Table of Contents
ISRN Mathematical Analysis
Volume 2013, Article ID 989475, 10 pages
http://dx.doi.org/10.1155/2013/989475
Research Article

Asymptotic Smoothing and Global Attractors for a Class of Nonlinear Evolution Equations

1School of Mathematics and Computing Science, Changsha University of Science and Technology, Changsha 410004, China
2Hunan Provincial Center for Disease Control and Prevention, Changsha, Hunan 410005, China

Received 31 March 2013; Accepted 24 April 2013

Academic Editors: S. Liu and W. Shen

Copyright © 2013 Yongqin Xie 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 prove the asymptotic regularity of global solutions for a class of semilinear evolution equations in . Moreover, we study the long-time behavior of the solutions. It is proved that, under the natural assumptions, these equations possess the compact attractor which is bounded in , where the nonlinear term satisfies a critical exponential growth condition.

1. Introduction

In this paper, we study the asymptotic regularity and the long-time behaviors of the solutions for the following semilinear evolution equations: where is an open-bounded set of with smooth boundary , , and satisfies

Equation (1), which appears as a class of nonlinear evolution equations as , is used to represent the propagation problems of lengthways wave in nonlinear elastic rods and ion-sonic of space transformation by weak nonlinear effect (see, for instance, [14]): and as (named the Karman equation) is used to represent the flow of condensability airs in the across velocity of sound district (see, for instance, [5]) In [6], the authors have discussed the existence of global solutions in under the assumptions that the initial values are sufficiently small. In [7, 8], the authors have discussed the nonexistence of global week solutions for the following system: where , , and are nonlinear operators. As , in [9], the authors have discussed the long-time behaviors of solutions of (1) in ; specifically in [10], the authors have discussed the long-time behaviors of solutions of (1) in . However, an open question remains whether the global attractor regularizes in the critical case and as , the long-time behaviors of solutions of (1) have not been considered completely up to now. In this paper, we try to discuss the problem.

In the study of the global attractor regularization, the critical nonlinearity exponent brings a difficulty. About the regularity of attractor for the strongly damped wave equations, for the subcritical case, the authors in [11] have proved that the global attractor is bounded in . For the critical case, Pata and Zelik [12] have proved that the global attractor is bounded in when the nonlinearity satisfies , for all , and the authors also pointed out further that one can prove the regularity of the attractor when only satisfies the natural assumptions (which have been realized recently in [13, 14]). A general way is to obtain higher regularity of the solutions than their initial values (see, for instance, [11, 12]). Then we can get the global attractor regularization. However, since (1) contains terms and , they are essentially different from usual strongly damped wave equations and it is very difficult for us to obtain better regularity of the solutions of (1). Therefore, we must propose a new and more general way to study the smoothing of global attractors for (1).

In this paper, we will apply the techniques introduced in Zelik [15], Sun and Yang [13], and Yang and Sun [14] to overcome the difficulty due to the critical nonlinearity and establish the asymptotic regularity of solutions. Based on this regularity result, we obtain asymptotically compactness of the semigroup in , and the existence of the compact attractor has been proved. Moreover the compact attractor is bounded in .

2. Functional Setting

In what follows, we give some notations which will be used throughout this paper. Let be a bounded subset of with a sufficiently smooth boundary , and corresponding norms, , and , respectively; the norms in are denoted by . Let be the (strictly) positive operator on defined by

For the family of Hilbert spaces , , their inner products and norms are, respectively, Then we have the continuous embedding

The product Hilbert spaces endowed with the usual inner products and norms

Denote by any positive constant, which may be different from line to line and even in the same line.

Throughout the paper we assume that the function satisfies the following conditions: Also, let admit the decomposition , where , satisfying where is the first eigenvalue of in with the Dirichlet boundary condition. Without loss of generality, we can think is large enough, say, . Notice that by (13) and (15), there exists , such that

We denote by the function which is easily seen to satisfy the inequalities

We will complete our task exploiting the transitivity property of exponential attraction [16, Theorem 5.1], which we recall below for the readers’ convenience.

Lemma 1 (see [16]). Let , , and be subsets of such that for some and . Assume also that for all there holds for some and some . Then it follows that where and .

3. Global Solution in

Exploiting the dissipative conditions (16) and (18) and the standard energy estimates technique, it is easy to obtain the following result.

Lemma 2. Assume that for some . Then one has the following estimates: (i); (ii), for any ; here and are constants, which are dependent of .

Proof. Multiplying (1) by , integrating in over , and then integrating in on , we have
Multiplying (1) by and integrating by parts, we get
By Holder’s and Young’s inequality associated with the Sobolev embedding theorem, we find that (25) implies that where denotes the measure of .
Multiplying (1) by and integrating by parts, we get Combining with (11), we are led to the following estimation:
Applying (i), estimation (ii) follows by denoting .
Based on the estimations above, the existence and uniqueness of the global weak solutions for (1) with the initial conditions can be obtained by standard Faedo-Galerkin method, which we omit here (see examples in Evans [17]).

Theorem 3. Let be a bounded domain. The assumptions (11)–(19) hold. . . Then (1) have such weak solutions : for all   for all , Furthermore,

The weak solutions of (1) are unique and continuously dependent on initial conditions.

Lemma 4. Let be a bounded domain. The assumptions (11)–(19) hold. Let , , be two initial values, , and denote by the two corresponding solutions to problem (1) in time interval . Then the following estimate holds, for all , the constant is only dependent on but independent of  , , and .

From Theorem 3 and Lemma 4, the initial boundary value problem (1) is equivalent to a continuous semigroup defined by

4. Bounded Absorbing Set

We now deal with the dissipative feature of the semigroup . Namely, we show that the trajectories originating from any given bounded set eventually fall, uniformly in time, into a bounded absorbing set .

Theorem 5. has a bounded absorbing set in ; that is, for any bounded subset , there exists such that

Proof. We set and rewrite (1) as follows: and we take the inner product of (35) with and set small enough and then we obtain
Young’s inequality gives that there exist positive constants , , and only dependent on , such that here as small enough and .
By Lemma 2, this shows that
Then we get for any and .
Hence, there exists such that
According to Lemma 2, it follows that if , then there exists , such that which completes the proof of the desired results.

Corollary 6. For any , there exists , such that provided that , where is a positive constant only dependent on (given by (16) and (18)) and (given by (42)).

Proof. Multiplying (1) by , respectively, integrating in over , and then integrating in on and associated with Theorem 5 and Sobolev embedding , we can get the conclusions above.

Hereafter, we always assume that for some , is the bounded absorbing set of in obtained in Theorem 5.

5. Asymptotic Regularity of the Solutions

In this section, we will establish some a priori estimates about the solutions of (1), which are the basis of our analysis. Let be a unique weak solution of (1) corresponding to the initial data . We decompose into the sum where and are the solutions to the problems

It is convenient to denote

Lemma 7. For any , there exist and , such that whenever , it follows that the constant is independent of and .

Proof. Denoting and repeating word by word Theorem 5, that applies to the present case with in place of (with the further simplification that for now ).
Set Let be small enough and ; then Combining with (13) we get the differential inequality where and is the Sobolev embedded constant .
Multiplying (46) by , integrating in over , and then integrating in on , So we get where .
By (50), and by the above here . Then there exists a constant such that
Combining (51), (57), and Young’s inequality, we are led to the differential inequality By Gronwall Lemma, we get
Due to Lemma 2 and (46), then ; for any , there holds
From (57) we get where . Putting together (51), the proof is finished.

Lemma 8. For any , there exist and , such that whenever and for any time and every , the solutions of (47) satisfy the following estimates:

Proof. Due to Lemma 2, Theorem 5, and Corollary 6, there exists an increasing function in , such that
Choosing and multiplying (47) by , we are led to the identity where
Combining with and (2), we have
By virtue of (11), we get
Since , by (14) we deduce that
Finally,
Plugging (67)–(70) into (65), we obtain where is an increasing function in and the Gronwall lemma entails, there exists a constant such that
The proof is finished.

Lemma 9. Let be a bounded domain. The assumptions (11)–(15) hold, and let be the solution of (1) corresponding to the initial data . Then, for any , one can decompose as where and satisfy the following estimates: for all , with the constants and depending on , , and but both being independent of .

Proof. From Theorem 5, we know that there exists a constant which depends only on the -bound of , such that Now, taking and in every interval , we set where and are the solutions of (46) and (47), respectively, in the interval with the initial data and . And in interval , we set and , where and are the solutions of (46) and (47), respectively, in the interval with the initial data and .
Then from Lemma 7, we have and from Lemma 8, we have where is the characteristic function of set .

Remark 10. From the proof of Lemma 9, we observe that the decomposition and also satisfy further that In what follows we begin to establish the asymptotic regularity of the solutions. Now we can claim the following result.

Lemma 11. Let be a bounded domain with smooth boundary, and assume that satisfies (11)–(15); there exists constant which depends only on the -bounds of , such that for any and holds.

Proof. Let and multiplying (47) by , we get that
Let then here and where .
Combining with (2), (63), and , we have Using Lemma 8 to deal with the nonlinear term, we get
Using Remark 10, we have
Moreover, from Lemma 2, Remark 10, and Lemma 8, there exists a constant such that, for all , where is given in (75).
At the same time, we get
Similarly Since , by (14) and Lemma 7, we deduce that
Finally, Moreover, following Lemma 7, we can take large enough such that for all
Plugging (86)–(93) into (65), we obtain where and is an increasing function in .
Then using the Gronwall inequality and integrating over , we entail Letting , then we get Hence, by Lemma 8, (85) and noting that is fixed, the proof is finished.

Lemma 12. Assume that is bounded in . Then there exists a constant which depends only on the -bounds of such that, for any and

Proof. We set , take the inner product of (35) with , and set ; we have We only need to deal with the nonlinear term separately, and the remainders are same as in Lemma 11:
Using Lemma 9 and Remark 10, we have Substituting (101)–(102) into (100), we get that, for all , where . Using the Gronwall inequality and integrating over , we get that there exists a constant which depends only on the -bounds of such that, for any and ,

In the following, based on Lemmas 11 and 12, we perform a bootstrap argument, whose proof is similar to that of Lemmas 11 and 12 (e.g., see [11, 13]). Here we only point out the results and omit the proof.

Lemma 13. For each , assume that is bounded in . Then there exists a constant which depends only on the -bounds of , such that holds for all and .

Lemma 14. For each , if the initial data set is bounded in , then the decomposed ingredient (the solutions of (47)) satisfies which holds for all and , where the constant depends only on the -bounds of .

Theorem 15 (asymptotic regularity of solutions). Let be a bounded domain in with smooth boundary and let satisfy (11)–(15). Assume further that . Then there exist a bounded (in ) set , a positive constant , and a monotonically increasing function such that, for any bounded (in ) set and , the following estimate holds: where denotes the usual Hausdorff semidistance in .

Proof. Let be a bounded (in ) absorbing set obtained in Theorem 5.
From Lemmas 11 and 7, we know that the set satisfies where the constant is given in Lemma 11 corresponding to . Applying Lemmas 14 and 7 to , we see that there is a set which is bounded in , such that where depends on the -bounds of (determined in Lemma 7).
Combining with Lemma 4, we know that the conditions in Lemma 1 are all satisfied. Hence, we have for two appropriate constants and . Since is fixed, by finite steps (e.g., at most by steps), we can infer that there is a bounded (not only in , but also bounded in ) set such that Note further that all the constants in (111) depend only on and . Now, for any bounded (in ) , from Theorem 5 we see that there is a such that Hence, where . Finally, we apply the attraction transitivity lemma, that is, Lemma 1, again to (111) and (113), and this completes the proof.

6. Global Attractor

Collecting now Theorem 5, Lemmas 7 and 8, and Theorem 15, we establish that is asymptotically compact. Therefore, by means of well-known results of the theory of dynamical systems we get the following.

Theorem 16. Let be a bounded domain with smooth boundary and assume that satisfies (11)–(15); the semigroup possesses a global attractor on . Moreover, this attractor is bounded in .

Acknowledgment

This work was partly supported by National Science and Technology Major Projects of China (no. 2012ZX10001001-006) and the Scientific Research Fund of Hunan Provincial Education Department (Grant no. 10C042, 11A008).

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