Abstract and Applied Analysis

Volume 2013 (2013), Article ID 805978, 6 pages

http://dx.doi.org/10.1155/2013/805978

Research Article

Global Attractor for Partial Functional Differential Equations with State-Dependent Delay

Zhimin He¹ and Bo Du²

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.

Abstract.

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.

1. Introduction

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.

2. Preliminaries

We recall some definitions and results from the integrated semigroup.

Definition 1 (see [6]). Let . A function is said to be an integral solution of (1) if

(i) for ,(ii), (iii).

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 [7]). generates a strongly continuous semigroup on .

Based on the previous abstract results, we give some concrete results for (1); see [6].

Definition 4. Let . For any given with , the function is said to be an integral solution of (1) with initial function at if

Lemma 5 (see [6]). Under the assumptions (H₁)–(H₃), if with , then (1) posseses a unique global integral solution with initial function at , which can be expressed by (7).

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 [8]). 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 [8]). A semigroup , is said to be point dissipative if there is a bounded set that attracts each point of under .

Lemma 8 (see [9]). 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 [10]). 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.

Lemma 11. Assume that the conditions of Lemma 10 are satisfied; furthermore, , where is the constant defined by Lemma 10. Then is point dissipative.

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.

Here, we state our main theorem of this paper, which is an immediate consequence of Lemmas 8, 11, and 12.

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.

Acknowledgments

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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