Global Attractors in for Nonclassical Diffusion Equations
We study the existence of global attractors for nonclassical diffusion equations in . The nonlinearity satisfies the arbitrary order polynomial growth conditions.
In this paper, we investigate the long-time behavior of the solutions for the following nonclassical diffusion equations: with the initial data where , and the nonlinearity satisfies () , and , () , and , and () ,
where , , , , , and are all positive constants. Moreover, without loss of generality, we also assume .
In 1980, Aifantis in [1–3] pointed out that the classical reaction-diffusion equation does not contain each aspect of the reaction-diffusion problem, and it neglects viscidity, elasticity, and pressure of medium in the process of solid diffusion and so forth. Furthermore, Aifantis found out that the energy constitutional equation revealing the diffusion process is different along with the different property of the diffusion solid. For example, the energy constitutional equation is different, when conductive medium has pressure and viscoelasticity or not. He constructed the mathematical model by some concrete examples, which contains viscidity, elasticity, and pressure of medium, that is the following nonclassical diffusion equation: This equation is a special form of the nonclassical diffusion equation used in fluid mechanics, solid mechanics, and heat conduction theory (see [1–4]). Recently, Aifantis presented a new model about this problem and scrutinized the concrete process of constructing model; the reader can refer to  for details.
The longtime behavior of (1.1) acting on a bounded domain has been extensively studied by several authors in [6–13] and references therein. In  the existence of a global attractor for the autonomous case has been shown provided that the nonlinearity is critical and . Furthermore, for the non-autonomous, the existence of a uniform attractor and exponential attractors has been scrutinized when the time-dependent forcing term only satisfies the translation bounded domain instead of translation compact, namely, . A similar problem was discussed in  by virtue of the standard method based on the so-called squeezing property. To our best knowledge, the dynamics of (1.1) acting on an unbounded domain has not been considered by predecessors.
As we know, if we want to prove the existence of global attractors, the key point is to obtain the compactness of the semigroup in some sense. For bounded domains, the compactness is obtained by a priori estimates and compactness of Sobolev embeddings. This method does not apply to unbounded domains since the embeddings are no longer compact. To overcome the difficulty of the noncompact embedding, in , using the idea of Ball , the author proved that the solutions are uniformly small for large space and time variables and then showed that the weak asymptotic compactness is equivalent to the strong asymptotic compactness in certain circumstances. In , the authors provided new a priori estimates for the existence of global attractors in unbounded domains and then applied this approach to a nonlinear reaction-diffusion equation with a nonlinearity having a polynomial growth for arbitrary order and with distribution derivatives in homogeneous term. More recently, in  the authors achieved the existence of global attractors for reaction-diffusion equations in , by using the methods presented in . Our purpose in this paper is to study the existence of global attractors of (1.1) on the unbounded domains , and we borrow the idea of [17, 18]. Our main result is Theorem 4.6.
This paper is organized as follows. In Section 2, we recall some basic definitions and related theorems that will be used later. In Section 3, we prove the existence of weak solution for nonclassical diffusion equations in . The main result is stated and proved in Section 4.
In this section, we iterate some notations and abstract results.
Definition 2.1 (see ). Let be a metric space, and let be bounded subsets of . The Kuratowski measure of noncompactness of defined by
Definition 2.2 (see ). Let be a Banach space, and let be a family of operators on . We say that is a continuous semigroup ( semigroup) (or norm-to-weak continuous semigroup) on , if satisfies(i) (the identity),(ii), for all ,(iii), if in (or , if in ).
Definition 2.3 (see ). A semigroup (or norm-to-weak continuous semigroup) in a complete metric space is called -limit compact if for every bounded subset of and for every , there is a , such that
Condition C (see ). For any bounded set of a Banach space , there exists a and a finite dimensional subspace of such that is bounded and where is a bounded projector.
Lemma 2.4 (see ). Let be a Banach space, and let be a semigroup (or norm-to-weak continuous semigroup) in .(1)If Condition C holds, the is -limit compact.(2)Let be a uniformly convex Banach space. Then is -limit compact if and only if Condition C holds.
Lemma 2.5 (see ). Let be a Banach space, and let be a semigroup (or norm-to-weak continuous semigroup) in .(1)If Condition C holds, the is -limit compact;(2)Let be a uniformly convex Banach space. Then is -limit compact if and only if Condition C holds.
Theorem 2.6 (see ). Let be a Banach space. Then the semigroup (or norm-to-weak continuous semigroup) has a global attractor in if and only if(1)there is a bounded absorbing set .(2) is -limit compact.
Lemma 2.7 (see ). Let be an absolutely continuous positive function on , which satisfies for some the differential inequality for almost every , where and are functions on such that for some and , and for some . Then for some and
Lemma 2.8 (see ). Let be Banach spaces, with reflexive. Suppose that is a sequence that is uniformly bounded in , and is uniformly bounded in , for some . Then there is a subsequence that converges strongly in .
3. Unique Weak Solution
Proof. We decompose our proof into three steps for clarity.
Step 1. For any , we consider the existence of the weak solution for the following problem in : Choose a smooth function with
Since is a bounded domain, so the existence and uniqueness of solutions can be obtained by the standard Faedo-Galerkin methods; see [6, 8, 11, 16]; we have the unique weak solution
Step 2. According to Step 1, we denote ; then satisfies For the mathematical setting of the problem, we denote , , , .
Multiplying (3.5) by in , using , and , we have By the Poincaré inequality, for some , we conclude that Hence, it follows that We get the following estimate: Similar to (3.9), using , , and , we get where is independent of .
and yield Choose such that ; then . Noting that , then , and we have the embedding . According to (3.12) and (3.13), we get where is independent of .
Thanks to (3.14), is bounded in , and is bounded in .
For , where is independent of . We can obtain that is bounded in .
Since , Therefore, there exists , such that , , , and are continuous embedding to .
According to (3.5) and (3.14)–(3.16), we obtain So has a subsequent (we also denote ) weak* convergence to in and ; has a subsequent (we also denote ) weak* convergence to . Let outside of ; we can extend to .
As introduced in [6, 20], is dense in the dual space of , , , and , so we can choose for all as a test function such that
Since for all , there exists bounded domain such that , . It follows that is uniformly bounded in , and . Since , according to Lemma 2.8, there is a subsequence (we also denote ) that converges strongly to in .
Using the continuity of and , we have
In line with (3.18) and (3.19), and let , we geting for all : Thus, is the weak solution of (3.2) and satisfies
Step 3 (uniqueness and continuous dependence). Let , be in , and setting , we see that satisfies Taking the inner product with of (3.22), using , , and , we obtain By the Gronwall Lemma, we get This is uniqueness and is continuous dependence on initial conditions.
Thanks to Theorem 3.1, and leting , is a semigroup.
4. Global Attractor in
Lemma 4.1. Assume , , and are satisfied. There is a positive constant such that for any bounded subset , there exists such that
Proof. Multiplying (1.1) by in , using , and , we have By virtue of the Poincaré inequality, for some , there holds Furthermore, By the Gronwall Lemma, we get We completed the proof.
Lemma 4.2. Assume , , and hold. Then for any and , there are some and such that whenever and .
Proof. Choose a smooth function with
where , , and there is a constant such that .
Multiplying (1.1) with and integrating on , we obtain Next we deal with the right hand side of (4.9) one by one: According to the condition and the bounded absorbing set in for , it follows that where is independent of . For any given, let
Hence, combining (4.10) with (4.11), when , we conclude that Using and , it yields Since , there exist , such that Then From the assumption , provide , such that Thus combining (4.9), (4.13), (4.16), and (4.17), we finally obtain Furthermore, there holds According to Lemma 2.7, we obtain Thus, we get provided and , we complete the proof.
Lemma 4.3. Assume , , and hold. There is a positive constant such that for any bounded subset , there exists such that
Proof. Multiplying (1.1) by in , we find Using , , and , we have the following estimates: Together with (4.6) and (4.19)–(4.21), by the Poincaré inequality, for some , this yields By the Gronwall Lemma, we get We complete the proof.
Remark 4.4. There is a constant , such that for any bounded subset , when , there holds
Proof. Denote , and we split as
where is a smooth function:
with , and there is a positive constant such that . Then
From Lemma 4.1, we know that as .
For any given, we can choose large enough; by Remark 4.4, we can assume So we conclude that For any bounded set , can be split as Then in line with the property of noncompact measure, it follows that On the other hand, From Lemma 4.3, we get Recall that On account of Remark 4.4, it yields Therefore, we have That is, is -limit compact in .
The authors would like to thank the referee for careful reading of the paper and for his or her many vital comments and suggestions. This work was partly supported by the NSFC (11061030,11101334) and the NSF of Gansu Province (1107RJZA223), in part by the Fundamental Research Funds for the Gansu Universities.
J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Springer, Berlin, Germany, 1972.
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, NY, USA, 1997.
Q. F Ma, S. H. Wang, and C. K. Zhong, “Necessary and sufficient conditions for the existence of global attractors for semigroups and applications,” Indiana University Mathematics Journal, vol. 51, no. 6, pp. 1541–1559, 2002.View at: Google Scholar
J. C. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Disspative Parabolic PDEs and the Theory of Global attractors, Cambridge University Press, Cambridge, Mass, USA, 2001.View at: Publisher Site