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Abstract and Applied Analysis

Volume 2014 (2014), Article ID 235937, 8 pages

http://dx.doi.org/10.1155/2014/235937

## Existence and Stability for Stochastic Partial Differential Equations with Infinite Delay

^{1}Department of Mathematics, Anhui Normal University, 1 East Beijing Road, Wuhu 241000, China^{2}Department of Mathematics, Donghua University, 2999 North Renmin Road, Songjiang, Shanghai 201620, China^{3}Department of Mathematics and Physics, Bengbu College, Bengbu 233030, China

Received 9 November 2013; Accepted 17 December 2013; Published 8 January 2014

Academic Editor: Weilin Xiao

Copyright © 2014 Jing Cui et al. 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

We consider a class of neutral stochastic partial differential equations with infinite delay in real separable Hilbert spaces. We derive the existence and uniqueness of mild solutions under some local Carathéodory-type conditions and also exponential stability in mean square of mild solutions as well as its sample paths. Some known results are generalized and improved.

#### 1. Introduction

The theory of stochastic partial differential equations (SPDEs) has recently become an important area of investigation stimulated by its numerous applications to problems arising in natural and social sciences. There is much current interest in studying qualitative properties for SPDEs (see, e.g., Caraballo et al. [1], Liu [2], Luo and Liu [3], Da Prato and Zabczyk [4], Peszat and Zabczyk [5], Wang and Zhang [6], and references therein). We would like to mention that the stochastic partial functional differential equations (SPFDEs) have been considered intensively. For example, under the global Lipschitz and linear growth conditions, Govindan [7] showed by the stochastic convolution the existence, uniqueness, and almost sure exponential stability of neutral SPDEs with finite delays; Taniguchi et al. [8] considered the existence and uniqueness of mild solutions to SPDEs with finite delays by Banach fixed point theorem; while by imposing a so-called Carathéodory condition on the nonlinearities, Jiang and Shen [9] studied the existence and uniqueness of mild solutions for neutral SPFDEs by successive approximation; Samoilenko et al. [10] investigated the existence, uniqueness, and controllability results for neutral SPFDEs.

On the other hand, it is well known that infinite delay (stochastic) equations have wide application in many areas [11, 12]. However, as for neutral SPDEs with infinite delay, as far as we know, there exist only a few results about the existence and asymptotic behavior of mild solutions. We mention here the recent papers by Ren and Sun [13] and Li and Liu [14] considering the existence of solutions of second-order stochastic evolution equations and neutral stochastic differential inclusions with infinite delay, respectively; Cui and Yan [15] investigated the existence and longtime behavior of mild solutions for a class of neutral stochastic partial differential equations with infinite delay in distribution, while Taniguchi [16] concerned the existence and asymptotic behavior for stochastic evolution equations with infinite delay.

In this paper, inspired by the aforementioned papers [13, 15], we consider a class of neutral stochastic partial differential equations (NSPDEs) with infinite delay. The space (see Section 2) with some axioms proposed by Hale and Kato [17] is employed as our phase space. We study the existence and uniqueness of mild solutions to SPDEs with infinite delay under some local Carathéodory conditions with the non-Lipschitz conditions in Bao and Hou [18] and Jiang and Shen [9] being regarded as special cases and investigate the longtime behavior of mild solutions as well.

The structure of this paper is as follows. In the next section, we introduce some necessary notations and preliminaries. The existence and uniqueness of mild solutions are discussed in Section 3. The exponential stability in mean square of mild solutions as well as its sample paths are presented in Section 4.

#### 2. Preliminaries

For more details on this section, we refer to Da Prato and Zabczyk [4] and Pazy [19]. Let and be two separable Hilbert spaces. stands for the set of all linear bounded operators from into , equipped with the usual operator norm . In this paper, we use the symbol to denote norms of operators regardless of the spaces involved when no confusion possibly arises.

Let be a filtered complete probability space satisfying the usual condition, which means that the filtration is a right continuous increasing family and contains all -null sets. Let be a -valued Wiener process defined on with covariance operator ; that is, where is a positive, self-adjoint, trace class operator on . Denote by the space of all -Hilbert-Schmidt operators from to with the norm Let be the infinitesimal generator of an analytic semigroup in . Then is invertible and generates a bounded analytic semigroup for large enough. Suppose that , where denotes the resolvent set of . Then, for , it is possible to define the fractional power operator as a closed linear invertible operator on its domain . Furthermore, the subspace is dense in and the expression defines a norm on .

Throughout this paper, we will employ an axiomatic definition of the phase space introduced by Hale and Kato [17].

*Definition 1. *The axioms of the phase space (denoted by simply) are established for -measurable, continuous functions mapping into endowed with a norm , and satisfies the following axioms:)If , , is continuous on and , then, for every , the following properties hold:(1);(2);(3), where is a constant, are independent of , and is continuous and is locally bounded.()The space is complete.

*Remark 2. *For convenience, we replace condition in by, where .

*Example 3. *Let be a Banach space and . Assume that is a continuous function with . Define
If is endowed with the norm
then is a Banach space [14] and satisfies the axioms in Definition 1 with , .

Consider the following NSPDEs with infinite delay in the form: where can be regarded as a -valued stochastic process. Assume that are appropriate mappings specified later. The initial value is an -measurable -valued random variable independent of with finite second moment.

Now we present the definition of the mild solution for (6).

*Definition 4. *An -adapted -valued stochastic process defined on , is called the mild solution for (6) if(a) is continuous and is a -valued stochastic process;(b) almost surely;(c)for arbitrary , satisfies the following integral equation:

We denote by the space of all -valued, continuous, and -adapted processes such that(1) and is continuous on ;(2)for all , It is obvious that the space is a Banach space with the norm defined by (9).

#### 3. The Existence and Uniqueness Theorem

In this section, we present our main results on the existence and uniqueness of the mild solution of (6). We first introduce the following assumptions.()Assume that is the infinitesimal generator of an analytic semigroup of bounded linear operators in , satisfying for some .()There exist some constants and such that, for any , , we have , and we further assume that , for all .()(a)There exists a function such that is locally integrable in for any fixed and is continuous, nondecreasing, and concave in for each fixed . Moreover, for any , , the following inequality holds: (b)For any , the differential equation has a global solution for any initial value .(global conditions)(a)There exists a function such that is locally integrable in for any fixed and is continuous nondecreasing and concave in for each fixed , for any . Moreover, for any , , the following inequality holds: (b)For any constant , if a nonnegative function satisfies that then for any .()(local conditions)(a)For any integer , there exists a function such that is locally integrable in for any fixed and is continuous nondecreasing and concave in for each fixed , for any . Moreover, for any , with , , the following inequality holds: (b)For any constant , if a nonnegative function satisfies that then for any . The following lemma that appeared in [19] is useful.

Lemma 5. *Under the assumption of , for any , the following equality holds:
**
and there exists a positive constant such that, for any ,
*

Lemma 6 (Liu [2]). *Let . Suppose generates a pseudocontraction -semigroup . That is, , , for some . Then the process has a continuous modification and there exists a constant such that
*

Theorem 7. *Let hold. Then the system (6) admits a unique mild solution provided that
*

*Proof. *The proof is similar to the proof of Theorem 3.1 in Jiang and Shen [9], we omit the detail.

We now state our main theorem in this section.

Theorem 8. *Let , and (21) hold. Then the system (6) admits a unique mild solution provided that
*

*Proof. *Let be a positive integer and . We introduce the sequence of the functions and , as follows:
Then the functions and satisfy assumption , and for any , , the following inequality holds:
As a consequence of Theorem 7, there exist the unique mild solutions and , respectively, to the following integral equations:
Define the stopping time
In view of Hölder’s inequality (25), we obtain
where we have used the fact that for ,
Note that
By assumption , we have
By virtue of Lemma 5, Hölder’s inequality together with assumption we have
Employing assumption , Hölder’s inequality, and Jensen’s inequality, it follows that
Combining Lemma 6 with Jensen’s inequality, there exists a positive constant such that
Therefore, for all , we have
The assumption indicates that
Thus, for a.e. ,
Note that for each , there exists an such that . Define by
Since , it holds that
Taking , we have
which completes the proof.

*Remark 9. *We obtain the existence and uniqueness of mild solution to (6) under local Carathéodory conditions with the non-Lipschitz conditions in [9, 18] being regarded as special cases, which makes it more feasible that the conditions of solution can be satisfied.

#### 4. Exponential Stability

In this section, we consider the exponential stability in mean square and almost sure exponential stability of the mild solutions of (6). For the sake of brevity, we denote by or similar notations the unique mild solution of (6) with the initial data .

*Definition 10. *The mild solution of (6) is said to be exponentially asymptotically stable in mean square if there exist a pair of positive constants and such that, for any mild solution of (6),

We need the following assumptions before we proceed further.For any , there exist some positive constants and such that for all .There exist some constant and a continuous function such that for any , , where satisfies , , .The following lemma is needed to consider our results.

Lemma 11 (see [16]). *Assume that the semigroup is exponentially stable; that is, , , for some . Then, for any -adapted predictable process with , , the following inequality holds:
*

Now, we state our main result of this section on the stability in mean square.

Theorem 12. *Let and be two mild solutions of (6) with the initial data and , respectively. Assume that , , and hold. Then
**
where , .*

*Proof. *Since by assumption, and are solutions of (6), we have
Then,
We now estimate the terms on the right-hand side of (46). From assumption and , we obtain
Noting that and
we have
Standard computations involving Hölder’s inequality and Lemma 5 yield that
Combing assumption with Hölder’s inequality it follows that
Applying first Lemma 7.2 in [4] and then Lemma 11 we obtain
Define
Recalling (46), from (47) to (52) we derive that
where , .

Invoking Gronwall’s Lemma we get
This completes the proof.

Corollary 13. *Suppose that all the conditions of Theorem 12 hold. Then for any mild solutions and of (6)
**
where and are defined in Theorem 12, is the constant in assumption . Consequently, if , then the mild solution is exponentially asymptotically stable in mean square.*

Corollary 14. *Suppose that all the conditions of Theorem 12 hold. If , for all and , then the trial solution of (6) is exponentially asymptotically stable in mean square.*

Finally, we consider the stability of sample path.

Theorem 15. *Suppose that all the conditions of Corollary 14 hold, then the sample path of the trial solution of (6) is exponentially asymptotically stable.*

*Proof. *The method is similar to the proof of Theorem 5.1 in [7], we omit it here.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The authors would like to express their sincere gratitude to the editor and the anonymous referees for their valuable comments and error corrections. Jing Cui is partially supported by the National Natural Science Foundation of China (11326171, 11271020), the Natural Science Foundation of Anhui Province (1208085MA11, 1308085QA14), the Key Natural Science Foundation of Anhui Educational Committee (KJ2013A133), and the PhD Start-up Fund of Anhui Normal University. Litan Yan is partially supported by National Natural Science Foundation of China (11171062) and Innovation Program of Shanghai Municipal Education Commission (12ZZ063). Xichao Sun is partially supported by the Natural Science Foundation of Anhui Province (1408085QA10).

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