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International Journal of Mathematics and Mathematical Sciences
Volume 2011 (2011), Article ID 178057, 27 pages
http://dx.doi.org/10.1155/2011/178057
Research Article

On the Dynamics of Nonautonomous Parabolic Systems Involving the Grushin Operators

1Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy, Cau Giay, 10307 Hanoi, Vietnam
2Faculty of Computer Science and Engineering, Hanoi Water Resources University, 175 Tay Son, Dong Da, 10508 Hanoi, Vietnam

Received 19 December 2010; Accepted 21 February 2011

Academic Editor: Feng Qi

Copyright © 2011 Anh Cung The and Toi Vu Manh. 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 long-time behavior of solutions to nonautonomous semilinear parabolic systems involving the Grushin operators in bounded domains. We prove the existence of a pullback -attractor in for the corresponding process in the general case. When the system has a special gradient structure, we prove that the obtained pullback -attractor is more regular and has a finite fractal dimension. The obtained results, in particular, extend and improve some existing ones for the reaction-diffusion equations and the Grushin equations.

1. Introduction

Nonautonomous equations are of great importance and interest as they appear in many applications in the natural sciences. One way of studying the long-time behavior of solutions of such equations is using the theory of pullback attractors. This theory has been developed for both the nonautonomous and random dynamical systems and has shown to be very useful in the understanding of the dynamics of such dynamical systems (see [1] and references therein). In recent years, the existence of pullback attractors for reaction-diffusion equations has been studied widely (see, e.g., [26]). However, to the best of our knowledge, little seems to be known for the asymptotic behavior of solutions of nonautonomous degenerate equations.

One of the classes of degenerate equations that has been studied widely in recent years is the class of equations involving an operator of the Grushin type [7] The global existence and long-time behavior of solutions to semilinear parabolic equations involving the Grushin operator, in both autonomous and nonautonomous cases, have been studied in some recent works [810].

In this paper we consider the following nonautonomous semilinear parabolic system: where , is given, is an unknown vector-function. Here , , and satisfy the following conditions:

(H1) has a positive symmetric part: ;(H2) is a -vector function such that: where , , , and are positive constants; (H3) such that where is the first eigenvalue of the operator in with the homogeneous Dirichlet boundary condition.

In order to study problem (1.2), we will use the natural energy space defined as the complete of in the following norm: From the results in [11], we know that the embedding is continuous if , where ; moreover, this embedding is compact if .

Notations
Denote , and the dual space of . For functions , we set so if , then where denotes the inner product in .

Noting that by assumption (H1), we have Hence

The aim of this paper is to study the long-time behavior of solutions to problem (1.2) by using the theory of pullback -attractors. We first prove, under assumptions (H1)–(H3), the existence of a pullback -attractor in for the process associated to problem (1.2). Then, with an additional condition that the system has a special gradient structure, namely, and there exists a function such that , we prove the existence of a pullback -attractor in the space for the process . Moreover, we prove the boundedness of the pulback -attractor in and in , and give estimates of the fractal dimension of the pulback -attractor. It is worth noticing that our results, in particular, extend and improve some recent results on the existence of pullback -attractors for the reaction-diffusion equations [35] and for the Grushin equations [8].

Let us explain the methods used in the paper. We first prove the existence of a family of pullback -absorbing sets in . Thanks to the compactness of the embedding , we immediately get the existence of a pullback -attractor in . When the system has a special gradient structure, we are able to prove the existence of a pullback -attractor in . To do this, we follow the general lines of the approach used in [8], with some modifications so that we can improve conditions imposed on the external force . In particular, we use the asymptotic a priori estimate method initiated in [12] to testify the pullback asymptotic compactness of the corresponding process. Moreover, in this case we also prove the regularity of the pullback -attractor in the spaces and . Finally, using the recent results in [13], we give an estimate of the fractal dimension of the pullback -attractor. It is noticed that we do not impose the restriction on the exponent in (H2) as in [13].

The rest of the paper is organized as follows. In Section 2, for the convenience of the reader, we recall some concepts and results on pullback -attractors which we will use. In Section 3, we prove the existence of a pullback -attractor in in the general case. In Section 4, under the additional assumption that the system has a gradient structure, we prove the regularity and fractal dimension estimates of the pullback -attractor.

2. Preliminaries

2.1. Pullback Attractors

For convenience of the reader, we recall in this section some concepts and results on the theory of pullback -attractors, which will be used in the paper.

Let be a metric space with metric . Denote by the set of all bounded subsets of . For , 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 verifying the norm-to-weak continuity of a process.

Proposition 2.1 (see [14]). Let be two Banach spaces, be, respectively, their dual spaces. Assume 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 a compact set of to be a bounded set of .

Suppose that is a nonempty class of parameterized sets .

Definition 2.2. The process is said to be pullback -asymptotically compact if for any , any , and any sequence with for all , and , any sequence , the sequence is relatively compact in .

Definition 2.3. A process is called pullback --limit compact if for any , any , and , there exists a such that where is the Kuratowski measure of noncompactness of ,

Lemma 2.4 (see [3]). A process is pullback -asymptotically compact if and only if it is --limit compact.

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

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

Theorem 2.7 (see [3]). Let be a norm-to-weak continuous process such that is pullback -asymptotically compact. If there exists a family of pullback -absorbing sets , then has a unique pullback -attractor and

2.2. Fractal Dimension of Pullback Attractors

Given a compact and , we denote by the minimum number of open balls in with radius which are necessary to cover .

Definition 2.8. For any nonempty compact set , the fractal dimension of is the number

Definition 2.9. A bounded subset is called a uniformly pullback absorbing set for process if for every is bounded, there exists a such that here, does not depend on the choice of .

Theorem 2.10 (see [13]). Let be a process in a separable Hilbert space , be a uniformly pullback absorbing set in , be a pullback attractor for , if there exists a finite dimensional projection in the space such that for all and some and for all , where , and are independent of the choice of . Then the family of pullback attractors possesses a finite fractal dimension especifically

3. Existence of Pullback -Attractors in

Denote where is the conjugate of (i.e., ).

Definition 3.1. Let and be given. A function is called a weak solution of problem (1.2) on if for all test functions .

One can prove that if and , then (see [10]). This makes the initial condition in (1.2) meaningful.

Theorem 3.2. Under assumptions (H1)–(H3), for any , , given, problem (1.2) has a unique weak solution on . Moreover, the solution exists on the interval and the following inequality holds:

Proof. The existence and uniqueness of a weak solution to problem (1.2) are proved similarly to the scalar case in [10], so it is omitted here.
We now prove inequality (3.3). Multiplying (1.2) by , integrating over , and using (1.11), we have Using condition (1.3) and the Cauchy inequality, we obtain Because , so (3.5) becomes Applying the Gronwall inequality we get (3.3).

Now, we can define the family of two-parameter mappings where is the unique weak solution of (1.2) with the initial datum at time . Then defines a continuous process on .

Let be the set of all functions such that and denote by the class of all families such that for some , where is the closed ball in with radius .

Lemma 3.3. Under assumptions (H1)–(H3), there exists a constant such that the solution of problem (1.2) satisfies the following inequality for all : where . This implies that there exists a family of pullback -absorbing sets in for the process .

Proof. We multiply (1.2) by and integrate over . After some standard transformations we obtain Without loss of generality, we may assume that . Otherwise we can replace by . The function satisfies the same conditions with modified constants , because (see (1.4)). Hence, since , we get where we have used condition (1.5). Similarly, we have Hence By the Cauchy inequality we have From (3.9)–(3.13) we obtain thus, where . Multiplying (3.15) by and integrating from to , we obtain Multiplying (3.3) by and integrating from to , we have Now, from (3.5) we get Multiplying this equation by and integrating from to , we deduce that Using (3.17), (3.19) becomes Substituting (3.20) into (3.16) we obtain Hence we get (3.8) with .
Let be the right-hand side of (3.8), and let be the closed ball in centered at 0 with radius . Obviously for any and any , by (3.8) there exists such that the solution with initial datum at time satisfies for all ; that is, is a family of bounded pullback -absorbing sets in .

From the above lemma we deduce that the process maps a compact set of to be a bounded set of , and thus by Proposition 2.1, the process is norm-to-weak continuous in . Since has a family of pullback -absorbing sets in and the embedding is compact, we immediately get the following.

Theorem 3.4. Under assumptions (H1)–(H3), the process associated to problem (1.2) has a pullback -attractor in .

4. Some Further Results in the Gradient Case

In this section, instead of (H1)–(H3), we assume that (H1bis), where is the unit matrix and ; (H2bis) satisfies (H2) and , where is a potential function satisfying with , , being positive constants (H3bis) satisfies where .

The aim of this section is to prove that the pullback -attractor obtained in Section 3 is more regular and has a finite fractal dimension.

4.1. Existence of a Pullback -Attractor in

Denote by the set of all functions such that and denote by the class of all families such that for some , where is the closed ball in with radius . Thanks to the above gradient structure, one can prove the existence of a pullback -attractor, not only in , but also in the space for the process .

We first prove the following.

Lemma 4.1. Under assumptions (H1bis)–(H3bis), the solution of problem (1.2) satisfies the following inequality for all : where . This implies that there exists a family of pullback -absorbing sets in for the process .

Proof. Using (3.5) with and the fact that , we have thus Integrating from to , and in particular, we have Furthermore, multiplying (4.5) from to and using (4.6) we obtain By assumption (H2bis), then (4.7) becomes Multiplying (1.2) by and integrating over , we have thus Using (4.8), (4.10), and the uniform Gronwall inequality, we get Now, using (H2bis) once again we have from (4.11) that Thus we obtain (4.3) with a suitable positive constant Hence, by the argument as in the end of the proof of Lemma 3.3, we obtain a family of bounded pullback -absorbing sets in .

To prove that the process is pullback -asymptotically compact in , we need the following lemma.

Lemma 4.2 (see [8, Lemma 3.6]). Let be a norm-to-weak continuous process in and , and let satisfy the following two conditions: (i) is pullback -asymptotically compact in ;(ii)for any , , there exist constants and such that Then is pullback -asymptotically compact in .

Theorem 4.3. Under assumptions (H1bis)–(H3bis), the process associated to problem (1.2) has a pullback -attractor in .

Proof. It is sufficient to show that the process satisfies the condition (ii) in Lemma 4.2. We will give some formal calculations, a rigorous proof is done by use of Galerkin approximations and Lemma 11.2 in [15].
Let be a positive number, we will write (or ) as any component of is greater than or equal to (or as any component of is less than or equal to ).
Using (1.3), (1.4), and for large enough, we have because .
Multiplying (1.2) by and integrating over we obtain where On the other hand, by the Cauchy inequality, we have which implies that Hence, from (4.15) and (4.19), we have From (4.16), using (4.20) and noting that we have Now, multiplying the above inequality by and integrating from to , we get Then On the other hand, integrating (4.5) from to , we have Therefore, substituting (4.25) into (4.24), we obtain Hence, for any , there exists and such that for any and any , we have Repeating the same step above, just taking instead of , we deduce that there exist and such that for any and any , where Let and , we obtain So, we have This completes the proof.

To prove the existence of a pullback -attractor in , we need the folowing lemma.

Lemma 4.4. Under assumptions (H1bis)–(H3bis), for any and any bounded subset , there exists a positive constant such that for all and all , where is independent of and .

Proof. We give some formal calculations, a rigorous proof is done by use of Galerkin approximations and Lemma 11.2 in [15].
Differentiating (1.2) in time and setting , we get Multiplying this inequality by and integrating over and using (1.11), we get By the Cauchy inequality and using (1.5), we obtain that Let . Using (3.5), we have By (H2bis) we then infer from the above inequality that Integrating this inequality from to , we obtain On the other hand, integrating (4.36) from to , we obtain So, substituting (4.39) into (4.38), we deduce Now multiplying (1.2) by and integrating over , we have So applying the uniform Gronwall inequality, we get Integrating (4.41) from to and by (4.40)–(4.42), we have Therefore, by (4.35), (4.43), using the uniform Gronwall inequality once again, we get Hence we get (4.32).

Theorem 4.5. Under assumptions (H1bis)–(H3bis), the process associated to problem (1.2) has a pullback -attractor in .

Proof. By Lemma 4.1, has a family of bounded pullback -absorbing sets in . It remains to show that is pullback -asymptotically compact in , that is, for any , any , and any sequence , any sequence , the sequence is precompact in . Thanks to Theorem 4.3, we need only to show that the sequence is precompact in .
Let . By Theorem 3.4, we can assume that is a Cauchy sequence in . We have where we have used condition (1.5). Because is a Cauchy sequence in and by Lemma 4.4, one gets The proof is complete.

4.2. and -Boundedness of the Pullback -Attractor

First, we prove the existence of a family of pullback -absorbing sets for process in .

Proposition 4.6. Under assumptions (H1bis)–(H3bis), then for any and any bounded subset , there exists a positive constant <