remark 4.4. There is a constant , such that for any bounded subset , when , there holds

Lemma 4.5. Assume , , and are satisfied. Then the semigroup associated with the initial value problems (1.1) and (1.2) is -limit compact.

Proof. Denote , and we split as

where is a smooth function:

with , and there is a positive constant such that . Then

From Lemma 4.1, we know that as .

For any given, we can choose large enough; by Remark 4.4, we can assume

So we conclude that

For any bounded set , can be split as

Then in line with the property of noncompact measure, it follows that

On the other hand,

From Lemma 4.3, we get

Recall that

On account of Remark 4.4, it yields

Therefore, we have

That is, is -limit compact in .

Theorem 4.6. Assume , , and hold. Then the semigroup associated with the initial value problems (1.1) and (1.2) has a global attractor in .

Acknowledgments

The authors would like to thank the referee for careful reading of the paper and for his or her many vital comments and suggestions. This work was partly supported by the NSFC (11061030,11101334) and the NSF of Gansu Province (1107RJZA223), in part by the Fundamental Research Funds for the Gansu Universities.

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