# The Asymptotic Behavior for Second-Order Neutral Stochastic Partial Differential Equations with Infinite Delay

Authors: Huabin Chen
Journal: Discrete Dynamics in Nature and Society
 (2011)

---

## Abstract

By establishing two Lemmas, the exponential stability and the asymptotical stability for mild solution to the second-order neutral stochastic partial differential equations with infinite delay are obtained, respectively. Our results can generalize and improve some existing ones. Finally, an illustrative example is given to show the effectiveness of the obtained results.


---

## Body

## 1. Introduction

The neutral stochastic differential equations can play an important role in describing many sophisticated dynamical systems in physical, biological, medical, chemical engineering, aero-elasticity, and social sciences [1–3], and the qualitative dynamics such as the existence and uniqueness, stability, and controllability for first-order neutral stochastic partial differential equations with delays have been extensively studied by many authors; see, for example, the existence and uniqueness for neutral stochastic partial differential equations under the non-Lipschitz conditions was investigated by using the successive approximation [4–6]; in [7], Caraballo et al. have considered the exponential stability of neutral stochastic delay partial differential equations by the Lyapunov functional approach; in [8], Dauer and Mahmudov have analyzed the existence of mild solutions to semilinear neutral evolution equations with nonlocal conditions by using the fractional power of operators and Krasnoselski-Schaefer-type fixed point theorem; in [9], Hu and Ren have established the existence results for impulsive neutral stochastic functional integrodifferential equations with infinite delays by means of the Krasnoselskii-Schaefer-type fixed point theorem; some sufficient conditions ensuring the controllability for neutral stochastic functional differential inclusions with infinite delay in the abstract space with the help of the Leray-Schauder nonlinear alterative have been given by Balasubramaniam and Muthukumar in [10]; Luo and Taniguchi, in [11], have studied the asymptotic stability for neutral stochastic partial differential equations with infinite delay by using the fixed point theorem. Very recently, in [12], the author has discussed the exponential stability for mild solution to neutral stochastic partial differential equations with delays by establishing an integral inequality.Although there are many valuable results about neutral stochastic partial differential equations, they are mainly concerned with the first-order case. In many cases, it is advantageous to treat the second-order stochastic differential equations directly rather than to convert them to first-order systems. The second-order stochastic differential equations are the right model in continuous time to account for integrated processes that can be made stationary. For instance, it is useful for engineers to model mechanical vibrations or charge on a capacitor or condenser subjected to white noise excitation through a second-order stochastic differential equations. The studies of the qualitative properties about abstract deterministic second-order evolution equation governed by the generator of a strongly continuous cosine family was proposed in [13–15]. Recently, Mahmudov and McKibben, in [16],  have established the approximate controllability of second-order neutral stochastic evolution equations; the existence and uniqueness for mild solution to second-order neutral impulsive stochastic evolution equations with delay under the non-Lipschitz condition was considered by the successive approximation [17]; Balasubramaniam and Muthukumar in [10] have also discussed the approximate controllability of second-order neutral stochastic distributed implicit functional differential equations with infinite delay; Sakthivel et al. in [18] have studied the asymptotic stability of second-order neutral stochastic differential equations by the fixed point theorem.However, the work done by Sakthivel et al. [18] is mainly in connection with no heredity case. Since many systems arising from realistic models heavily depend on histories [19] (i.e., there is the effect of infinite delay on state equations), there is a real need to proceed with studying the second-order neutral stochastic partial differential equations with infinite delay. Although Sakthivel et al. [18] have applied the fixed point theorem to discuss the asymptotic stability for mild solution to the second-order neutral stochastic partial differential equations, the method proposed by Sakthivel et al. [18] is not suitable for such equations with infinite delay. Obviously, the Lyapunov functional method utilized by Caraballo et al. [7] fails to deal with the asymptotic behavior for mild solution to the second-order neutral stochastic partial differential equations with infinite delay since the mild solutions do not have stochastic differentials. Besides, to the best of author’s knowledge, there is no paper which is involved with the exponential stability and the asymptotic stability for mild solution to second-order neutral stochastic partial differential equations with infinite delay. So, in this paper, we will make the first attempt to close this gap.The format of this work is organized as follows. In Section2, some necessary definitions, notations, and Lemmas used in this paper are introduced; in Section 3, the main results in this paper are given. Finally, an illustrative example is provided to demonstrate the effectiveness of our obtained results.


## 2. Preliminaries

LetX and Y be two real, separable Hilbert spaces and L(Y,X) the space of bounded linear operators from Y to X. For the sake of convenience, we will use the same notation ∥·∥ to denote the norms in X,Y and L(Y,X) when no confusion possibly arises. Let (Ω,ℑ,{ℑt}t≥0,P) be a complete probability space equipped with some filtration ℑt(t≥0) satisfying the usual conditions, that is, the filtration is right continuous and ℑ0 contains all P-null sets. Let C((-∞,0],X) be the space of all bounded and continuous functions φ from (-∞,0] to X with the sup-norm ∥·∥C=sup-∞<θ≤0∥φ(θ)∥, and the space 𝔅 present the family of all ℑt(t≥0)-measurable and C((-∞,0],X)-valued random variables.Letβn(t)(n=1,2,…) be a sequence of real-valued one-dimensional standard Brownian motions mutually independent over (Ω,ℑ,{ℑt}t≥0,P). Set w(t)=∑n=1+∞λnβn(t)en,t≥0, where λn≥0(n=1,2,…) are nonnegative real numbers and {en}(n=1,2,…) is a complete orthonormal basis in Y. Let Q∈L(Y,Y) be an operator defined by Qen=λnen with finite trace trQ=∑n=1+∞λn<+∞. Then, the above Y-valued stochastic process w(t) is called a Q-Wiener process.Definition 2.1 (see [20]).
Letσ∈L(Y,X) and  define
(2.1)‖σ‖L202:=tr⁡(σQσ*)={∑n=1+∞‖λnσen‖2}.
If ∥σ∥L202<+∞, then σ is  called a Q-Hilbert—Schmidt operator, and let L20(Y,X) denote the space of all Q-Hilbert—Schmidt operators σ:Y→X.
Now, for the definition of anX-valued stochastic integral of an L20(Y,X)-valued and ℑt-adapted predictable process Φ(t) with respect to the Q-Wiener process w(t), the readers can refer to [20].
In this paper, we consider the following second-order neutral stochastic partial differential equations with infinite delay:(2.2)d[x′(t)-f0(t,xt)]=[Ax(t)+f1(t,xt)]dt+f2(t,xt)dw(t),t∈[0,+∞),x0(⋅)=φ,x′(0)=ξ,
where φ∈𝔅 and ξ is also an ℑ0-measurable X-valued random variable independent of the Wiener process w(t). A:D(A)⊂X→X is the infinitesimal generator of a strongly continuous cosine family on X; fi:[0,+∞)×𝔅→X(i=0,1), f2:[0,+∞)×𝔅→L20(Y,X) are three approximate mappings. In this sequel, the history xt:(-∞,0]→X, xt(θ)=x(t+θ)(t≥0) belongs to the space 𝔅.
At the end of this section, let us introduce the following Lemmas and definitions that are useful for the development of our results. The one parameter cosine family{C(t):t∈R}⊂L(X,X) satisfying(i)
C(0)=I,(ii)
C(t)x is in continuous in t on R for all x∈R,(iii)
C(t+s)+C(t-s)=2C(t)C(s) for all t,s∈Ris called a strongly continuous cosine family.
The corresponding strongly continuous sine family{S(t):t∈R}⊂L(X) is defined by S(t)x=∫0tC(s)xds,t∈R,x∈X. The generator A:X→X of {C(t):t∈R} is given by Ax=(d2/dt2)C(t)x|t=0 for all x∈D(A)={x∈X:C(·)x∈C2(R,X)}. It is well known that the infinitesimal generator A is a closed, densely defined operator on X. Such cosine and the corresponding sine families and their generators satisfy the following properties.Lemma 2.2 (see [21]).
Suppose thatA is the infinitesimal generator of a cosine family of operators {C(t):t∈R}. Then, the following holds:(i)
there existsM*≥1 and α≥0 such that ∥C(t)∥≤M*eαt and hence ∥S(t)∥≤M*eαt,(ii)
A∫srS(u)xdu=[C(r)-C(u)]x for all 0≤s≤r<+∞,(iii)
there existsN*≥1 such that ∥S(s)-S(r)∥≤N*|∫rseα|θ|dθ| for all 0≤r≤s<+∞.Lemma 2.3 (see [20]).
For  anyr≥1 and  for  arbitrary L20(Y,X)-valued predictable process ϕ(·) such that
(2.3)sup⁡s∈[0,t]E‖∫0sϕ(u)dw(u)‖2r≤Cr(∫0t(E‖ϕ(s)‖L202r)1/rds)r,t∈[0,+∞),
where Cr=(r(2r-1))r.Definition 2.4.
AnX-value  stochastic  process x(t)(t∈R) is  called  a  mild solution of the system (2.2) if(i)
x(t) is  adapted  to ℑt(t≥0) and has  càdlàg path on t≥0 almost surely,(ii)
for  arbitraryt∈[0,+∞), P{ω:∫0t∥x(t)∥2dt<+∞}=1 and  almost  surely(2.4)x(t)=C(t)φ+S(t)(ξ-f0(0,x0))+∫0tC(t-s)f0(s,xs)ds+∫0tS(t-s)f1(s,xs)ds+∫0tS(t-s)f2(s,xs)dw(s),
where x0(·)=φ∈𝔅.Definition 2.5.
The solution of integral equation (2.4) is  said  to  be  exponentially stable  in p(p≥2) moment, if there exists  a  pair  of  positive  constants γ>0 and M1>0 such that
(2.5)E‖x(t)‖p≤M1e-γt,t≥0,p≥2,
for  any  initial  value φ∈𝔅.Definition 2.6.
The solution of integral equation (2.4) is said to be stable in p(p≥2) moment, if for arbitrarily given ε>0, there exists a δ>0 such that E∥ξ∥C2<δ guarantees that
(2.6)E{sup⁡t≥0‖x(t)‖p}<ε,p≥2.Definition 2.7.
The  solution  of  integral equation (2.4) is said to be asymptotically stable in p(p≥2) moment, if it is stable in mean  square and for any φ∈𝔅, a.s., we have
(2.7)lim⁡T→+∞E{sup⁡t≥T∥x(t)∥p}=0,p≥2.


## 3. Main Results

In order to obtain our main results, we need the following assumptions.(H1)
The cosine family of operators{C(t):t≥0} on X and the corresponding sine family {S(t):t≥0} satisfy the conditions ∥C(t)∥≤Me-bt and ∥S(t)∥≤Me-at, t≥0 for some constants M≥1, a>0 and b>0.(H2)
The mappingsfi(i=0,1,2) satisfy the following conditions: there exist three positive constants Ci>0(i=0,1,2) and a function: k:(-∞,0]→[0,+∞) with two important properties: ∫-∞0k(t)dt=1 and ∫-∞0k(t)e-νtdt<+∞(ν>0), such that(3.1)‖fi(t,x)-fi(t,y)‖≤Ci∫-∞0k(θ)‖x(t+θ)-y(t+θ)‖dθ,fi(t,0)=0,i=0,1,‖f2(t,x)-f2(t,y)‖L20≤C2∫-∞0k(θ)‖x(t+θ)-y(t+θ)‖dθ,f2(t,0)=0,
for any x,y∈𝔅 and t≥0.(H3)
5p-1Mp[b-pC0p+a-pC1p+C2pa-(p/2)(2(p-1)p-2)1-(p/2)(p(p-1)2)p/2]<1,(p≥2).Remark 3.1.
Obviously, under the conditions: (H1)-(H2), the existence and uniqueness of mild solution to the system (2.2) can be shown by using the Picard iterative method, and the proof is very similar to that proposed in [4, 17]. Here, we omit it. In particular, the system (2.2) has one unique trivial mild solution when the initial value φ=0.Lemma 3.2.
Forγ1,γ2∈(0,ν], there exist  some  positive  constants: λi>0(i=1,2,3,4) and a function y:(-∞,+∞)→[0,+∞). If (λ3/γ1)+(λ4/γ2)<1, the following inequality:
(3.2)y(t)≤{λ1e-γ1t+λ2e-γ2t+λ3∫0te-γ1(t-s)∫-∞0k(θ)y(s+θ)dθds+λ4∫0te-γ2(t-s)∫-∞0k(θ)y(s+θ)dθds,t≥0,λ1e-γ1t+λ2e-γ2t,t∈(-∞,0],
holds. Then, one has y(t)≤M2e-μt,t∈(-∞,+∞), where μ∈(0,γ1∧γ2) is a positive root of the algebra equation: ((λ3/(γ1-μ))+(λ4/(γ2-μ)))∫-∞0k(θ)e-μθdθ=1 and M2=max{λ1+λ2,(λ1(γ1-μ)/(λ3∫-∞0k(θ)e-μθdθ)),(λ2(γ2-μ)/(λ4∫-∞0k(θ)e-μθdθ))}>0.Proof.
LettingF(λ)=((λ3/(γ1-λ))+(λ4/(γ2-λ)))∫-∞0k(θ)e-λθdθ-1, we have F(0)F(γ-)<0 holds, that is, there exists a positive constant μ∈(0,γ1∧γ2), such that F(μ)=0.
For anyε>0 and letting
(3.3)Cε=max⁡{λ1+λ2+ε,(λ1+ε)γ1-μλ3∫-∞0k(θ)e-μθdθ,(λ2+ε)γ2-μλ4∫-∞0k(θ)e-μθdθ}>0.
Now, in order to show this Lemma, we only claim that (3.2) implies
(3.4)y(t)≤Cεe-μt,t∈(-∞,+∞).
It is easily seen that (3.4) holds for any t∈(-∞,0]. Assume, for the sake of contradiction, that there exists a t1>0 such that
(3.5)y(t)<Cεe-μt,t∈(-∞,t1),y(t1)=Cεe-μt1.
Then, it from (3.2) follows that
(3.6)y(t1)≤λ1e-γ1t1+λ2e-γ2t1+λ3Cε∫0t1e-γ1(t1-s)∫-∞0k(θ)e-μ(s+θ)ds+λ4Cε∫0t1e-γ2(t1-s)∫-∞0k(θ)e-μ(s+θ)dθds=(λ1-Cελ3γ1-μ∫-∞0k(θ)e-μθdθ)e-γ1t1+(λ2-Cελ4γ2-μ∫-∞0k(θ)e-μθdθ)e-γ2t1+(λ3γ1-μ∫-∞0k(θ)e-μθdθ+λ4γ2-μ∫-∞0k(θ)e-μθdθ)Cεe-μt1.
From the definitions of μ and Cε, we obtain
(3.7)λ3γ1-μ∫-∞0k(θ)e-μθdθ+λ4γ2-μ∫-∞0k(θ)e-μθdθ=1,λ1-λ3Cεγ1-μ∫-∞0k(θ)e-μθdθ=λ1-λ3γ1-μ∫-∞0k(θ)e-μθdθ(λ1+ε)γ1-μλ3∫-∞0k(θ)e-μθdθ<0,λ2-Cελ4γ2-μ∫-∞0k(θ)e-μθdθ=λ2-λ4γ2-μ∫-∞0k(θ)e-μθdθ(λ2+ε)γ2-μλ4∫-∞0k(θ)e-μθdθ<0.
Thus, (3.6) yields
(3.8)y(t1)<Cεe-μt1,
which contradicts (3.5), that is, (3.4) holds.
Asε>0 is arbitrarily small, in view of (3.4), it follows
(3.9)y(t)≤M2e-μt,t≥0,
where M2=max{λ1+λ2,(λ1(γ1-μ)/(λ3∫-∞0k(θ)e-μθdθ)), (λ2(γ2-μ)/(λ4∫-∞0k(θ)e-μθdθ))}>0. The proof of this Lemma is completed.Lemma 3.3.
Forγ1,γ2>0, there exist  some  positive  constants: λi>0(i=1,2,3,4) and a function y:(-∞,+∞)→[0,+∞). If (λ3/γ1)+(λ4/γ2)<1, the following inequality:
(3.10)y(t)≤{λ1e-γ1t+λ2e-γ2t+λ3∫0te-γ1(t-s)∫-∞0k(θ)y(s+θ)dθds+λ4∫0te-γ2(t-s)∫-∞0k(θ)y(s+θ)dθds,t≥0,λ1+λ2,t∈(-∞,0],
holds. Then, one has limt→+∞y(t)=0.Proof.
In order to show the conclusion of this Lemma, the proof is divided into two steps as follows.Step 1.
We show that there exists a positive constantdε>0 such that
(3.11)y(t)≤dε,
for any t∈(-∞,+∞). Firstly, for for all ε>0, letting
(3.12)dε=max⁡{λ1+λ2+ε,γ1λ3(λ1+ε),γ2λ4(λ2+ε)}>0.
It is obviously seen that y(t)≤dε for any t∈(-∞,0]. Assumed that there exists a t1>0 such that
(3.13)y(t)<dε,t∈(-∞,t1),y(t1)=dε.
Then, it from (3.10) implies that
(3.14)y(t1)≤λ1e-γ1t1+λ2e-γ2t1+λ3dε∫0t1e-γ1(t1-s)ds+λ4dε∫0t1e-γ2(t1-s)ds=(λ1-λ3dεγ1)e-γ1t1+(λ2-λ4dεγ2)e-γ2t1+(λ3γ1+λ4γ2)dε.
From the definition of dε, we have
(3.15)λ1-λ3dεγ1=λ1-λ3γ1γ1λ3(λ1+ε)<0,λ2-λ4dεγ2=λ2-λ4γ2γ2λ4(λ2+ε)<0.
Thus, (3.14) yields
(3.16)y(t1)<dε,
which contradicts (3.13), that is, (3.11) holds.
Asε>0 is arbitrarily small, in view of (3.11), it follows
(3.17)y(t)≤d,t∈(-∞,+∞),
where d=max{λ1+λ2,(γ1/λ3)λ1,(γ2/λ4)λ2}.Step 2.
We prove thatlimt→+∞y(t)=0.
From the inequality (3.17), it has shown that y(t) is a bounded function defined on the interval (-∞,+∞). Thus, as t→+∞, the upper limit (denoted by l≥0) of y(t) exists, namely,
(3.18)lim⁡¯t→+∞y(t)=l,
the remaining work is to prove l=0.
Supposed thatl>0. From (3.18), there must exist arbitrary positive scalar ε>0 and constant T1>0 such that
(3.19)y(t)<l+ε,∀t≥T1.
On the other hand, since ∫-∞0k(s)ds=1, there must exist T2>0 such that
(3.20)∫-∞-T2k(θ)dθ<ε,∀t≥T2.
LettingT=max{T1,T2}, (3.19) and (3.20) hold for t>T. Thus, it from (3.10) follows that
(3.21)y(t)≤λ1e-γt+λ2e-γ2t+λ3∫0te-γ1(t-s)∫-∞s-Tk(u-s)y(u)duds+λ3∫0te-γ1(t-s)∫s-Tsk(u-s)y(u)duds+λ4∫0te-γ2(t-s)∫-∞s-Tk(u-s)y(u)duds+λ4∫0te-γ2(t-s)∫s-Tsk(u-s)y(u)duds≤[λ1+λ3∫02Teγ1s∫s-Tsk(u-s)y(u)duds]e-γ1t+[λ2+λ4∫02Teγ2s∫s-Tsk(u-s)y(u)duds]e-γ2t+(λ3dγ1+λ4dγ2)ε+(λ3γ1+λ4γ2)(l+ε).
By virtue of (3.18), we have
(3.22)l≤(λ3dγ1+λ4dγ2)ε+(λ3γ1+λ4γ2)(l+ε).
From the arbitrary property ofε, it follows l≤((λ3/γ1)+(λ4/γ2))l, that is,(λ3/γ1)+(λ4/γ2)≥1, which contradicts the condition: (λ3/γ1)+(λ4/γ2)<1. Thus, l=0. The proof of this Lemma is completed.Theorem 3.4.
Suppose  that  the  conditions:   (H1)–(H3) are  satisfied and a,b∈(0,ν], then the mild solution to system (2.2) is exponentially  stable  in p(p≥2) moment.Proof.
In view of (2.4) and the elementary inequality,  we have
(3.23)‖x(t)‖p=‖C(t)φ+S(t)(ξ-f0(0,x0))+∫0tC(t-s)f0(s,xs)ds+∫0tS(t-s)f1(s,xs)ds+∫0tS(t-s)f2(s,xs)dw(s)‖p≤5p-1Mp‖φ‖pe-bt+5p-1Mp‖ξ-f0(0,x0)‖pe-at+5p-1Mpb1-p∫0te-b(t-s)‖f0(s,xs)‖pds+5p-1Mpa1-p∫0te-a(t-s)‖f1(s,xs)‖pds+5p-1‖∫0tS(t-s)f2(s,xs)dw(s)‖p.
From the condition (H2), it from (3.23) concludes that
(3.24)E‖x(t)‖p≤5p-1MpE‖φ‖pe-bt+5p-1MpE‖ξ-f0(0,x0)‖pe-at+5p-1Mpb1-pC0p∫0te-b(t-s)E(∫-∞0k(θ)‖x(s+θ)‖dθ)pds+5p-1Mpa1-pC1p∫0te-a(t-s)E(∫-∞0k(θ)‖x(s+θ)‖dθ)pds+5p-1E‖∫0tS(t-s)f2(s,xs)dw(s)‖p.
From Lemma2.3, we obtain
(3.25)E‖∫0tS(t-s)f2(s,xs)dw(s)‖p≤Mp(∫0t(e-ap(t-s)E‖f2(s,xs)‖L20p)2/pds)p/2(p(p-1)2)p/2=Mp(∫0te-2a(t-s)(E‖f2(s,xs)‖L20p)2/pds)p/2(p(p-1)2)p/2≤Mp(∫0te-(2a(p-1)/(p-2))(t-s)ds)(p/2)-1∫0te-a(t-s)E‖f2(s,xs)‖L20pds(p(p-1)2)p/2≤MpC2p(2a(p-1)p-2)1-(p/2)(p(p-1)2)p/2∫0te-a(t-s)E(∫-∞0k(θ)‖x(s+θ)‖dθ)pds.
Substituting (3.25) into (3.24), it follows
(3.26)E‖x(t)‖p≤5p-1MpE‖φ‖pe-bt+5p-1MpE‖ξ-f0(0,x0)‖pe-at+5p-1Mpb1-pC0p∫0te-b(t-s)∫-∞0k(θ)E‖x(s+θ)‖pdθds+5p-1Mpa1-pC1p∫0te-a(t-s)∫-∞0k(θ)E‖x(s+θ)‖pdθds+5p-1MpC2p(2a(p-1)p-2)1-(p/2)(p(p-1)2)p/2∫0te-a(t-s)∫-∞0k(θ)E‖x(s+θ)‖pdθds.
And it is easily verified that there exists two positive numberM′>0 and M′′>0 such that E∥x(t)∥2≤M′e-bt+M′′e-at, for any t∈(-∞,0].
By Lemma3.2, we can derive that E∥x(t)∥2≤M1e-μt,t∈[0,+∞)(μ∈(0,γ1∧γ2)), where
(3.27)M1={(p(p-1)2)p/25p-1Mp(E‖φ‖p+E‖ξ-f0(0,x0)‖p),5p-1Mpb-pC0p,5p-1Mp(a-pC1p+C2pa-p/2)×(2(p-1)p-2)1-(p/2)(p(p-1)2)p/2}>0.
The proof of this  Theorem is completed.Theorem 3.5.
Suppose that the conditions:   (H1)–(H3) are  satisfied, then the mild solution to system (2.2) is asymptotically  stable  in p(p≥2) moment.Proof.
Similarly, we can obtain the conclusion as follows:(3.28)E‖x(t)‖p≤5p-1MpE‖φ‖pe-bt+5p-1MpE‖ξ-f0(0,x0)‖pe-at+5p-1Mpb1-pC0p∫0te-b(t-s)∫-∞0k(θ)E‖x(s+θ)‖pdθds+5p-1Mpa1-pC1p∫0te-a(t-s)∫-∞0k(θ)E‖x(s+θ)‖pdθds+5p-1MpC2p(2a(p-1)p-2)1-(p/2)(p(p-1)2)p/2∫0te-a(t-s)∫-∞0k(θ)E‖x(s+θ)‖pdθds,
and it is easily verified that there exists two positive number M*′>0 and M*′′>0 such that E∥x(t)∥2≤M*′+M*′′, for any t∈(-∞,0].
By Lemma3.3, we can derive that
(3.29)lim⁡t→+∞E‖x(t)‖p=0.
To obtain the asymptotical stability inp(p≥2)-moment, we need to prove that mild solution of system (2.2) is stable in p(p≥2)-moment. Let ε>0 be given and choose δ>0(δ<ε) such that
(3.30)Ce-(a∧b)t*+Δε<ε,
where
(3.31)C=5p-1MpE‖φ‖p+5p-1MpE‖ξ-f0(0,x0)‖p,Δ=5p-1Mp[b-pC0p+a-pC1p+C2pa-p/2(2(p-1)(p-2))1-(p/2)(p(p-1)2)p/2].
Ifx(t,0,φ) is a mild  solution of system (2.2) with supθ∈(-∞,0]E∥φ(θ)∥p<δ, then x(t) is defined in (2.4). Now, we claim that E∥x(t)∥p<ε for all t≥0. Notice that supθ∈(-∞,0]E∥φ(θ)∥p<ε. If there exists t*>0 such that E∥x(t*)∥p=ε and E∥x(t)∥p<ε, for all t∈(-∞,t*), then it follows from (2.4) that
(3.32)E‖x(t*)‖p≤5p-1Mp(E‖φ‖p+E‖ξ-f0(0,x0)‖p)e-(a∧b)t*+5p-1Mpb1-pC0p∫0t*e-b(t*-s)∫-∞0k(θ)E‖x(s+θ)‖pdθds+5p-1Mpa1-pC1p∫0t*e-a(t*-s)∫-∞0k(θ)E‖x(s+θ)‖pdθds+5p-1MpC2p(2a(p-1)p-2)1-(p/2)(p(p-1)2)p/2∫0t*e-a(t*-s)∫-∞0k(θ)E‖x(s+θ)‖pdθds<Ce-(a∧b)t*+Δε<ε,
which contradicts the definition of t*. This shows that the mild solution of system (2.2) is asymptotically stable in p(p≥2)-moment. The proof of this Theorem is completed.


## 4. An Illustrative Example

In this section, we provide an example to illustrate the obtained results above. LetX=L2[0,π] and Y=R1 with the norm ∥·∥. And let en:=2/πsin(nξ)(n=1,2,…) denote the completed orthonormal basis in X. Let w(t):=∑n=1+∞λnβn(t)en,(λn>0), where {βn(t)} are one-dimensional standard Brownian motions mutually independent on a usual complete probability space (Ω,ℑ,ℑt,P). Define A:X→X by A=(∂2/∂ξ2) with the domain D(A)={h∈X:h,(∂/∂ξ)h are absolutely continuous, (∂2/∂ξ2)h∈X,h(0)=h(π)=0}. Then, (4.1)Ah=∑n=1+∞n2(h,en)en,h∈D(A),
where en,n=1,2,3,…, is also the orthonormal set of eigenvector of A. It is well known that ∥C(t)∥≤exp(-π2t) and ∥S(t)∥≤exp(-π2t),t≥0.Now, we consider the following second-order neutral stochastic partial differential equations with infinite delays:(4.2)d[∂∂tz(t,y)-a0ππ∫-∞0(-θ)-1/2eπ2θz(t+θ,ξ)dθ]=[∂2∂ξ2z(t,ξ)+α1ππ∫-∞0(-θ)-1/2eπ2θz(t+θ,ξ)dθ]dt+α2ππ∫-∞0(-θ)-1/2eπ2θz(t+θ,ξ)dθdw(t),t≥0,ξ∈[0,π],x(t,0)=x(t,π)=0,t≥0,z(θ,ξ)=φ(θ,ξ),θ∈(-∞,0],ξ∈[0,π],∂∂tz(0,ξ)=ζ(ξ),ξ∈[0,π].Define(4.3)f0(t,zt)=α0ππ5∫-∞0(-θ)-1/2eπ2θz(t+θ,ξ)dθ,f0(t,0)=0,f1(t,zt)=α1ππ5∫-∞0(-θ)-1/2eπ2θz(t+θ,ξ)dθ,f1(t,0)=0,f2(t,zt)=α2π5∫-∞0(-θ)-1/2eπ2θz((t+θ),ξ)dθ,f2(t,0)=0,
for any zt∈𝔅.It is easily verified that(4.4)‖f0(t,zt1)-f0(t,zt2)‖≤α0ππ5∫-∞0(-θ)-1/2eπ2θ‖z1(t+θ,ξ)-z2(t+θ,ξ)‖dθ,f0(t,0)=0,‖f1(t,zt1)-f1(t,zt2)‖≤α1ππ5∫-∞0(-θ)-1/2eπ2θ‖z1(t+θ,ξ)-z2(t+θ,ξ)‖dθ,f1(t,0)=0,‖f2(t,zt1)-f2(t,zt2)‖≤α2π5∫-∞0(-θ)-1/2eπ2θ‖z1(t+θ,ξ)-z2(t+θ,ξ)‖dθ,f2(t,0)=0,
for any zt1,zt2∈B.By virtue of Theorems3.4 and 3.5, the exponential stability in p(p≥2)-moment and the asymptotical stability in p(p≥2)-moment for mild solution to system (4.2) are obtained, provided that the following inequality: (4.5)α0p+α1p+α2(2(p-1)p-2)1-(p/2)(p(p-1)2)p/2<5,p≥2,
holds.Remark 4.1.
Obviously, the result in [18] is ineffective in dealing with this example, and our results are more general than those proposed in [18]. Besides, our results can be easily extended to investigate two cases: (1) the exponential stability and the asymptotic stability for the second-order neutral stochastic partial differential equations with infinite delay and impulses and (2) the exponential stability for the second-order neutral stochastic partial differential equations with time-varying delays; the readers can refer to [12, 22]. Here, we omit them.



---

*Source: 584510-2011-08-17.xml*