Abstract and Applied AnalysisVolume 2012 (2012), Article ID 272145, 20 pagesdoi:10.1155/2012/272145

Research Article

Regularity and Exponential Growth of Pullback Attractors for Semilinear Parabolic Equations Involving the Grushin Operator

Nguyen Dinh Binh

Faculty of Applied Mathematics and Informatics, Hanoi University of Technology, 1 Dai Co Viet, Hai Ba Trung, Hanoi, Vietnam

Received 24 November 2011; Accepted 28 December 2011

Academic Editor: Shaoyong Lai

Copyright © 2012 Nguyen Dinh Binh. 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.

Considered here is the first initial boundary value problem for a semilinear degenerate parabolic equation involving the Grushin operator in a bounded domain . We prove the regularity and exponential growth of a pullback attractor in the space for the nonautonomous dynamical system associated to the problem. The obtained results seem to be optimal and, in particular, improve and extend some recent results on pullback attractors for reaction-diffusion equations in bounded domains.

1. Introduction

Let be a bounded domain in , with smooth boundary . In this paper, we consider the following problem: where is the Grushin operator, is given, the nonlinearity and the external force satisfy the following conditions.(H1) The nonlinearity satisfies where , , () are positive constants. Relation (1.3) and (1.4) imply that where , and are positive constants.(H2) satisfies where is the first eigenvalue of the operator in with the homogeneous Dirichlet boundary condition.

The Grushin operator was first introduced in [1]. Noting that if , then is not elliptic in domains of which intersect the hyperplane . In the last few years, the existence and long-time behavior of solutions to parabolic equations involving the Grushin operator have been studied widely in both autonomous and nonautonomous cases (see, e.g., [2–7]). In particular, the existence of a pullback attractor in for the process associated to problem (1.1) is considered in [2].

In this paper we continue the study in the paper [2]. First, we will prove the existence of pullback attractors in (see Section 2 for its definition) and . As we know, if the external force is only in , then solutions of problem (1.1) are at most in and have no higher regularity. Therefore, there are no compact embedding results that hold for this case. To overcome the difficulty caused by the lack of embedding results, we exploit the asymptotic a priori estimate method which was initiated in [8, 9] for autonomous equations and developed recently for nonautonomous equations in the case of pullback attractors in [10]. Noting that, to prove the existence of pullback attractors in , we only need assumption (H2) of the external force ; however, to prove the existence of pullback attractors in and , we need an additional assumption of , namely, (3.18) in Section 3. Next, following the general lines of the approach in [11], we give exponential growth conditions in for the pullback attractors. It is noticed that, as far as we know, the best known results on the pullback attractors for nonautonomous reaction-diffusion equations are the boundedness and exponential growth in of the pullback attractors [11, 12]. Therefore, the obtained results seem to be optimal and, in particular when , improve the recent results on pullback attractors for the nonautonomous reaction-diffusion equations in [11–15].

The content of the paper is as follows. In Section 2, for the convenience of the reader, we recall some concepts and results on function spaces and pullback attractors which we will use. In Section 3, we prove the existence of pullback attractors in the spaces and by using the asymptotic a priori estimate method. In Section 4, under additional assumptions of , an exponential growth in for the pullback attractors is deduced.

2. Preliminaries

2.1. Operator and Function Spaces

In order to study the boundary value problem for equations involving the Grushin operator, we have usually used the natural energy space defined as the completion of in the following norm:

and the scalar product The following lemma comes from [16].

Lemma 2.1. Assume that is a bounded domain in . Then the following embeddings hold: (i) continuously;(ii) compactly if , where, .

Now, we introduce the space defined as the closure of with the norm

The following lemma comes directly from the definitions of and .

Lemma 2.2. Assume that is a bounded domain in , with smooth boundary . Then continuously.

It is known that (see, e.g., [3]) for the operator , there exist

such that

and is a complete orthonormal system in .

2.2. Pullback Attractors

Let be a Banach space with the norm . denotes all bounded sets of . The Hausdorff semidistance between and is defined by

Let be a process in , that is, such that and for all , . The process is said to be norm-to-weak continuous if , as in , for all , . The following result is useful for proving the norm-to-weak continuity of a process.

Proposition 2.3 (see [9]). Let be two Banach spaces, and let be, respectively, their dual spaces. Suppose that is dense in , the injection is continuous, and its adjoint is dense, and is a continuous or weak continuous process on . Then is norm-to-weak continuous on if and only if for , , maps compact sets of into bounded sets of .

Definition 2.4. The process is said to be pullback asymptotically compact if for any , any , any sequence , and any sequence , the sequence is relatively compact in .

Definition 2.5. A family of bounded sets is called a pullback absorbing set for the process if for any and any , there exist and such that

Definition 2.6. The family is said to be a pullback attractor for if (1) is compact for all , (2) is invariant, that is, (3) is pullback attracting, that is, (4)if is another family of closed pullback attracting sets, then , for all .

Theorem 2.7 (see [13]). Let be a norm-to-weak continuous process which is pullback asymptotically compact. If there exists a pullback absorbing set , then has a unique pullback attractor and

In the rest of the paper, we denote by , the norm and inner product in , respectively, and by the norm in . By we denote the norm in . For a Banach space , will be the norm. We also denote by an arbitrary constant, which is different from line to line, and even in the same line.

3. Existence of Pullback Attractors in

It is well known (see, e.g., [2] or [14]) that under conditions , problem (1.1) defines a processwhere is the unique weak solution of (1.1) with initial datum at time . The process has a pullback attractor in .

In this section, we will prove that the pullback attractor is in fact in .

Lemma 3.1. Assuming that and satisfy (H 1)-(H 2), is a weak solution of (1.1). Then the following inequality holds for : where is a positive constant.

Proof. Multiplying (1.1) by and then integrating over , we get Using hypothesis (H1) and the inequality , we have Letting , by (H1), we have Now multiplying (3.4) by and using (3.5), we get Integrating (3.6) from to and to , respectively, we obtain Multiplying (1.1) by and integrating over , we have Thus Combining (3.8) and (3.10), and using the uniform Gronwall inequality, we have Using (H1) once again and thanks to as , we get the desired result from (3.11).

Lemma 3.2. Assume that (H 1), (H 2) hold. Then for any and any that is bounded, there exists such that for any and any , where .

Proof. Integrating (3.10) from to and using (3.8) and (3.11), in particular we find On the other hand, differentiating (1.1) and denoting  , we have Taking the inner product of (3.14) with in , we get Using (1.5) and Young’s inequality, after a few computations, we see that Combining (3.16) and (3.13) and using the uniform Gronwall inequality, we obtain The proof is now complete because as .

3.1. Existence of a Pullback Attractor in

In this section, following the general lines of the method introduced in [9], we prove the existence of a pullback attractor in . In order to do this, we need an additional condition of where , are defined as in (3.30).

Lemma 3.3. The process associated to problem (1.1) has a pullback absorbing set in .

Proof. Multiplying (1.1) by and integrating over , we get From (1.3) and the fact that continuously, we have On the other hand, by Cauchy’s inequality, we see that Combining (3.19)–(3.22) imply that Applying (3.2) and Lemma 3.2, we conclude the existence of a pullback absorbing set in for the process .

Lemma 3.4. For any , any , and any bounded set , there exists such that where depends on , but not on , and .

Proof. We will prove the lemma by induction argument. Letting  and denoting we prove that for , there exist and such that where depends on and and depends only on .For , we have from (3.17). Integrating (3.16) and using continuously, we get .Assuming that (), () hold, we prove so are and . Multiplying (3.14) by and integrating over , we obtain Using the imbedding once again, we get Combining Holder’s and Young’s inequalities, we see that where . Choose , such that thus Hence Then from (3.27), we infer that Applying (3.26) and (3.31) in (3.25), we find that Combining () and (3.32), using the uniform Gronwall inequality and taking into account assumption (3.18), we get . On the other hand, integrating (3.32) from to , we find . Now since , and taking , we get the desired estimate.

We will use the following lemma.

Lemma 3.5 (see [15]). If there exists such that , for all , then

Let in , and let be the orthogonal projection, where are the eigenvectors of operator . For any , we write

Lemma 3.6. For any , any and any , there exist and such that

Proof. Multiplying (3.14) by and then integrating over , using and Cauchy’s inequality we get We multiply (3.36) by and use assumption (1.4). We get Integrating (3.37) from to , Now integrating (3.38) with respect to from to , we infer that Thus By Lemma 3.5 and since as , there exist and such that for all and . For the second term of the right-hand side of (3.40), using Holder’s inequality we have From Lemmas (3.5)–(3.7), we see that there exist and such that Let and , from (3.40), taking into account (3.41) and (3.43), we obtain (3.35).

Lemma 3.7 (see [9]). Let be a bounded subset in . If has a finite -net in , then there exists an , such that for any , the following estimate is valid:

Using Lemma 3.7 and taking into account Lemmas 3.2 and 3.6 we conclude that the set has a finite -net in . Therefore, we get the following result.

Lemma 3.8. For any , any that is bounded, and any , there exists and such that

Lemma 3.9 (see [9]). For any , any bounded set , and any , there exist and such that where is the Lebesgue measure in and .

Lemma 3.10 (see [2]). Let be a norm-to-weak continuous process in and . Then is pullback asymptotically compact in if (i) is pullback asymptotically compact in ; (ii)for any , any bounded set , and any , there exist and where is independent of , , , and .

We are now ready to prove the existence of a pullback attractor in .

Theorem 3.11. Assume that assumptions (1.3)–(1.7) and (3.18) hold. Then the process associated to problem (1.1) possesses a pullback attractor in .

Proof. Because of Lemma 3.10, since has a pullback absorbing set in , we only have to prove that for any , any , and any , there exist and such that Taking the inner product of (1.1) with in , where we have Some standard computations give us Combining (3.50)–(3.53), we find Applying Lemmas 3.7 and 3.8 to (3.54) we find there exist and such that Repeating the above arguments with in place of , we have for some and , where Letting and we have This completes the proof.

3.2. Existence of a Pullback Attractor in

In this section, we prove the existence of a pullback attractor in .

Lemma 3.12. The process associated to (1.1) has a pullback absorbing set in .

Proof. We multiply (1.1) by ; then, using , we have Using , Cauchy’s inequality, and argument as in Lemma 3.3, from (3.59) we have Taking into account (3.11), the proof is complete.

In order to prove the existence of the pullback attractor in , we will verify so-called “(PDC) condition”, which is defined as follow

Definition 3.13. A process is said to satisfy (PDC) condition in if for any , any bounded set and any , there exists and a finite dimensional subspace of such that (i) is bounded in ; and(ii), for all and , where is a canonical projection and is the identity.

Lemma 3.14 (see [13]). If a process satisfies (PDC) condition in then it is pullback asymptotically compact in . Moreover, if is convex then the converse is true.

Lemma 3.15 (see [9]). Assume that satisfies (1.3) and (1.5). Then for any subset , if in , then we have where the Kuratowski noncompactness measure in a Banach space defined as

Theorem 3.16. Assume that satisfies (1.3)–(1.5), satisfies (1.7) and (3.18). Then the process generated by (1.1) has a pullback attractor in .

Proof. We consider a complete trajectory lies on pullback attractor in for , that is, and , for all . Denoting and multiplying (1.1) by we have Using Holder’s inequality we get Thanks to Lemmas 3.6 and 3.15 and the fact that , we see that satisfies condition (PDC) in . Now from Lemmas 3.3 and 3.14 we get the desired result.

4. Exponential Growth in of Pullback Attractors

In this section, we will give an exponential growth condition in for the pullback attractor .

First, we recall a result in [17] which is necessary for the proof of our results.

Lemma 4.1. Let , be Banach spaces such that is reflexive, and the inclusion is continuous. Assume that is bounded sequence in such that weakly in for some and . Then, for all and

In the following theorem, instead of evaluating the functions which are differentiable enough and then using Lemma 4.1, we will formally evaluate the function .

Theorem 4.2. Assume that satisfies (1.3)-(1.5), satisfies (H 2), (3.18) and the following conditions Then satisfies

Proof. We differentiate with respect to time in (1.1), then multiply by , we get Integrating in the last inequality, in particular, we get for all . Now, integrating with respect to , between and for all , in paricular, for all . Multiplying (1.1) by and then integrating on , we get Using hypothesis (H1) and the fact that , we have Integrating (4.9) from to , we have Thus, Multiplying (1.1) by then integrating over , we have Integrating now between and , we obtain From (4.13) and using hypothesis (H1), we get Hence Integrating inequality (4.15) with respect to from to , we obtain for all , . From (4.16) and (4.11), we obtain that for all .From (4.7), taking and we have for all . Analogously, and if we take and in inequlity (4.15), then From (4.18) and (4.19), we obtain for all . Owing to this inequality and (4.17), we have for all . From (3.23), (3.60) and using Young’s inequality, we have for all . From (4.22) and thank to (4.21), we have for all . From (4.23) and thanks to (4.17), we have for all . Now, observe that by Cauchy’s inequality for all , . Thus, from (4.23), we have for all , . From this inequlity, and the fact that , we obtain for all , and any . Now, thank to (4.2), (4.3), we obtain (4.3) from (4.27).

Acknowledgment

The authors thanks the reviewers very much for valuable comments and suggestions.

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