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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 673605, 15 pages
Asymptotic Behavior of a Class of Degenerate Parabolic Equations
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.
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 .
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 ) the coefficient is allowed to have “essential” zeroes at some points or even to be unbounded. In , 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 , under the same assumption as in , 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 , 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 ) 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 ). 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
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 ). 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 .
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  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).
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  which can guarantee that is invariant.
Therefore, the desired claim follows immediately.
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).
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