Abstract and Applied Analysis

Volume 2012 (2012), Article ID 673605, 15 pages

doi:10.1155/2012/673605

Research Article

Asymptotic Behavior of a Class of Degenerate Parabolic Equations

Hongtao Li¹ and Shan Ma²

¹School of Traffic and Transportation, Lanzhou Jiaotong University, Lanzhou 730070, China

²School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China

Received 19 August 2012; Revised 19 November 2012; Accepted 23 November 2012

Academic Editor: Andrew Pickering

Copyright © 2012 Hongtao Li and Shan 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.

We investigate the asymptotic behavior of solutions of a class of degenerate parabolic equations in a bounded domain () with a polynomial growth nonlinearity of arbitrary order. The existence of global attractors is proved in , , and , respectively, when can be just compactly embedded into () but not .

1. Introduction

Let us consider the following degenerate parabolic equations:

where is a bounded domain in (), with smooth boundary , is a given nonnegative function, and is a function satisfying

both for all .

For the long-time behavior problems of the classical evolutionary equations, especially, the classical reaction-diffusion equation, much has been accomplished in recent years (see, e.g., [1–9] and the references therein), whereas for degenerated evolutionary equations such information is by comparison very incomplete. The main feature of the problem (1.1) is that the differential operator is degenerate because of the presence of a nonnegative diffusion coefficient which is allowed to vanish somewhere (the physical meaning, see [10–12]). Actually, in order to handle media which have possibly somewhere “perfect” insulators (see [10]) the coefficient is allowed to have “essential” zeroes at some points or even to be unbounded. In [13], the authors considered the existence of positive solutions when nonlinearity is superlinear and subcritical function for a semilinear degenerate elliptic equation under the assumption that , for some , satisfies

Recently, motivated by [13], under the same assumption as in [13], the authors of [11, 12, 14–20] proved the existence of global attractors of a class of degenerate evolutionary equations for the case of .

The present paper is devoted to the case of which is essentially different from the case of , and which will cause some technical difficulties. In [13], the authors pointed out that the number plays the role of critical exponent. It is well known that some kind of compactness of the semigroup associated with (1.1) is necessary to prove the existence of the global attractor in . However, there is no corresponding compact embedding result in this case since is compactly embedded only into but not . Hence, the existence of the global attractor in cannot be obtained by usual methods.

In this paper, we assume the weighted function satisfies the following.

and for every .

We will firstly obtain the existence and uniqueness of weak global solutions by use of the singular perturbation then use the asymptotic a priori estimate (see [9]) to verify that the semigroup associated with our problem is asymptotically compact and establish the existence of the global attractor in , and , respectively.

2. Preliminary Results

In this section, we firstly present some notation and preliminary facts on functional spaces then review some necessary concepts and theorems that will be used to prove compactness of the semigroup. For convenience, hereafter let be the norm of , the modular (or the absolute value) of , and an arbitrary positive constant, which may vary from line to line and even in the same line.

2.1. Functional Spaces

The appropriate Sobolev space for (1.1) is , defined as a completion of with respect to the norm

The dual space is denoted by , that is, .

The next proposition refers to continuous and compact inclusion of .

Proposition 2.1 (see [13]). Let be bounded domain in () and let satisfy (1.4) for some . Then the following embeddings hold:

(i) is continuously embedded in ; (ii) is continuously embedded in ; (iii) is compactly embedded in as .

Remark 2.2. when , when , which plays the role of the critical exponent in the Sobolev embedding.

In this paper we only consider the case of when .

2.2. Some Results on Existence of Global Attractors

In this subsection, we review briefly some basic concepts and results on the existence of global attractors; see [2, 5, 7, 9] for more details.

Definition 2.3. Let be a semigroup on Banach space . is called asymptotically compact if for any bounded sequence and , as , and has a convergent subsequence in .

Theorem 2.4. Suppose is a semigroup on . Assume further is a continuous or weak continuous semigroup on for some and possesses a global attractor in , where is bounded. Then possesses a global attractor in if and only if

(i) has a bounded absorbing set in , and (ii)for any and any bounded subset , there exist positive constants and such that

Theorem 2.5. Let be a semigroup on and have a bounded absorbing set in . Then for any and any bounded subset , there exist positive constants and such that

where (sometimes we also write it as ) denotes the Lebesgue measure of and .

Theorem 2.6. For any , the bounded subset of has a finite -net in if there exists a positive constant which depends on such that

(i) has a finite -net in for some , ; (ii)

3. Existence and Uniqueness of the Weak Global Solutions

In this paper, throughout we denote , and , respectively, where is the conjugate exponent of , that is, . In addition, we always assume that satisfies (1.2)-(1.3) and the external forcing term belongs only to .

Definition 3.1. A function is called a weak solution of (1.1) on if and only if

and almost everywhere in such that

holds for all test functions .

The following lemma makes the initial condition in problem (1.1) meaningful.

Lemma 3.2 (see [16]). If and , then .

Theorem 3.3. Assume () is a bounded open domain with smooth boundary, satisfies (1.2)-(1.3), and . Then for any and there exists a unique solution of (1.1) which satisfies

The mapping is continuous in .

Proof. For any , we choose such that are uniformly bounded with respect to , and

Consider the problem

where

According to the standard Galerkin methods (see, e.g., [2, 6, 7]), we know the problem (3.5) admits a unique weak solution ; . Here is called a weak solution of the problem (3.5), if, for any , we have

and almost everywhere in .

Now we do some estimates on in the following.

Multiplying (3.5) by and integrating over , we get

By (1.3) and the Hölder’s inequality, we can deduce that

where .

Using the Gronwall lemma, for any , we have the following:

Integrating (3.8) and (3.9), both sides between and , and using the Young’s inequality, we may get by a standard procedure (see, e.g., [2, 6, 7]) that

with independent of .

Noting that (1.3), we obtain

So we have the following:

We now extract a weakly convergent subsequence, denoted also by for convenience, with

Since , it follows that

Now we show that is a weak solution of Problem (1.1). Multiply (3.5) by and let to derive

for .

Therefore, in order to obtain the existence we need only to prove

for is dense in .

From (3.8) we can obtain

Let . It is obvious that

Therefore,

Taking in the above inequality and noticing that

we arrive at

On the other hand, choosing in (3.7) leads to

Then, it follows from (3.22) and (3.23) that

Choosing with in the above inequality, we get

which implies by letting that

If we choose , we achieve the inequality with opposite sign. Thus

which leads to (3.17). Then follows from Lemma 3.2.

Now we will show that . Choosing some ; with as a test function and integrating by parts in the variable we have

Doing the approximations as above yields

taking limits to conclude that

since . Thus .

Thanks to (1.2), uniqueness and continuous dependence on initial conditions can easily be obtained.

We can therefore use these solutions to define a semigroup on by setting

which is continuous on in .

4. Existence of Global Attractors

In this section, we prove the existence of the global attractors in , , and , respectively. The following result is the existence of bounded absorbing sets which has been established in [18].

Theorem 4.1. The semigroup possesses bounded absorbing sets in , , and , respectively; that is, for any bounded subset in , there exists a constant , such that

for all and , where both and are positive constants independent of , .

In order to obtain the existence of a global attractor in we need to verify that possesses some kind of compactness in , which, however, we cannot obtain by usual methods for lack of the corresponding Sobolev compact embedding results for this case. Here, the new method introduced in [9] is used.

Let be the bounded absorbing set in , then we can consider our problem only in . For is compactly continuous into for some , we know that is compact in , and has a finite -net in .

Firstly, we give the following useful a priori estimate.

Theorem 4.2. For any and bounded subsets , there exist and such that

Proof. For any fixed , there exists such that for any and we have

Moreover, from Theorem 2.5, we know that there exist and such that

In addition, thanks to (1.3), we know when . In the following we assume and .

Multiplying (1.1) by and integrating over , we have

where denotes the positive part of , that is,

Let , then

By the Cauchy’s and Hölder’s inequality, we deduce that

Combining with (4.3)-(4.4) and , we get

We apply the Gronwall lemma to infer

Replacing with and using the same method as above, we obtain

Hence, by (4.10) and (4.11), we have (4.2).

According to Theorem 2.6, we know is compact in ; hence, Theorem 4.1 implies the existence of an attractor in , immediately.

Theorem 4.3. The semigroup associated with (1.1) possesses a global attractor in , that is, is compact and invariant in and attracts the bounded sets of in the topology of .

We now establish the existence of global attractor in .

Theorem 4.4. The semigroup associated with (1.1) possesses a global attractor in , that is, is compact and invariant in and attracts the bounded sets of in the topology of .

Proof . From Theorems 2.4, 4.1, and 4.3, we need only to verify that for any and bounded subset there exist and such that

Letting , from (1.3), we deduce that

So,

On account of the standard Cauchy’s and Hölder’s inequalities, it follows from (4.7) that

Taking , integrating the last equality between and , and combining with (4.2)–(4.4), we have

On the other hand, let , multiplying (1.1) by and integrating over , then we have

From (4.16) and (4.17) we apply the uniform Gronwall lemma to obtain

Hence

Replacing and with and , respectively, and repeating the same steps as above we obtain

Then, from (4.19)-(4.20), we have

Thus, by (4.4) and (4.14), (4.21) implies (4.12). The proof is finished.

4.1. Global Attractor in

In order to prove the existence of a global attractor in , we need the following lemma.

Lemma 4.5. For any bounded subset in , there exists a constant such that

where , is independent of .

Proof. Multiplying (1.1) by and integrating over , we get

Let and differentiate (1.1) with respect to to get

Multiplying the above equality by and integrating over , by (1.2), we obtain

Taking , integrating (4.23) from to , and considering Theorem 4.1, we get

as is large enough.

Combing with (4.25) and (4.26) and using the uniform Gronwall lemma, we have

as is large enough.

Next, we verify is asymptotically compact in .

Theorem 4.6. The semigroup is asymptotically compact in .

Proof. Let be an absorbing set in obtained in Theorem 4.1, then we need only to verify that possesses a convergent subsequence in for any sequence .

In fact, by Theorems 4.3 and 4.4, we know is precompact in and . So we can assume that the subsequence is a Cauchy sequence in and . Now we prove that is a Cauchy sequence in .

Noticing that the prime operator is strong monotone, that is,

we have

which, combining with Lemma 4.5, yields the respected result immediately.

In order to establish the existence of global attractor in we need some continuity of the semigroup to guarantee the invariance of global attractor. However, it is difficult to obtain the continuity of semigroup in since we do not impose any restriction on . Here we use the norm-to-weak continuity instead of the norm-to-norm (or weak-to-weak) continuity of the semigroup in the usual criterions for the existence of global attractors.

Theorem 4.7. The semigroup possesses a global attractor in , that is, is compact and invariant in and attracts every bounded subset of in the topology of .

Proof. Let be an absorbing set in obtained in Theorem 4.1. Set

Then from Theorems 4.1 and 4.6 we know that is nonempty and compact in and attracts every bounded subset of in the topology of . In addition, it is easy to obtain the norm-to-weak continuity of the semigroup from Theorem 3.2 in [9] which can guarantee that is invariant.

Therefore, the desired claim follows immediately.

Acknowledgments

The authors would like to thank the referee for their many helpful comments and suggestions. This work was partially supported by the National Science Foundation of China Grants (11031003 and 11201203).

References

J. W. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems, vol. 278, Cambridge University Press, Cambridge, UK, 2000.

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, vol. 25, North-Holland Publishing, Amsterdam, The Netherland, 1992.

M. Marion, “Attractors for reaction-diffusion equations: existence and estimate of their dimension,” Applicable Analysis, vol. 25, no. 1-2, pp. 101–147, 1987.

M. Marion, “Approximate inertial manifolds for reaction-diffusion equations in high space dimension,” Journal of Dynamics and Differential Equations, vol. 1, no. 3, pp. 245–267, 1989.

Q. Ma, S. Wang, and C. Zhong, “Necessary and sufficient conditions for the existence of global attractors for semigroups and applications,” Indiana University Mathematics Journal, vol. 51, no. 6, pp. 1541–1557, 2002.

J. C. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, Cambridge, UK, 2001.

R. Temam, Infinite Dimensional Dynamical Dystems in Mechanics and Physics, vol. 68, Springer, Berlin, Germany, 2nd edition, 1997.

J. Valero and A. Kapustyan, “On the connectedness and asymptotic behaviour of solutions of reaction-diffusion systems,” Journal of Mathematical Analysis and Applications, vol. 323, no. 1, pp. 614–633, 2006.

C.-K. Zhong, M.-H. Yang, and C.-Y. Sun, “The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations,” Journal of Differential Equations, vol. 223, no. 2, pp. 367–399, 2006.

R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Volume 1: Physical origins and classical methods, Springer, Berlin, Germany, 1990.

N. I. Karachalios and N. B. Zographopoulos, “Convergence towards attractors for a degenerate Ginzburg-Landau equation,” Zeitschrift für Angewandte Mathematik und Physik, vol. 56, no. 1, pp. 11–30, 2005.

N. I. Karachalios and N. B. Zographopoulos, “Global attractors and convergence to equilibrium for degenerate Ginzburg-Landau and parabolic equations,” Nonlinear Analysis: Theory, Methods and Applications, vol. 63, no. 5–7, pp. e1749–e1768, 2005.

P. Caldiroli and R. Musina, “On a variational degenerate elliptic problem,” NoDEA Nonlinear Differential Equations and Applications, vol. 7, no. 2, pp. 187–199, 2000.

C. T. Anh and P. Q. Hung, “Global attractors for a class of degenerate parabolic equations,” Acta Mathematica Vietnamica, vol. 34, no. 2, pp. 213–231, 2009.

C. T. Anh and P. Q. Hung, “Global existence and long-time behavior of solutions to a class of degenerate parabolic equations,” Annales Polonici Mathematici, vol. 93, no. 3, pp. 217–230, 2008.

C. T. Anh, N. M. Chuong, and T. D. Ke, “Global attractor for the m-semiflow generated by a quasilinear degenerate parabolic equation,” Journal of Mathematical Analysis and Applications, vol. 363, no. 2, pp. 444–453, 2010.

C. T. Anh and T. D. Ke, “Long-time behavior for quasilinear parabolic equations involving weighted p-Laplacian operators,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 71, no. 10, pp. 4415–4422, 2009.

C. T. Anh, N. D. Binh, and L. T. Thuy, “On the global attractors for a class of semilinear degenerate parabolic equations,” Annales Polonici Mathematici, vol. 98, no. 1, pp. 71–89, 2010.

C. T. Anh and L. T. Thuy, “Notes on global attractors for a class of semilinear degenerate parabolic equations,” Journal of Nonlinear Evolution Equations and Applications, vol. 4, pp. 1–16, 2012.

N. I. Karachalios and N. B. Zographopoulos, “On the dynamics of a degenerate parabolic equation: global bifurcation of stationary states and convergence,” Calculus of Variations and Partial Differential Equations, vol. 25, no. 3, pp. 361–393, 2006.