Asymptotic Behavior of a Class of Degenerate Parabolic Equations

and Applied Analysis 3 2.1. Functional Spaces The appropriate Sobolev space for 1.1 is H 0 Ω , defined as a completion of C ∞ 0 Ω with respect to the norm


Introduction
Let us consider the following degenerate parabolic equations: where Ω is a bounded domain in R n n 2 , with smooth boundary ∂Ω, a is a given nonnegative function, and f is a C 1 function satisfying f s −l, 1.2 both for all s ∈ R.

Abstract and Applied Analysis
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 − div a x ∇u is degenerate because of the presence of a nonnegative diffusion coefficient a x 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 a 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 a ∈ L 1 loc Ω , for some α ∈ 0, 2 , satisfies lim inf x → z |x − z| −α a x > 0, for every z ∈ Ω.

1.4
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 α ∈ 0, 2 . The present paper is devoted to the case of α 2 which is essentially different from the case of α ∈ 0, 2 , and which will cause some technical difficulties. In 13 , the authors pointed out that the number 2 * α 2n/ n − 2 α 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 L 2 Ω . However, there is no corresponding compact embedding result in this case since H 1,a 0 Ω is compactly embedded only into L r Ω r < 2 but not L 2 Ω . Hence, the existence of the global attractor in L 2 Ω cannot be obtained by usual methods.
In this paper, we assume the weighted function a satisfies the following.
A 1 a ∈ L ∞ Ω and lim inf x → z |x − z| 2 a x > 0 for every z ∈ Ω.
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 L 2 Ω , L p Ω p 2 and H 1,a 0 Ω , respectively.

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 · p be the norm of L p Ω p 1 , |u| the modular or the absolute value of u, and C an arbitrary positive constant, which may vary from line to line and even in the same line.

Functional Spaces
The appropriate Sobolev space for 1.1 is H 1,a 0 Ω , defined as a completion of C ∞ 0 Ω with respect to the norm The dual space is denoted by H −1,a Ω , that is, H 1,a 0 Ω * H −1,a Ω . The next proposition refers to continuous and compact inclusion of H 1,a 0 Ω .
In this paper we only consider the case of α 2 when n 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 {S t } t 0 be a semigroup on Banach space X. {S t } t 0 is called asymptotically compact if for any bounded sequence {x n } ∞ n 1 and t n 0, t n → ∞ as n → ∞, and {S t n x n } ∞ n 1 has a convergent subsequence in X.
where m e (sometimes we also write it as |e|) denotes the Lebesgue measure of e ⊂ Ω and Ω |u|

Existence and Uniqueness of the Weak Global Solutions
In this paper, throughout we denote where q is the conjugate exponent of p, that is, 1/p 1/q 1. In addition, we always assume that f satisfies 1.2 -1.3 and the external forcing term g belongs only to L 2 Ω . and u| t 0 u 0 almost everywhere in Ω such that The following lemma makes the initial condition in problem 1.1 meaningful.
The mapping u 0 → u t is continuous in L 2 Ω .

Abstract and Applied Analysis 5
Proof. For any 0 < ε < 1, we choose u ε,0 ∈ C ∞ c Ω such that u ε,0 L ∞ Ω are uniformly bounded with respect to ε, and Consider the problem According to the standard Galerkin methods see, e.g., 2, 6, 7 , we know the problem 3.5 admits a unique weak solution u ε ∈ C 0, T ; and u ε | t 0 u ε,0 almost everywhere in Ω. Now we do some estimates on u ε in the following. Multiplying 3.5 by u ε and integrating over Ω, we get By 1.3 and the Hölder's inequality, we can deduce that where |Ω| Ω 1 dx. Using the Gronwall lemma, for any T > 0, we have the following: u ε is uniformly bounded in L ∞ 0, T; L 2 Ω with respect to ε.
3.10 6 Abstract and Applied Analysis Integrating 3.8 and 3.9 , both sides between 0 and T , and using the Young's inequality, we may get by a standard procedure see, e.g., 2, 6, 7 that Noting that 1.3 , we obtain

3.12
So we have the following: f u ε is uniformly bounded in L q 0, T; L q Ω with respect to ε.

3.13
We now extract a weakly convergent subsequence, denoted also by u ε for convenience, with

3.14
Since f ∈ C R , it follows that Now we show that u is a weak solution of Problem 1.1 . Multiply 3.5 by ϕ and let ε → 0 to derive for ϕ ∈ V .
Abstract and Applied Analysis 7 Therefore, in order to obtain the existence we need only to prove Let Therefore,

3.20
Taking ε → 0 in the above inequality and noticing that On the other hand, choosing ϕ u in 3.7 leads to Then, it follows from 3.22 and 3.23 that Choosing v u − λϕ with λ > 0 in the above inequality, we get If we choose λ < 0, we achieve the inequality with opposite sign. Thus which leads to 3.17 . Then u ∈ C 0, T ; L 2 Ω follows from Lemma 3.2. Now we will show that u 0

as a test function and integrating by parts in the t variable we have
Doing the approximations as above yields taking limits to conclude that since u ε0 → u 0 . Thus u 0 u 0 . 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 {S t } t 0 on L 2 Ω by setting

Existence of Global Attractors
In this section, we prove the existence of the global attractors in L 2 Ω , L p Ω , and H 1,a 0 Ω , respectively. The following result is the existence of bounded absorbing sets which has been established in 18 . In order to obtain the existence of a global attractor in L 2 Ω we need to verify that {S t } t 0 possesses some kind of compactness in L 2 Ω , 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 B 0 be the bounded absorbing set in H 1,a 0 Ω , then we can consider our problem only in B 0 . For H 1,a 0 Ω is compactly continuous into L r Ω for some 1 r < 2, we know that B 0 is compact in L r Ω , and B 0 has a finite ε-net in L r Ω .
Firstly, we give the following useful a priori estimate. In addition, thanks to 1.3 , we know f s 0 when s > C 0 /C 1 1/p . In the following we assume M max{M 1 , C 0 /C 1 1/p } and t T .
where u − M denotes the positive part of u − M, that is, Let Ω 1 Ω u t M , then By the Cauchy's and Hölder's inequality, we deduce that Combining with 4.3 -4.4 and L p Ω → L 2 Ω p 2 , we get We apply the Gronwall lemma to infer Cε.

4.10
Replacing u − M with u M − and using the same method as above, we obtain Cε.
According to Theorem 2.6, we know B 0 is compact in L 2 Ω ; hence, Theorem 4.1 implies the existence of an attractor in L 2 Ω , immediately.
Letting F s s 0 f τ dτ, from 1.3 , we deduce that So, On account of the standard Cauchy's and Hölder's inequalities, it follows from 4.7 that

4.15
Taking t T , integrating the last equality between t and t 1, and combining with 4.  Let v u t and differentiate 1.1 with respect to t to get v t − div a x ∇v f u v 0. 4.24 Multiplying the above equality by v and integrating over Ω, by 1.2 , we obtain

4.25
Taking t T , integrating 4.23 from t to t 1, and considering Theorem 4.1, we get In fact, by Theorems 4.3 and 4.4, we know {u n t n } ∞ n 1 is precompact in L 2 Ω and L p Ω . So we can assume that the subsequence {u n k t n k } ∞ k 1 is a Cauchy sequence in L 2 Ω and L p Ω . Now we prove that {u n k t n k } ∞ k 1 is a Cauchy sequence in H 1,a