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

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.


Introduction
The neutral stochastic differential equations can play an important role in describing many sophisticated dynamical systems in physical, biological, medical, chemical engineering, aeroelasticity, 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-Schaefertype 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 secondorder 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 Section 2, 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.

Preliminaries
Let X 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, {I t } t≥0 , P be a complete probability space equipped with some filtration I t t ≥ 0 satisfying the usual conditions, that is, the filtration is right continuous and I 0 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 I t t ≥ 0 -measurable and C −∞, 0 , X -valued random variables.
Let β n t n 1, 2, . . .be a sequence of real-valued one-dimensional standard Brownian motions mutually independent over Ω, I, {I t } t≥0 , P .Set w t ∞ n 1 λ n β n t e n , t ≥ 0, where λ n ≥ 0 n 1, 2, . . .are nonnegative real numbers and {e n } n 1, 2, . . . is a complete orthonormal basis in Y .Let Q ∈ L Y, Y be an operator defined by Qe n λ n e n with finite trace trQ Now, for the definition of an X-valued stochastic integral of an L 0 2 Y, X -valued and I t -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: where ϕ ∈ B and ξ is also an I 0 -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; 2 Y, X are three approximate mappings.In this sequel, the history 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 is called a strongly continuous cosine family.
The corresponding strongly continuous sine family 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 that A is the infinitesimal generator of a cosine family of operators {C t : t ∈ R}.Then, the following holds: i there exists M * ≥ 1 and α ≥ 0 such that C t ≤ M * e αt and hence S t iii there exists Lemma 2.3 see 20 .For any r ≥ 1 and for arbitrary L 0 2 Y, X -valued predictable process φ • such that

Main Results
In order to obtain our main results, we need the following assumptions.
H 1 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.
H 2 The mappings f i i 0, 1, 2 satisfy the following conditions: there exist three positive constants C i > 0 i 0, 1, 2 and a function: for any x, y ∈ B and t ≥ 0.
Remark 3.1.Obviously, under the conditions: H 1 -H 2 , 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.

3.6
From the definitions of μ and C ε , we obtain
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 constant d ε > 0 such that for any t ∈ −∞, ∞ .Firstly, for for all ε > 0, letting It is obviously seen that y t ≤ d ε for any t ∈ −∞, 0 .Assumed that there exists a t 1 > 0 such that Then, it from 3.10 implies that

3.14
From the definition of d ε , we have
As ε > 0 is arbitrarily small, in view of 3.11 , it follows where Step 2. We prove that lim t → ∞ 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, lim t → ∞ y t l, 3.18 the remaining work is to prove l 0. Supposed that l > 0. From 3.18 , there must exist arbitrary positive scalar ε > 0 and constant T 1 > 0 such that y t < l ε, ∀t ≥ T 1 .

3.19
On the other hand, since 0

3.20
Letting T max{T 1 , T 2 }, 3.19 and 3.20 hold for t > T. Thus, it from 3.10 follows that
Proof.In view of 2.4 and the elementary inequality, we have

3.23
From the condition H 2 , it from 3.23 concludes that

3.24
From Lemma 2.3, we obtain

3.26
And it is easily verified that there exists two positive number M > 0 and M > 0 such that E x t 2 ≤ M e −bt M e −at , for any t ∈ −∞, 0 .

3.27
The proof of this Theorem is completed.
Theorem 3.5.Suppose that the conditions: (H 1 )-(H 3 ) 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: To obtain the asymptotical stability in p 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 Ce − a∧b t * Δε < ε, 3.30 where If x t, 0, ϕ is a mild solution of system 2.2 with sup θ∈ −∞,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 sup θ∈ −∞,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 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.

An Illustrative Example
In this section, we provide an example to illustrate the obtained results above.Let X L 2 0, π and Y R 1 with the norm e π 2 θ z 1 t θ, ξ − z 2 t θ, ξ dθ, f 0 t, 0 0, e π 2 θ z 1 t θ, ξ − z 2 t θ, ξ dθ, f 1 t, 0 0, By virtue of Theorems 3.4 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: 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.

5 p− 1 M p b 1−p C p 0 t 0 e
−b t−s 0 −∞ k θ E x s θ p dθ ds 5 p−1 M p a 1−p C x s θ p dθ ds.
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 lim 3 where C r r 2r − 1 r .Definition 2.4.An X-value stochastic process x t t ∈ R is called a mild solution of the system 2.2 if i x t is adapted to I t t ≥ 0 and has càdlàg path on t ≥ 0 almost surely, ii for arbitrary t ∈ 0, ∞ , P {ω : t 0 x t 2 dt < ∞} 1 and almost surely x t C t ϕ S t ξ − f 0 0, x 0 t 0 C t − s f 0 s, x s ds t 0 S t − s f 1 s, x s ds t 0 S t − s f 2 s, x s dw s , 2.4 where x 0 • 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 M 1 > 0 such that E x t p ≤ M 1 e −γt , t ≥ 0, p ≥ 2, 2.5 for any initial value ϕ ∈ B.
• .And let e n : 2/π sin nξ n 1, 2, . . .denote the completed orthonormal basis in X.Let w t : n β n t e n , λ n > 0 , where {β n t } are one-dimensional standard Brownian motions mutually independent on a usual complete probability space Ω, I, I t , P .Define A : X → X by A ∂ 2 /∂ξ 2 with the domain D A {h ∈ X : h, ∂/∂ξ h are absolutely continuous, ∂ 2 /∂ξ 2 h ∈ X, h 0 . .., is also the orthonormal set of eigenvector of A. It is well known that C t ≤ exp −π 2 t and S t ≤ exp −π 2 t , t ≥ 0. Now, we consider the following second-order neutral stochastic partial differential equations with infinite delays: λ 2 e π 2 θ z 1 t θ, ξ − z 2 t θ, ξ dθ, f 2 t, 0 t ∈ B.