Existence and Stability for Stochastic Partial Differential Equations with Infinite Delay

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.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Jing Cui 1 and Litan Yan 2 and Xichao Sun 3

Academic Editor:Weilin Xiao

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; 8 January 2014

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.

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 [Bernoulli]((-∞,0],...) (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 (...,_|·|... ,Y9;·,·_YA;... ) and (...,_|·|... ,_{Y9;·,·YA;...} ) be two separable Hilbert spaces. [Lagrangian (script capital L)](...,...) 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 (Ω,...,{_...t}t...5;0 ,P) be a filtered complete probability space satisfying the usual condition, which means that the filtration is a right continuous increasing family and _...0 contains all P -null sets. Let W=(_Wt)t...5;0 be a ... -valued Wiener process defined on (Ω,...,{_...t}t...5;0 ,P) with covariance operator Q ; that is, [figure omitted; refer to PDF] where Q is a positive, self-adjoint, trace class operator on ... . Denote by ^{[Lagrangian (script capital L)]20} (...,...) the space of all Q -Hilbert-Schmidt operators from ... to ... with the norm [figure omitted; refer to PDF] Let A be the infinitesimal generator of an analytic semigroup S(t) in ... . Then (A-αI) is invertible and generates a bounded analytic semigroup for α>0 large enough. Suppose that 0∈ρ(A) , where ρ(A) denotes the resolvent set of A . Then, for α∈(0,1] , it is possible to define the fractional power operator (-A^)α as a closed linear invertible operator on its domain ...9F;((-A^)α ) . Furthermore, the subspace ...9F;((-A^)α ) is dense in ... and the expression [figure omitted; refer to PDF] defines a norm on _...α :=...9F;((-A^)α ) .

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

Definition 1.

The axioms of the phase space [Bernoulli]((-∞,0],...) (denoted by [Bernoulli] simply) are established for _...0 -measurable, continuous functions mapping (-∞,0] into ... endowed with a norm _{||·||[Bernoulli]} , and [Bernoulli] satisfies the following axioms:

(_H1 ): If x:(-∞,T][arrow right]... , T>0 , is continuous on [0,T] and _x0 ∈[Bernoulli] , then, for every t∈[0,T] , the following properties hold:

(1) _xt :=x(t+·)∈[Bernoulli] ;

(2) |x(t)|...4;H_{||xt ||[Bernoulli]} ;

(3) _{||xt ||[Bernoulli]} ...4;M(t)_{sup0...4;s...4;t} |x(s)|+N(t)_{||x0 ||[Bernoulli]} , where H>0 is a constant, M,N:[0,+∞)[arrow right][1,+∞) are independent of x(·) , and M is continuous and N is locally bounded.

(_H2 ): The space [Bernoulli] is complete.

Remark 2.

For convenience, we replace condition (3) in (_H1 ) by

(^{3[variant prime]} ) : _{||xt ||[Bernoulli]} ...4;_{sup0...4;s...4;t} |x(s)|+N_{||x0 ||[Bernoulli]} , where N=_{sup0...4;s...4;T} N(s) .

Example 3.

Let (...,_|·|... ) be a Banach space and [straight phi]∈C((-∞,0],...) . Assume that h:(-∞,0][arrow right](0,+∞) is a continuous function with l=^∫-∞0 ...h(t)dt<+∞ . Define [figure omitted; refer to PDF] If _...9E;h is endowed with the norm [figure omitted; refer to PDF] then (_...9E;h ,_{||·||...9E;h} ) is a Banach space [14] and satisfies the axioms in Definition 1 with M=l , N=1 .

Consider the following NSPDEs with infinite delay in the form: [figure omitted; refer to PDF] where _xt ={x(t+θ):-∞<θ...4;0} can be regarded as a [Bernoulli] -valued stochastic process. Assume that [figure omitted; refer to PDF] are appropriate mappings specified later. The initial value [varphi]={[varphi](θ):-∞<θ...4;0} is an _...0 -measurable [Bernoulli] -valued random variable independent of W with finite second moment.

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

Definition 4.

An _...t -adapted ... -valued stochastic process x(t) defined on -∞<t...4;T , 0...4;T<∞ is called the mild solution for (6) if

(a) x(t) is continuous and {_xt :0...4;t...4;T} is a [Bernoulli] -valued stochastic process;

(b) ^∫0T ...|x(u)^|...2 du<∞ almost surely;

(c) for arbitrary t∈[0,T] , x(t) satisfies the following integral equation: [figure omitted; refer to PDF]

We denote by ^{[physics M-matrix]2} ((-∞,T],...) the space of all ... -valued, continuous, and _...t -adapted processes x={x(t),-∞<t...4;T} such that

(1) _x0 =[varphi]∈[Bernoulli] and x(t) is continuous on [0,T] ;

(2) for all x∈^{[physics M-matrix]2} ((-∞,T],...) , [figure omitted; refer to PDF]

It is obvious that the space ^{[physics M-matrix]2} ((-∞,T],...) 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.

(_A1 ): Assume that A is the infinitesimal generator of an analytic semigroup of bounded linear operators {S(t),t...5;0} in ... , satisfying [figure omitted; refer to PDF] for some γ>0 .

(_A2 ): There exist some constants α∈(1/2,1] and _MG >0 such that, for any ξ,[varphi]∈[Bernoulli] , t...5;0 , we have G(t,ξ),G(t,[varphi])∈...9F;((-A^)α ) , and [figure omitted; refer to PDF] we further assume that G(t,0)...1;0 , for all t...5;0 .

(_A3 ):

(a) There exists a function F:^...+ ×^...+ [arrow right]^...+ such that F(t,u) is locally integrable in t for any fixed u...5;0 and is continuous, nondecreasing, and concave in u for each fixed t∈[0,T] . Moreover, for any t∈[0,T] , x∈[Bernoulli] , the following inequality holds: [figure omitted; refer to PDF]

(b) For any C>0 , the differential equation [figure omitted; refer to PDF] has a global solution for any initial value _u0 .

(_A4 ) : (global conditions)

(a) There exists a function Z:^...+ ×^...+ [arrow right]^...+ such that Z(t,u) is locally integrable in t for any fixed u...5;0 and is continuous nondecreasing and concave in u for each fixed t∈[0,T] , Z(t,0)=0 for any t∈[0,T] . Moreover, for any t∈[0,T] , x,y∈[Bernoulli] , the following inequality holds: [figure omitted; refer to PDF]

(b) For any constant D>0 , if a nonnegative function u(t) satisfies that [figure omitted; refer to PDF] then u(t)...1;0 for any t∈[0,T] .

(^{A4[variant prime]} ): (local conditions)

(a) For any integer N>0 , there exists a function _ZN :^...+ ×^...+ [arrow right]^...+ such that _ZN (t,u) is locally integrable in t for any fixed u...5;0 and is continuous nondecreasing and concave in u for each fixed t∈[0,T] , _ZN (t,0)=0 for any t∈[0,T] . Moreover, for any t∈[0,T] , x,y∈[Bernoulli] with _{||x||[Bernoulli]} ...4;N , _{||y||[Bernoulli]} ...4;N , the following inequality holds: [figure omitted; refer to PDF]

(b) For any constant D>0 , if a nonnegative function u(t) satisfies that [figure omitted; refer to PDF] then u(t)...1;0 for any t∈[0,T] .

The following lemma that appeared in [19] is useful.

Lemma 5.

Under the assumption of (_A1 ) , for any 0<β...4;1 , the following equality holds: [figure omitted; refer to PDF] and there exists a positive constant _Mβ such that, for any t>0 , [figure omitted; refer to PDF]

Lemma 6 (Liu [2]).

Let T>0 . Suppose A generates a pseudocontraction _C0 -semigroup S(t) . That is, ||S(t)||...4;^eαt , t...5;0 , for some α∈... . Then the process ^WA[varphi] (t)=^∫0t ...S(t-s)[varphi](s)dW(s) has a continuous modification and there exists a constant K>0 such that [figure omitted; refer to PDF]

Theorem 7.

Let (_A1 )-(_A4 ) hold. Then the system (6) admits a unique mild solution x(t)∈^{[physics M-matrix]2} ((-∞,T],...) provided that [figure omitted; refer to PDF]

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 (_A1 )-(_A3 ) , (^{A4[variant prime]} ) and (21) hold. Then the system (6) admits a unique mild solution x(t)∈^{[physics M-matrix]2} ((-∞,T],...) provided that [figure omitted; refer to PDF]

Proof.

Let N be a positive integer and _T0 ∈(0,T) . We introduce the sequence of the functions {^bN (t,u)} and {^σN (t,u)} , (t,u)∈[0,T]×[Bernoulli] as follows: [figure omitted; refer to PDF] Then the functions {^bN (t,u)} and {^σN (t,u)} satisfy assumption (_A3 ) , and for any x,y∈[Bernoulli] , t∈[0,T] , the following inequality holds: [figure omitted; refer to PDF] As a consequence of Theorem 7, there exist the unique mild solutions ^xN (t) and ^xN+1 (t) , respectively, to the following integral equations: [figure omitted; refer to PDF] Define the stopping time [figure omitted; refer to PDF] In view of Hölder's inequality (25), we obtain [figure omitted; refer to PDF] where we have used the fact that for 0...4;u...4;_τN , [figure omitted; refer to PDF] Note that [figure omitted; refer to PDF] By assumption (_A2 ) , we have [figure omitted; refer to PDF] By virtue of Lemma 5, Hölder's inequality together with assumption (_A2 ) we have [figure omitted; refer to PDF] Employing assumption (^{A4[variant prime]} ) , Hölder's inequality, and Jensen's inequality, it follows that [figure omitted; refer to PDF] Combining Lemma 6 with Jensen's inequality, there exists a positive constant K such that [figure omitted; refer to PDF] Therefore, for all t∈[0,_T0 ] , we have [figure omitted; refer to PDF] The assumption (^{A4[variant prime]} ) indicates that [figure omitted; refer to PDF] Thus, for a.e. ω , [figure omitted; refer to PDF] Note that for each ω∈Ω , there exists an _N0 (ω)>0 such that 0<_T0 ...4;_τN0 . Define x(t) by [figure omitted; refer to PDF] Since x(t⋀_τN )=^xN (t⋀_τN ) , it holds that [figure omitted; refer to PDF] Taking N[arrow right]∞ , we have [figure omitted; refer to PDF] 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 ^x[varphi] or similar notations the unique mild solution of (6) with the initial data [varphi] .

Definition 10.

The mild solution ^x[varphi] of (6) is said to be exponentially asymptotically stable in mean square if there exist a pair of positive constants α and β such that, for any mild solution ^{y[straight phi]} of (6), [figure omitted; refer to PDF]

We need the following assumptions before we proceed further.

(_A5 ) : For any x,y∈[Bernoulli] , there exist some positive constants _Qb and _Qσ such that [figure omitted; refer to PDF] for all t...5;0 .

(_A6 ) : There exist some constant α∈(1/2,1] and a continuous function μ such that for any ξ,[varphi]∈[Bernoulli] , t...5;0 , [figure omitted; refer to PDF] where μ:^...+ [arrow right]^...+ satisfies μ(t)...4;_QG^e-δt , _QG >0 , δ>γ>0 .

The following lemma is needed to consider our results.

Lemma 11 (see [16]).

Assume that the semigroup {S(t),t...5;0} is exponentially stable; that is, ||S(t)||...4;M^e-γt , t...5;0 , for some M,γ>0 . Then, for any _...t -adapted predictable process Φ with ^∫0t ...E|Φ(s)^{|[Lagrangian (script capital L)]20 2} ds<∞ , t...5;0 , the following inequality holds: [figure omitted; refer to PDF]

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

Theorem 12.

Let ^x[varphi] and ^{y[straight phi]} be two mild solutions of (6) with the initial data [varphi] and [straight phi] , respectively. Assume that (_A1 ) , (_A5 )-(_A6 ) , and 5^{QG2||(-A)-α ||2} <1 hold. Then [figure omitted; refer to PDF] where β = 10(1+^{QG2||(-A)-α ||2} )/(1-5^{QG2||(-A)-α ||2} ) , σ-=(5^{M1-α2QG2γ1-2α} Γ(2α-1)+5^γ-1_Qb +20_Qσ ) / (1-5^{QG2||(-A)-α ||2} ) .

Proof.

Since by assumption, x=^x[varphi] and y=^{y[straight phi]} are solutions of (6), we have [figure omitted; refer to PDF] Then, [figure omitted; refer to PDF] We now estimate the terms on the right-hand side of (46). From assumption (_A1 ) and (_A6 ) , we obtain [figure omitted; refer to PDF] Noting that δ>γ>0 and [figure omitted; refer to PDF] we have [figure omitted; refer to PDF] Standard computations involving Hölder's inequality and Lemma 5 yield that [figure omitted; refer to PDF] Combing assumption (_A5 ) with Hölder's inequality it follows that [figure omitted; refer to PDF] Applying first Lemma 7.2 in [4] and then Lemma 11 we obtain [figure omitted; refer to PDF] Define [figure omitted; refer to PDF] Recalling (46), from (47) to (52) we derive that [figure omitted; refer to PDF] where β = 10(1+^{QG2||(-A)-α ||2} )/(1-5^{QG2||(-A)-α ||2} ) , σ- = (5^{M1-α2QG2γ1-2α} Γ(2α-1)+5^γ-1_Qb +20_Qσ ) / (1-5^{QG2||(-A)-α ||2} ) .

Invoking Gronwall's Lemma we get [figure omitted; refer to PDF] This completes the proof.

Corollary 13.

Suppose that all the conditions of Theorem 12 hold. Then for any mild solutions ^x[varphi] and ^{y[straight phi]} of (6) [figure omitted; refer to PDF] where β and σ- are defined in Theorem 12, γ is the constant in assumption (_A1 ) . Consequently, if γ>σ- , then the mild solution ^x[varphi] is exponentially asymptotically stable in mean square.

Corollary 14.

Suppose that all the conditions of Theorem 12 hold. If G(t,0)=b(t,0)=σ(t,0)=0 , for all t...5;0 and γ>σ- , 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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