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

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Volume 2011, Article ID 584510,15pages doi:10.1155/2011/584510

Research Article

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

Infinite Delay

Huabin Chen

Department of Mathematics, Nanchang University, Jiangxi, Nanchang 330031, China

Correspondence should be addressed to Huabin Chen,chb [email protected] Received 17 March 2011; Accepted 19 May 2011

Academic Editor: Her-Terng Yau

Copyrightq2011 Huabin Chen. 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.

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.

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 sciences1–3, 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 approximation4–6; in 7, Caraballo et al. have considered the exponential stability of neutral stochastic delay partial differential equations by the Lyapunov functional approach; in8, 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

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infinite delay in the abstract space with the help of the Leray-Schauder nonlinear alterative have been given by Balasubramaniam and Muthukumar in10; Luo and Taniguchi, in11, have studied the asymptotic stability for neutral stochastic partial diﬀerential equations with infinite delay by using the fixed point theorem. Very recently, in12, 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, in16, 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 10have 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 Section 2, some necessary definitions, notations, and Lemmas used in this paper are introduced; inSection 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 andY be two real, separable Hilbert spaces and LY, Xthe space of bounded linear operators fromY toX. For the sake of convenience, we will use the same notation · to

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denote the norms inX, YandLY, Xwhen no confusion possibly arises. LetΩ,I,{It}_t≥0, P be a complete probability space equipped with some filtration Itt ≥ 0 satisfying the usual conditions, that is, the filtration is right continuous and I0 contains all P-null sets.

LetC−∞,0, Xbe the space of all bounded and continuous functions ϕ from−∞,0 to X with the sup-norm · C sup_{−∞<θ≤0}ϕθ, and the spaceB present the family of all It t≥0-measurable andC−∞,0, X-valued random variables.

Letβnt n 1,2, . . .be a sequence of real-valued one-dimensional standard Brown- ian motions mutually independent overΩ,I,{It}_t≥0, P. Setwt _∞

n 1

λnβnten, t≥0, whereλn≥0 n 1,2, . . .are nonnegative real numbers and{en}n 1,2, . . .is a complete orthonormal basis inY. LetQ∈LY, Ybe an operator defined byQen λnenwith finite trace

trQ _∞

n 1λn <∞. Then, the aboveY-valued stochastic processwtis called aQ-Wiener process.

Definition 2.1see20. Letσ∈LY, Xand define σ²_L0

2: trσQσ^∗ _∞

n 1

λnσen²

. 2.1

Ifσ²_L0

2 < ∞, thenσ is called aQ-Hilbert—Schmidt operator, and letL⁰₂Y, Xdenote the space of allQ-Hilbert—Schmidt operatorsσ:Y → X.

Now, for the definition of anX-valued stochastic integral of anL⁰₂Y, X-valued and It-adapted predictable processΦtwith respect to theQ-Wiener processwt, the readers can refer to20.

In this paper, we consider the following second-order neutral stochastic partial diﬀerential equations with infinite delay:

d

xt−f0t, xt

Axt f1t, xt dtf2t, xtdwt, t∈0,∞, x0· ϕ,

x0 ξ,

2.2

where ϕ ∈ B and ξ is also an I0-measurable X-valued random variable independent of the Wiener processwt.A : DA ⊂ X → X is the infinitesimal generator of a strongly continuous cosine family on X; fi : 0,∞×B → X i 0,1, f2 : 0,∞×B → L⁰₂Y, X are three approximate mappings. In this sequel, the history xt : −∞,0 → X, xtθ xtθ t≥0belongs to the spaceB.

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{Ct: t ∈ R} ⊂LX, Xsatisfying

iC0 I,

iiCtxis in continuous intonRfor allx∈R, iiiCts Ct−s 2CtCsfor allt, s∈R is called a strongly continuous cosine family.

The corresponding strongly continuous sine family{St : t∈ R} ⊂LXis defined byStx _t

0Csx ds, t∈R, x∈X. The generatorA:X → Xof{Ct :t∈R}is given by

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Ax d²/dt²Ctx|t 0for allx∈DA {x∈X : C·x∈C²R, X}. It is well known that the infinitesimal generatorAis a closed, densely defined operator onX. Such cosine and the corresponding sine families and their generators satisfy the following properties.

Lemma 2.2see21. Suppose thatAis the infinitesimal generator of a cosine family of operators {Ct:t∈R}. Then, the following holds:

ithere existsM^∗≥1 andα≥0 such thatCt ≤M^∗e^αtand henceSt ≤M^∗e^αt, iiA_r

sSux du Cr−Cuxfor all 0≤s≤r <∞, iiithere existsN^∗≥1 such thatSs−Sr ≤N^∗|_s

r e^α|θ|dθ|for all 0≤r≤s <∞.

Lemma 2.3see20. For anyr≥1 and for arbitraryL⁰₂Y, X-valued predictable processφ·such that

sup

s∈0,tE _s

0

φudwu

^2r ≤Cr

_t

0

Eφs^2r

L⁰₂

1/r

ds _r

, t∈0,∞, 2.3

whereCr r2r−1^r.

Definition 2.4. An X-value stochastic process xt t ∈ R is called a mild solution of the system2.2if

ixtis adapted toIt t≥0and has c`adl`ag path ont≥0 almost surely, iifor arbitraryt∈0,∞,P{ω: _t

0xt²dt <∞} 1 and almost surely

xt CtϕSt

ξ−f00, x0

_t

0

Ct−sf0s, xsds

_t

0

St−sf1s, xsds _t

0

St−sf2s, xsdws,

2.4

wherex0· ϕ∈B.

Definition 2.5. The solution of integral equation 2.4 is said to be exponentially stable in pp≥2moment, if there exists a pair of positive constantsγ >0 andM1>0 such that

Ext^p≤M1e^−γt, t≥0, p≥2, 2.5

for any initial valueϕ∈B.

Definition 2.6. The solution of integral equation2.4is said to be stable inpp≥2moment, if for arbitrarily givenε >0, there exists aδ >0 such thatEξ²_C < δguarantees that

E

sup

t≥0xt^p

< ε, p≥2. 2.6

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Definition 2.7. The solution of integral equation2.4is said to be asymptotically stable in pp≥2moment, if it is stable in mean square and for anyϕ∈B, a.s., we have

Tlim→∞E

sup

t≥Txt^p

0, p≥2. 2.7

3. Main Results

In order to obtain our main results, we need the following assumptions.

H1The cosine family of operators {Ct : t ≥ 0} onX and the corresponding sine family{St:t≥0}satisfy the conditionsCt ≤Me^−btandSt ≤Me^−at,t≥0 for some constantsM≥1,a >0 andb >0.

H2The mappings fi i 0,1,2 satisfy the following conditions: there exist three positive constantsCi > 0i 0,1,2and a function:k : −∞,0 → 0,∞with two important properties:₀

−∞ktdt 1 and₀

−∞kte^−νtdt <∞ ν >0, such that fit, x−fi

t, y≤Ci

₀

−∞kθxtθ−ytθdθ, fit,0 0, i 0,1, f2t, x−f2

t, y

L⁰₂≤C2

₀

−∞kθxtθ−ytθdθ, f2t,0 0,

3.1

for anyx, y∈Bandt≥0.

H35^p−1M^p

b^−pC₀^pa^−pC^p₁C^p₂a^−p/2

2p−1 p−2

_1−p/2pp−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 system2.2can be shown by using the Picard iterative method, and the proof is very similar to that proposed in4,17. Here, we omit it. In particular, the system2.2has one unique trivial mild solution when the initial valueϕ 0.

Lemma 3.2. Forγ1, γ2 ∈ 0, ν, there exist some positive constants: λi > 0i 1,2,3,4and a functiony:−∞,∞ → 0,∞. Ifλ3/γ1 λ4/γ2<1, the following inequality:

yt≤

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

λ1e^−γ¹^tλ2e^−γ²^tλ3

_t

0

e^−γ¹^t−s ₀

−∞kθysθdθ ds λ4

_t

0

e^−γ²^t−s ₀

−∞kθysθdθ ds, t≥0,

λ1e^−γ¹^tλ2e^−γ²^t, t∈−∞,0,

3.2

holds. Then, one hasyt≤ M2e^−μt, t ∈−∞,∞, whereμ∈0, γ1∧γ2is a positive root of the algebra equation:λ3/γ1−μλ4/γ2−μ₀

−∞kθe^−μθdθ 1 andM2 max{λ1λ2,λ1γ1− μ/λ3

₀

−∞kθe^−μθdθ,λ2γ2−μ/λ4

₀

−∞kθe^−μθdθ}>0.

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Proof. LettingFλ λ3/γ1−λ λ4/γ2−λ₀

−∞kθe^−λθdθ−1, we haveF0Fγ−<0 holds, that is, there exists a positive constantμ∈0, γ1∧γ2, such thatFμ 0.

For anyε >0 and letting

Cε max

⎧⎨

⎩λ1λ2ε,λ1ε γ1−μ λ3

₀

−∞kθe^−μθdθ,λ2ε γ2−μ λ4

₀

−∞kθe^−μθdθ

⎫⎬

⎭>0. 3.3

Now, in order to show this Lemma, we only claim that3.2implies

yt≤Cεe^−μt, t∈−∞,∞. 3.4

It is easily seen that3.4holds for anyt∈−∞,0. Assume, for the sake of contradiction, that there exists at1>0 such that

yt< Cεe^−μt, t∈−∞, t1, yt1 Cεe^−μt¹. 3.5

Then, it from3.2follows that

yt1≤λ1e^−γ¹^t¹λ2e^−γ²^t¹λ3Cε

_t₁

0

e^−γ¹^t¹^−s ₀

−∞kθe^−μsθds λ4Cε

_t₁

0

e^−γ²^t¹^−s ₀

−∞kθe^−μsθdθ ds

λ1− Cελ3

γ1−μ ₀

e^−γ¹^t¹

λ2− Cελ4

γ2−μ ₀

e^−γ²^t¹

λ3

γ1−μ ₀

−∞kθe^−μθdθ λ4

γ2−μ ₀

Cεe^−μt¹.

3.6

From the definitions ofμandCε, we obtain

λ3

γ1−μ ₀

−∞kθe^−μθdθ λ4

γ2−μ ₀

−∞kθe^−μθdθ 1, λ1− λ3Cε

γ1−μ ₀

−∞kθe^−μθdθ λ1− λ3

γ1−μ ₀

−∞kθe^−μθdθλ1ε γ1−μ λ3

₀

−∞kθe^−μθdθ <0, λ2− Cελ4

γ2−μ ₀

−∞kθe^−μθdθ λ2− λ4

γ2−μ ₀

−∞kθe^−μθdθλ2ε γ2−μ λ4

₀

−∞kθe^−μθdθ <0.

3.7

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Thus,3.6yields

yt1< Cεe^−μt¹, 3.8

which contradicts3.5, that is,3.4holds.

Asε >0 is arbitrarily small, in view of3.4, it follows

yt≤M2e^−μt, t≥0, 3.9

whereM2 max{λ1λ2,λ1γ1−μ/λ3

₀

−∞kθe^−μθdθ,λ2γ2−μ/λ4

₀

−∞kθe^−μθdθ}>

0. The proof of this Lemma is completed.

Lemma 3.3. Forγ1, γ2 >0, there exist some positive constants:λi>0 i 1,2,3,4and a function y:−∞,∞ → 0,∞. Ifλ3/γ1 λ4/γ2<1, the following inequality:

yt≤

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

λ1e^−γ¹^tλ2e^−γ²^tλ3

_t

0

e^−γ¹^t−s ₀

−∞kθysθdθ ds λ4

_t

0

e^−γ²^t−s ₀

−∞kθysθdθ ds, t≥0,

λ1λ2, t∈−∞,0,

3.10

holds. Then, one has lim_t→_∞yt 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

yt≤dε, 3.11

for anyt∈−∞,∞. Firstly, for for allε >0, letting

dε max

λ1λ2ε,γ1

λ3λ1ε,γ2

λ4λ2ε

>0. 3.12

It is obviously seen thatyt≤dεfor anyt∈−∞,0. Assumed that there exists at1>0 such that

yt< dε, t∈−∞, t1, yt1 dε. 3.13

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Then, it from3.10implies that

yt1≤λ1e^−γ¹^t¹λ2e^−γ²^t¹λ3dε

_t₁

0

e^−γ¹^t¹^−sdsλ4dε

_t₁

0

e^−γ²^t¹^−sds

λ1− λ3dε

γ1

e^−γ¹^t¹

λ2−λ4dε

γ2

e^−γ²^t¹

λ3

γ1 λ4

γ2

dε.

3.14

From the definition ofdε, we have λ1−λ3dε

γ1 λ1−λ3

γ1

λ3λ1ε<0,

λ2−λ4dε

γ2 λ2−λ4

γ2

λ4λ2ε<0.

3.15

Thus,3.14yields

yt1< dε, 3.16

which contradicts3.13, that is,3.11holds.

Asε >0 is arbitrarily small, in view of3.11, it follows

yt≤d, t∈−∞,∞, 3.17

whered max{λ1λ2,γ1/λ3λ1,γ2/λ4λ2}.

Step 2. We prove that lim_t_→∞yt 0.

From the inequality3.17, it has shown thatytis a bounded function defined on the interval−∞,∞. Thus, ast → ∞, the upper limitdenoted byl≥0ofytexists, namely,

lim_t→_∞yt l, 3.18

the remaining work is to provel 0.

Supposed thatl > 0. From3.18, there must exist arbitrary positive scalarε >0 and constantT1>0 such that

yt< lε, ∀t≥T1. 3.19

On the other hand, since₀

−∞ksds 1, there must existT2 >0 such that _−T₂

−∞ kθdθ < ε, ∀t≥T2. 3.20

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LettingT max{T1, T2},3.19and3.20hold fort > T. Thus, it from3.10follows that

yt≤λ1e^−γtλ2e^−γ²^tλ3

_t

0

e^−γ¹^t−s _s−T

−∞ ku−syudu ds λ3

_t

0

e^−γ¹^t−s _s

s−Tku−syudu ds λ4

_t

0

e^−γ²^t−s _s−T

−∞ ku−syudu ds λ4

_t

0

e^−γ²^t−s _s

s−Tku−syudu ds

≤

λ1λ3

_2T

0

e^γ¹^s _s

e^−γ¹^t

λ2λ4

_2T

0

e^γ²^s _s

e^−γ²^t

λ3d γ1 λ4d

γ2

ε

λ3

γ1 λ4

γ2

lε.

3.21

By virtue of3.18, we have

l≤ λ3d

γ1 λ4d γ2

ε

λ3

γ1 λ4

γ2

lε. 3.22

From the arbitrary property ofε, it followsl ≤ λ3/γ1 λ4/γ2l, 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 anda, b ∈ 0, ν, then the mild solution to system2.2is exponentially stable inpp≥2moment.

Proof. In view of2.4and the elementary inequality, we have

xt^p

CtϕSt

ξ−f00, x0

_t

0

Ct−sf0s, xsds

_t

0

St−sf1s, xsds _t

0

St−sf2s, xsdws

p

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≤5^p−1M^pϕ^pe^−bt5^p−1M^pξ−f00, x0^pe^−at5^p−1M^pb^1−p _t

0

e^−bt−sf0s, xs^pds 5^p−1M^pa^1−p

_t

0

e^−at−sf1s, xs^pds5^p−1

_t

0

St−sf2s, xsdws

p

.

3.23

From the conditionH2, it from3.23concludes that

Ext^p≤5^p−1M^pEϕ^pe^−bt5^p−1M^pEξ−f00, x0^pe^−at 5^p−1M^pb^1−pC₀^p

_t

0

e^−bt−sE ₀

−∞kθxsθdθ

p

ds

5^p−1M^pa^1−pC^p₁ _t

0

e^−at−sE 0

−∞kθxsθdθ

p

ds

5^p−1E

_t

0

St−sf2s, xsdws

p

.

3.24

FromLemma 2.3, we obtain

E

_t

0

St−sf2s, xsdws

p

≤M^p _t

0

e^−apt−sEf2s, xs^p

L⁰₂

2/p

ds _p/2

p p−1

2

_p/2

M^p _t

0

e^−2at−s

Ef2s, xs^p

L⁰₂

_2/p ds

p/2 p

p−1 2

p/2

≤M^p _t

0

e−2ap−1/p−2t−sds

_p/2−1_t

0

e^−at−sEf2s, xs^p

L⁰₂ds p

p−1 2

_p/2

≤M^pC^p₂ 2a

p−1 p−2

_1−p/2

p p−1

2

_p/2_t

0

e^−at−sE ₀

−∞kθxsθdθ

_p ds.

3.25

Substituting3.25into3.24, it follows

Ext^p

≤5^p−1M^pEϕ^pe^−bt5^p−1M^pEξ−f00, x0^pe^−at 5^p−1M^pb^1−pC^p₀

_t

0

e^−bt−s ₀

−∞kθExsθ^pdθ ds

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5^p−1M^pa^1−pC₁^p _t

0

e^−at−s ₀

−∞kθExsθ^pdθ ds 5^p−1M^pC^p₂

2a p−1 p−2

_1−p/2

p p−1

2

p/2_t

0

e^−at−s ₀

−∞kθExsθ^pdθ ds.

3.26 And it is easily verified that there exists two positive numberM >0 andM>0 such thatExt²≤Me^−btMe^−at, for anyt∈−∞,0.

ByLemma 3.2, we can derive thatExt² ≤ M1e^−μt, t ∈ 0,∞ μ ∈ 0, γ1∧γ2, where

M1

⎧⎨

⎩5^p−1M^p

Eϕ^pEξ−f00, x0^p

,5^p−1M^pb^−pC₀^p,5^p−1M^p

a^−pC₁^pC^p₂a^−p/2

× 2

p−1 p−2

_1−p/2

p p−1

2

_p/2⎫

⎬

⎭>0.

3.27

The proof of this Theorem is completed.

Theorem 3.5. Suppose that the conditions: (H1)–(H₃) are satisfied, then the mild solution to system 2.2is asymptotically stable inp p≥2moment.

Proof. Similarly, we can obtain the conclusion as follows:

Ext^p

≤5^p−1M^pEϕ^pe^−bt5^p−1M^pEξ−f00, x0^pe^−at 5^p−1M^pb^1−pC^p₀

_t

0

e^−bt−s ₀

−∞kθExsθ^pdθ ds 5^p−1M^pa^1−pC^p₁

_t

0

e^−at−s ₀

−∞kθExsθ^pdθ ds 5^p−1M^pC^p₂

2a p−1 p−2

_1−p/2

p p−1

2

_p/2_t

0

e^−at−s ₀

−∞kθExsθ^pdθ ds, 3.28 and it is easily verified that there exists two positive numberM_∗ >0 andM_∗ > 0 such that Ext²≤M_∗M_∗, for anyt∈−∞,0.

ByLemma 3.3, we can derive that

t→∞lim Ext^p 0. 3.29

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To obtain the asymptotical stability inpp ≥2-moment, we need to prove that mild solution of system2.2is stable in pp ≥ 2-moment. Letε > 0 be given and chooseδ >

0δ < εsuch that

Ce^−a∧bt^∗ Δε < ε, 3.30

where

C 5^p−1M^pEϕ^p5^p−1M^pEξ−f00, x0^p, Δ 5^p−1M^p

⎡

⎣b^−pC₀^pa^−pC^p₁C^p₂a^−p/2 2

p−1 p−2

_1−p/2

p p−1

2

_p/2⎤

⎦. 3.31

If xt,0, ϕ is a mild solution of system 2.2 with sup_{θ∈−∞,0}Eϕθ^p < δ, then xt is defined in 2.4. Now, we claim that Ext^p < ε for all t ≥ 0. Notice that sup_{θ∈−∞,0}Eϕθ^p < ε. If there exists t^∗ > 0 such thatExt^∗^p ε and Ext^p < ε, for allt∈−∞, t^∗, then it follows from2.4that

Ext^∗^p

≤5^p−1M^p

Eϕ^pEξ−f00, x0^p e^−a∧bt^∗ 5^p−1M^pb^1−pC^p₀

_t∗

0

e^−bt^∗^−s ₀

−∞kθExsθ^pdθ ds 5^p−1M^pa^1−pC^p₁

_t^∗

0

e^−at^∗^−s ₀

−∞kθExsθ^pdθ ds 5^p−1M^pC₂^p

2a p−1 p−2

_1−p/2

p p−1

2

_p/2_t∗

0

e^−at^∗^−s ₀

−∞kθExsθ^pdθ ds

< Ce^−a∧bt^∗ Δε

< ε,

3.32 which contradicts the definition oft^∗. This shows that the mild solution of system2.2is asymptotically stable inp 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. Let X L²0, πand Y R¹ with the norm · . And leten :

2/πsinnξ n 1,2, . . . denote the completed orthonormal basis inX. Letwt: _∞

n 1

λnβnten, λn >0, where{βnt}

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

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DA {h ∈ X : h, ∂/∂ξhare absolutely continuous,∂²/∂ξ²h∈ X, h0 hπ 0}.

Then,

Ah ∞ n 1

n²h, enen, h∈DA, 4.1

whereen, n 1,2,3, . . ., is also the orthonormal set of eigenvector ofA. It is well known that Ct ≤exp−π²tandSt ≤exp−π²t, t≥0.

Now, we consider the following second-order neutral stochastic partial diﬀerential equations with infinite delays:

d ∂

∂tz t, y

− a0

π√ π

₀

−∞−θ^−1/2e^π²^θztθ, ξdθ

∂²

∂ξ²zt, ξ α1

π√ π

₀

−∞−θ^−1/2e^π²^θztθ, ξdθ

dt α2

π√ π

₀

−∞−θ^−1/2e^π²^θztθ, ξdθ dwt, t≥0, ξ∈0, π, xt,0 xt, π 0, t≥0,

zθ, ξ ϕθ, ξ, θ∈−∞,0, ξ∈0, π,

∂

∂tz0, ξ ζξ, ξ∈0, π.

4.2

Define

f0t, zt α0π√ π 5

₀

−∞−θ^−1/2e^π²^θztθ, ξdθ, f0t,0 0, f1t, zt α1π√

π 5

₀

−∞−θ^−1/2e^π²^θztθ, ξdθ, f1t,0 0, f2t, zt α2π

5 ₀

−∞−θ^−1/2e^π²^θztθ, ξdθ, f2t,0 0,

4.3

for anyzt∈B.

It is easily verified that f0

t, z¹_t

−f0

t, z²_t≤ α0π√ π 5

₀

−∞−θ^−1/2e^π²^θz¹tθ, ξ−z²tθ, ξdθ, f0t,0 0, f1

t, z¹_t

−f1

t, z²_t ≤ α1π√ π 5

₀

−∞−θ^−1/2e^π²^θz¹tθ, ξ−z²tθ, ξdθ, f1t,0 0, f2

t, z¹_t

−f2

t, z²_t≤ α2π 5

₀

−∞−θ^−1/2e^π²^θz¹tθ, ξ−z²tθ, ξdθ, f2t,0 0, 4.4 for anyz¹_t, z²_t ∈B.

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By virtue of Theorems3.4and3.5, the exponential stability inp p ≥2-moment and the asymptotical stability inp p≥2-moment for mild solution to system4.2are obtained, provided that the following inequality:

α^p₀α^p₁α2

2 p−1 p−2

_1−p/2

p p−1

2

_p/2

<5, p≥2, 4.5

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 in18. Besides, our results can be easily extended to investigate two cases:1the 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.

References

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