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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 805978, 6 pages
Global Attractor for Partial Functional Differential Equations with State-Dependent Delay
1College of Science, Zhejiang A and F University, Hangzhou, Zhejiang 311300, China
2Department of Mathematics, Huaiyin Normal University, Huaian, Jiangsu 223300, China
Received 25 March 2013; Accepted 28 June 2013
Academic Editor: Ferenc Hartung
Copyright © 2013 Zhimin He and Bo Du. 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.
This work aims to investigate the existence of global attractors for a class of partial functional differential equations with state-dependent delay. Using the classic theory about global attractors in infinite dimensional dynamical systems, we obtain some sufficient conditions for guaranteeing the existence of a global attractor.
Partial functional differential equations with state-dependent delay appear frequently in applications as models of various phenomena, such as biological, chemical, and physical systems, which are characterized by both spatial and temporal variables. For this reason, the study of this kind of equation has received much attention in recent years. For more details, see for instance [1–5] and the references therein. However, it is worth pointing out that all of the papers mentioned above are mainly devoted to the existence of solutions or mild solutions. The literature related to global attractors is limited.
It is known that the global attractor is a very useful tool, which is valid for more general situations than those for stability to study the asymptotical behavior. In the present paper, we are devoted to investigating the existence of a global attractor for a type of partial functional differential equations with state-dependent delay as follows: where , is the space of continuous functions from to the Banach space , equipped with the uniform norm , and is a linear operator on a Banach space satisfying the following well-known Hille-Yosida condition: there exist and such that where is the resolvent set of . Consider that satisfies the following properties. Let . There exists a continuous and bounded function such that
Consider that satisfies the following properties. (i) For every , the function is strongly measurable. For each , is continuous. There exist a positive constant and a bounded function such that
For every , the history function is defined by In the present paper, we will obtain some sufficient conditions for guaranteeing the existence of a global attractor to (1) with being a Hille-Yosida operator but not necessarily densely defined.
We recall some definitions and results from the integrated semigroup.
Remark 2. Clearly, if is an integral solution of (1), then for . So , which is a necessary condition for the existence of an integral solution.
Let us introduce the part of the operator in which is defined by
Lemma 3 (see ). generates a strongly continuous semigroup on .
Definition 4. Let . For any given with , the function is said to be an integral solution of (1) with initial function at if
According to Remark 2, denote . Then from Lemma 5, for each , we define the following operator on by where is a unique global integral solution of (1) in Lemma 5. Clearly, is a strongly continuous semigroup on .
Definition 6 (see ). An invariant set is said to be a global attractor if is a maximal compact invariant set which attracts each bounded set .
Definition 7 (see ). A semigroup , is said to be point dissipative if there is a bounded set that attracts each point of under .
Lemma 8 (see ). If(i)there is a such that is compact for ,(ii) is point dissipative in ,
then there exists a nonempty global attractor in .
3. The Global Attractor
In this section, we will obtain the existence of a global attractor to (7) by using Lemma 8. For the convenience of the proof, we give some assumptions and lemmas. For -semigroup , there exists positive constant such that is compact for .
Lemma 9 (see ). If where all the functions involved are continuous on , , and , then satisfies
Lemma 10. Assume that assumptions (H1)–(H4) hold. Then, for each , if , there exists a constant such that the integral solution of (1) satisfies the following inequality: where and .
Proof. By , for each , we have
In the following proof, for simplicity, take in ; then Instead of considering the norm directly, we firstly estimate for some constant .
Case 1. For , by (7), we have
Case 2. For , we have Therefore, from (15) and (16), for , we get On the other hand, we have which combines with (17) and yields So we get Using Lemma 9, we have Thus, (12) holds.
Proof. From Lemma 10, we find that for each , since , there exits a such that for , Therefore, attracts each point of , where denotes the open ball in with center 0 and radius .
Now, we show the compactness of the operator .
Lemma 12. Assume that assumptions (H1)–(H5) hold. Then, is compact for .
Proof. Let and let be any bounded sequence of . We will use the Ascoli-Arzelà theorem to show that is precompact in by two steps.
Step 1. Show that for any , the set is precompact. For and , by (8), we have where is the integer solution of (1) with initial function . From and the boundedness of , we know that are precompact. Now, considering the second term in (25), for sufficiently small , we have Note that from Lemma 10, we have By and , we get Therefore, there exist some constants such that which yields where is a compact set. Thus, is precompact.
Step 2. Show the equicontinuity of . Let ; we have which leads to Since the mapping is norm continuous for , for some , put Then Thus By the boundedness of , then Obviously, belongs to a compact subset of ; we have Hence is equicontinuity.
Theorem 13. Assume that assumptions (H1)–(H5) hold. If , then (1) has a nonempty global attractor .
As applications, we give the following example.
Let and . where is a constant, , and satisfies . Define the operator by and with Then satisfies the Hille-Yosida condition in and Moreover, the part of in is the infinitesimal generator of a strongly continuous semigroup on such that According to Theorem 13, if there exist , and such that , then (38) has a global attractor.
This paper is supported by the Natural Science Foundation of Jiangsu Education Office (11KJB110002), Postdoctoral Foundation of Jiangsu (1102096C), and Postdoctoral Foundation of China (2012M511296).
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