ISRN Applied MathematicsVolume 2013 (2013), Article ID 204270, 14 pageshttp://dx.doi.org/10.1155/2013/204270

Research Article

Asymptotic Behavior for a Class of Nonclassical Parabolic Equations

Yanjun Zhang and Qiaozhen Ma

School of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China

Received 18 May 2013; Accepted 18 June 2013

Academic Editors: Y.-K. Chang, X. Xue, and K.-V. Yuen

Copyright © 2013 Yanjun Zhang and Qiaozhen Ma. 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.

This paper is devoted to the qualitative analysis of a class of nonclassical parabolic equations with critical nonlinearity, where and are two parameters. Firstly, we establish some uniform decay estimates for the solutions of the problem for , which are independent of the parameter . Secondly, some uniformly (with respect to ) asymptotic regularity about the solutions has been established for , which shows that the solutions are exponentially approaching a more regular, fixed subset uniformly (with respect to ). Finally, as an application of this regularity result, a family of finite dimensional exponential attractors has been constructed. Moreover, to characterize the relation with the reaction diffusion equation (), the upper semicontinuity, at , of the global attractors has been proved.

1. Introduction

We study the long-time behavior of the following class of nonclassical parabolic equations: where is a bounded domain with smooth boundary , and are two parameters, the external force is time independent, and the nonlinearity satisfies some specified conditions later.

When for the fixed constant , equation is a usual reaction-diffusion equation, and its asymptotic behavior has been studied extensively in terms of attractors by many authors; see [1–5].

For each fixed , equation is a nonclassical reaction-diffusion equation, which arises as models to describe physical phenomena such as non-Newtonian flow, soil mechanics, heat conduction; see [6–8] and references therein. Aifantis in [6] provided a quite, general approach for obtaining these equations. The asymptotic behavior of the solutions for this equation has been studied by many authors; see [9–16].

For the fixed constant , any , and the long-time behavior of the solutions of has been considered by some researchers; see [10, 13]. In [10] the author proved the existence of a class of attractors in with initial data and the upper semicontinuity of attractors in under subcritical assumptions and in the case of . In [13] similar results have been shown when and .

In this paper, inspired by the ideas in [17, 18] and motivated by the dynamical results in [19–22], we study the uniform (with respect to the parameter ) qualitative analysis (a priori estimates) for the solutions of the nonclassical parabolic equations and then give some information about the relation between the solutions of and those of . Our main difficulty comes from the critical nonlinearity and the uniformness with respect to .

This paper is organized as follows. In Section 2, we introduce basic notations and state our main results. In Section 3, we recall some abstract results that we will use later. In Section 4, we present several dissipative estimates about the solution of when , which hold uniformly with respect to . The main results are proved for in Section 5. Moreover, in Section 6, as an application, we construct a finite dimensional exponential attractor and prove the upper semicontinuity of the global attractor obtained in Section 5.

2. Main Results

Before presenting our main results, we first state the basic mathematical assumptions for considering the long-time behaviors of the nonclassical parabolic equations and then introduce some notations that we will use throughout this paper.(i) with and satisfies the following conditions: where is a positive constant and is the first eigenvalue of on . The number is called the critical exponent. is not compact in this case, and this is one of the essential difficulties in studying the asymptotic regularity.(ii)Assumption on the parameters and . From the work in [18, 19], we know that a very large damping has the effect of freezing the system, if the damping acts only on the velocity , and this prevents the squeezing of the component . Therefore, the most dissipative situation occurs in between, that is, for a certain damping , which depends on the other coefficient of the equation. Therefore, in our frame, we choose such that as in order to obtain the uniformly (with respect to ) asymptotic regularity about the solutions of .(iii) with domain , and consider the family of Hilbert space , with the standard inner products and norms, respectively, In particular, and mean the inner product and norm, respectively.(iv), with the usual norm In particular, we denote and .(v)For each , we define () as and define as Then is a Banach Space for every .

The global well-posedness of solutions and its asymptotic behavior for have been studied extensively under assumptions (1)-(2) by many authors in [9–14] and references therein in fact note that for each fixed .

The main results of this paper are the following asymptotic regularity.

Theorem 1. Under assumptions (1), (2), and , there exist a positive constant , a bounded (in ) subset , and a continuous increasing function such that, for any bounded (in  ) subset , where , and are all independent of , and is the semigroup generated by in .

This result says that asymptotically, for each , the solutions are exponentially approaching a more regular fixed subset uniformly (with respect to ) for . Moreover, it implies the following results.(1)For each , has a global attractor in , and (2)Based on Theorem 1, applying the abstract result devised in [23, 24], for each we can prove the existence of a finite dimensional exponential attractor in . Moreover, our attraction is uniform (with respect to ) under the -norm (not only with the -norm); see Lemma 19.(3)Since the global attractor , it also implies that the fractal dimension of the global attractor is finite. Moreover, in line with Theorem 1, we prove the upper semicontinuity of at ; see Lemma 20.

For the proof of Theorem 1, the main difficulty comes from the critical nonlinearity and the uniformness with respect to .

Hereafter, we will also use the following notation: denote by the space of continuous increasing functions and by the space of continuous decreasing functions such that . Moreover, , , and are the generic constants, and , are generic functions, which are all independent of ; otherwise we will point out clearly.

3. Preliminaries

In this section, we recall some results used in the main part of the paper.

The first result comes from [17], which will be used to prove the asymptotic regularity for the case .

Lemma 2 (see [17]). Let and be two Banach spaces and a -semigroup on with a bounded absorbing set . For every , assume that there exist two solution operators on and on satisfying the following properties. (i)For any two vectors and satisfying , (ii)There exists such that (iii)There are and such that Then, there exist positive constants , , and such that where .

Next, we recall a criterion for the upper semicontinuity of attractors.

Lemma 3 (see [25, 26]). Let be a family of semigroups defined on the Banach space , and for each , let have a global attractor . Assume further that is a nonisolated point of and that there exist , , and a compact set such that Then the global attractors are upper semicontinuous on at ; that is,

Lemma 4 (see [27]). 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

For the proof, we refer the reader to [27, Lemma].

A standard Gronwall-type lemma will also be needed.

Lemma 5. Let be an absolutely continuous positive function on , which satisfies for some the differential inequality for some and some . Then,

4. Uniformly Decaying Estimates in

In this section, we always assume that (1), (2), and such that as hold and only belongs to , so all results in this section certainly hold for the case .

The main purpose of this section is to deduce some dissipative estimates about the semigroups associated with in . Here, using the method in [19, 20, 22] for a strongly damped wave equation and a semilinear second order evolution equation, we will show that the radius of the absorbing set of associated with in can be chosen to be independent of .

Lemma 6. There exists a positive constant , which depends only on , , and coefficients of (1)-(2), satisfying that for any and any bounded (in ) subset , there is a (which depends only on the bound of ) such that where both and are independent of .

Proof. Throughout the proof, the generic constants are independent of . For clarity, we separate the proof into three claims.Claim  1. There exists an which depends on , , (but independent of and ) such that where depends on but not on .Multiplying by , we have By virtue of (2), we conclude that there exists , such that At the same time, by the Hölder inequality, we get Substituting (26) and (27) into (25) and noticing as , we obtain where and is a small positive constant such that .And then applying Lemma 4 to above inequality, it follows that where , .Then, Claim 1 follows from (29) immediately.Claim  2. There exists an which depends on , ,  and (but is independent of and ) such that where is given in Claim 1.Noting (25) and taking in (26), it yields Then, for any , integrating (31) over and using Claim  1, we can complete this claim immediately. Claim  3. Multiplying by , we find furthermore, Then, from assumptions (1)-(2), Claim 1, and using Hölder inequality, there holds On the other hand, from Claim 2 we know that for each there is a time such that where depends on .When , for each , integrating (33) over and applying (34)–(36), we obtain that Now, taking (is independent of ), we can complete our proof.

Remark 7. Observing that above process of proof, we can also deduce that, for any and any , where is independent of and .Moreover, if is bounded in , then we can obtain for some constant which depends on ,  . Indeed, from the fact that there is a constant such that for any , (39) can be obtained just by repeating the proof of Lemma 6 and taking in (35) since is bounded in .On the other hand, from the proof of Claim 3 as follows, we can get further estimates about

Lemma 8. There exists a positive constant such that for any and any bounded (in ) subset , where , , is the time given in Claim  1, and only depends on but is independent of and .

Proof. By differentiation of , we can obtain the following equation: Multiplying (42) by , we have When , using Lemma 6, there holds So, we obtain Therefore, as , for , integrating (45) over and substituting (40), we can complete our proof at once.

For later applications, we present some Hölder continuity of in .

Lemma 9. For any bounded subset , there exists a constant which depends only on and such that

Proof. Let   and   be the solutions of corresponding to the initial data   and  . Then the difference satisfies with initial data .For (46), multiplying (48) by , we have where we used (38). Then, when applying Gronwall lemma, we can obtain (46).For (47), when , multiplying (48) by and combining with Lemma 8, we have Hence, by (47) we complete the proof.

Hereafter, we denote the uniformly (with respect to ) bounded absorbing set obtained in Lemma 6 as , that is, and denote the time by such that Lemmas 6 and 8 hold for ; that is, holds for any and all . Moreover, similar to Remark 7, noting now that is bounded in , we have

5. Proof of the Main Results

Throughout this section, we always assume that (1), (2), and hold for .

5.1. Decomposition of the Equation

For the nonlinear function satisfying (1)-(2), from [12, 17, 19, 22] for our situation we know that allows the following decomposition , where and satisfy

Now, decomposing the solution into the sum for any and any , where and are the solutions of the following equations:

Applying the general results in [9, 12, 14], we know that both (59) and (60) are global well-posed in , and also forms a semigroup.

Moreover, as in Section 4, we can deduce a similar estimate for in , and so . There exist constants ( is given in Lemma 6) and such that for any and any ,

5.2. The First A Priori Estimate

We begin with the decay estimates for the solution of (59).

Lemma 10. There exists a constant and such that where both and are independent of .

Proof. Multiplying (59) by , we have By means of (55), it follows that .Therefore, there exists such that for all and any .As a result, we multiply (59) by and obtain Then integrating with (55), (61), (62), and (65), we conclude Thus, using the following Lemma 11 with (67), allows us to complete our proof by taking and some increasing function .

Lemma 11. Let be a continuous semigroup on the Banach space , satisfying Then

Its proof is obvious and we omit it here.

The next estimate is about the solution of (60).

Lemma 12. For every (given) and any , there is a positive constant which only depends on ,  and   such that the solutions of (60) satisfy where both are independent of , and .

Proof. Multiplying (60) by and integrating over , Then the proof is completely similar to that in [12, Lemma], so, we omit it.

Based on Lemmas 10 and 12, following the idea in Zelik [21], we can now decompose as follows.

Lemma 13. Let be the solution of corresponding to the initial data . Then, for any , we can decompose as where and satisfy the following estimates: with the constants and depending on , and , but both independent of .

Proof. The proof is completely similar to that of [12, Lemma ] and [22, Lemma], since the estimates in Lemmas 10 and 12 hold uniformly with respect to .

Note that in the above decomposition in Lemma 13, we can require further that satisfies the following: there is a constant which depends only on , such that

5.3. The Second A Priori Estimate

The main purpose of this subsection is to deduce some uniformly asymptotic (with respect to and ) the a priori estimates about the solution of .

Lemma 14. There exists positive constants , , and such that for each , there is a subset satisfying and the exponential attraction where all , and are independent of , and denotes the Hausdorff semidistance with respect to the -norm.

Proof. It is convenient to separate our proof into three steps. We emphasize, especially, that all the generic constants in the proof are independent of .Step  1. We first claim that (recall ): , and such that for each , there is a subset satisfying and the exponential attraction We will apply Lemma 2 with and (note that for any ). From (54), we can write For any and , satisfying , we decompose the solution of as , where which uniquely solves the following equations, respectively: with and , and is the solution of (59) corresponding to the initial data .For (80), from (54), (56), (78), and Lemmas 10 and 12, we can directly calculate that where , is given in Lemma 10.Multiplying by (80), we have Furthermore, using the similar estimates of Lemma 6, we get where is a small positive constant such that for all .And then applying Lemma 5 to above inequality, there holds For (81), since then Using Hölder inequality we get where we used (53), (62), and Lemmas 10 and 12.Hence, multiplying by (83), we have Furthermore, we have where is a small positive constant given in (84).Then, using Lemma 5 we obtain Therefore, combining (85) and (91), we can verify that all the conditions of Lemma 2 are satisfied for the cases , , and . Moreover, since there is a (independent of ) such that for any and the constants in our estimates are all independent of ; consequently, , , and are all independent of , and then we can deduce our claim.Step  2. We claim that there exists a constant which depends only on such that Multiplying by , we only need to note the following: First, since , we have and then while where we used (73).Moreover, since , we have and then where is given in Lemma 13.Hence, substituting the above estimates into (93), applying the Poincaré inequality we have Then using the Gronwall inequality and integrating over (from Lemma 12), we obtain Taking (in Lemma 13) small enough such that , we have Thus, Substituting above (100) and (102) into (99), we get that for all Step  3. Based on Step 1 and Step 2, applying the attraction transitivity lemma given in [28, Theorem] and noticing the Holder continuity Lemma 9, we can prove our lemma by performing a standard bootstrap argument, whose proof is now simple since Step 1 makes the nonlinear term become subcritical to some extent.

5.4. Proof of Theorem 1

Lemma 14 has shown some asymptotic regularities; however, the radius of depends on and the distances only under the -norm.

To prove Theorem 1, we first give two lemmas as preliminary.

Lemma 15. There exsits a constant such that for any bounded (in ) subset , there exsits such that

Proof. Multiplying by , we find Noting , from Lemma 6, yields hence, we obtain where is a small, positive constant.Similarly, with using Lemma 4 we finally complete the proof.

Lemma 16. There exists a constant such that for any bounded (in ) subset , there is a such that

Proof. From Lemma 15, we only need to estimate that the bound of is independent of .Applying Lemma 15 again, we have Taking which may provide that and , integrating (109) on , and from Lemma 15, when we yield Hence, multiplying by , we can complete our proof by applying the uniform Gronwall lemma.

Now, we are ready to prove Theorem 1.

Proof of Theorem 1. Set where the constant comes from Lemma 16.From Lemmas 16 and 14, we know that there is a such that (recall that is given in (78)) for all and any .On the other hand, note that such that Then, from Lemma 9, there exists which depends only on and (so only on , ) such that Therefore, from Lemma 14, we have Hence, noting that , and are all fixed, we can complete the proof by taking and applying Lemma 11.

6. Applications of Theorem 1

As for the applications of Theorem 1, in this subsection, we consider the existence of finite dimensional exponential attractors and the upper semicontinuity of global attractors for problem under assumptions (1), (2), and .

6.1. A Priori Estimates

For the subset defined in (113), and from Lemmas 6 and 8 we know that there is a such that where .

Now, for each , define as follows: where is the time given in Lemma 16 corresponding to . Then, for each we have as a positive invariant under (i.e., , for all ) (from Lemma 16)

Moreover, we have the following results.

Lemma 17. Under assumptions (1), (2), and , there exists a constant such that for every , the semigroup satisfies the following properties: admits a decomposition of the form where   and   satisfy the estimates where the constants are independent of and

Proof. For any two initial data with solution (), we decompose the difference as follows: where solves Next, for clarity, we decompose the remainder proof into two steps.Step  1. For , multiplying (123) by , we have where is a small positive constant such that .Using Lemma 4 we can deduce that Hence, by taking large enough, we get Step  2. For , multiplying (124) by (where is given in (121)), we obtain Case  1 ( ). Then by using the embedding for any , we have where we have used (118) and (46).Case  2 ( ). Since and embedding , we also have Case  3 ( ). Noticing that embedding and , we have therefore, for any , we have Hence, taking then, from (127) and (132), we can see that , ,  and   satisfy Lemma 17.

Lemma 18. Under assumptions (1), (2), and , for an arbitrary fixed time and any , the semigroup is Lipchitz continuous on in the following sense: there exists a positive constant such that for any , , ,

Proof. For and we have The second term of above inequality is handled by estimate (46). Concerning the first one Then from (116) and (117) we can deduce So, the proof is completed immediately.

6.2. Exponential Attractors

We are now ready to prove the following result about the existence of exponential attractors.

Lemma 19. Under assumptions (1), (2), and , for every , there exists a compact subset , uniformly bounded in , which satisfies the following conditions: (i) is semi-invariant with respect to the semigroup , that is, (ii)the fractal dimension of is finite, that is, (iii)for each enjoys a uniform exponential attraction property of the following form: for any bounded subset ,

Here, and may depend on , but is independent of .

Proof. For each , we know that is invariant and compact in . Hence, applying the abstract results established in [23, 24], from Lemmas 17 and 18 we can first construct an exponential attractor on with respect to the -norm. Then, we can complete the proof by using the attraction transitivity lemma given in [28,Theorem] from Lemma 14 and the Hölder continuity (47).

6.3. Upper Semicontinuity of Global Attractors

Since , (ii) of Lemma 19 implies that the fractal dimension of the global attractor is finite too. Moreover, we have the following upper semicontinuity result of at .

Lemma 20. Under assumptions (1), (2), and , the global attractors are upper semicontinuous at .

Proof. Since the global attractor is strictly invariant, that is, for all , it is obvious to see that Therefore, to apply Lemma 3, we can take and we only need to verify condition (14). Let and with ; also let with . Denote . Then solves the following equation: Multiplying (143) by , we have where we used Lemma 19 and (118) and noticed the process of Step  2 of Lemma 17 for . On the other hand, for (145), using Poincaré inequality, we have So, we obtain Noticing , using (24), Lemma 15, and Gronwall inequality, yields Hence, we know that there exists such that which implies Therefore, from (142) and (150), we can directly apply Lemma 3 to complete the proof.

Acknowledgments

The authors would to essentially thank the referee for his/her comments and suggestions, which have improved the original version of this paper. This work was partly supported by the NSFC (11101334), the NSF of Gansu Province (1107RJZA223) and the Fundamental Research Funds for The Gansu Universities.

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