Random Attractors of Stochastic Non-Autonomous Nonclassical Diffusion Equations with Linear Memory on a Bounded Domain

In this article, we discuss the long-time dynamical behavior of the stochastic non-autonomous nonclassical diffusion equations with linear memory and additive white noise in the weak topological space ( ) ( ) ( ) 1 2 1 0 0 , H L H μ + Ω × Ω  ). By decomposition method of the solution, we give the necessary condition of asymptotic compactness of the solutions, and then prove the existence of -random attractor, while the time-dependent forcing term ( ) ( ) 2 2 ; b g L L ∈ Ω  only satisfies an integral condition.

( ) ( ) ( ) By decomposition method of the solution, we give the necessary condition of asymptotic compactness of the solutions, and then prove the existence of -random attractor, while the time-dependent forcing term ( ) ( )

Introduction
In this article, we investigate the asymptotic behavior of solutions to the following stochastic nonclassical diffusion equations driven by additive noise and linear memory: where Ω is a bounded domain in ( ) , the initial data ( ) ω ω = = for all t ∈  .To consider system (1.1), we assume that the memory kernel satisfies and there exists a positive constant 0 δ > such that the function ( ) ( ) , 0, 0, 0. L s s s s µ µ µ δµ And suppose that the nonlinearity satisfies as follows: , , , f x s f x u f x s = + , s ∈  and for every fixed x ∈ Ω , ( ) ( ) , , , and , , , , α β γ δ = and l are positive constants, and q is a conjugate exponent of p.
We assume that the time-dependent external force term ( ) and for some constant 0 σ > to be specified later.
Equation (1.1) has its physical background in the mathematical description of viscoelastic materials.It's well known that the viscoelastic material exhibit natural damping, which according to the special property of these materials to retain a memory of their past history.And from the materials point of view, the property of memory comes from the memory kernel ( ) k s , which decays to zero with exponential rate.Many authors have constructed the mathematical model by some concrete examples, see [1]- [7].In [8] the authors considered the nonclassical diffusion equation with hereditary memory on a 3D bounded domains for a very general class of memory kernels  ; setting the problem both in the classical past history framework and in the more recent minimal state one, the related solution semigroups are shown to possess finite-dimensional regular exponential attractors.Equation (1.1) is a special case of the nonclassical diffusion equation used in fluid mechanics, solid mechanics, and heat conduction theory (see [1] [4] [5]).In [1] Aifantis, Urbana and Illinois discussed some basic mathematical results concerning certain new types of some equations, and in particular results showing how solutions of some equations can be expressed at in terms of solutions of the heat equation, also discussed diffusion in general viscoelastic and plastic solids.In [4] Kuttler and Aifantis presented a class of diffusion models that arise in certain nonclassical physical situations and discuss existence and uniqueness of the resulting evolution equations.
The long-time behavior of Equation (1.1) without white additive noise and 0 µ ≡ has been considered by many researchers; on a bounded domain see, e.g.
[9] [10] [11] [12] [13] and the references therein.In [10] the authors proved the existence and the regularity of time-dependent global attractors for a class of nonclassical reaction-diffusion equations when the forcing term ( ) ( ) and the nonlinear function satisfies the critical exponent growth.In [11] Sun and Yang proved the existence of a global attractor for the autonomous case provided that the nonlinearity is critical and ( ) ( ) The researchers in [12] obtained the Pullback attractors for the nonclassical diffusion equations with the variable delay on a bounded domain, where the nonlinearity is at most two orders growth.As far as the unbounded case for the system (1.1) the long-time behavior of solutions is concerned, most recently, by the tail estimate technique and some omega-limit compactness argument, for more details (see [14] [15] [16] [17] [18]).In [14] Ma studied the existence of global attractors for nonclassical diffusion equations with the arbitrary order polynomial growth conditions.By a similar technique, Zhang in [16] obtained the Pullback attractors for the non-autonomous case in ( ) where the growth order of the nonlinearity is assumed to be controlled by the space dimension N, such that the Sobolev embedding is continuous.However, it is regretted that some terms in the proof of [16] Lemma 3.4 are lost.Anh et al. [17] established the existence of pullback attractor in the space ( ) ( ) , where the nonlinearity satisfied an arbitrary polynomial growth, but some additional assumptions on the primitive function of the nonlinearity were required.And the case of 0 µ ≠ with additive noise on a bounded domain, Cheng used the de- composition method of the solution operator to consider the stochastic nonclassical diffusion equation with fading memory.For the case of 0 µ ≡ , Zhao studied the dynamics of stochastic nonclassical diffusion equations on unbounded domains perturbed by a -random term "intension of noise".(For more details see [2] [19] [20] [21] [22]).
To our best knowledge, Equation (1.1) on a bounded domain in the weak topological space and the time-dependent forcing term has not been considered by any predecessors.
The article is organized as follows.In Section two, we recall the fundamental results related to some basic function spaces and the existence of random attractors.In Section three, firstly, we define a continuous random dynamical system to proving the existence and uniqueness of the solution, then prove the existence of a closed random absorbing set and establish the asymptotic compactness of the random dynamical system finally prove the existence of -random attractor.

Preliminaries
In this section, we recall some basic concepts and results related to function spaces and the existence of random attractors of the RDSs.For a comprehensive exposition on this topic, there is a large volume of literature, see [2] [3] [19] [23]- [29].
We also introduce the family of Hilbert space ( ) In the following of this article, we denote Definition 2.2.A continuous random dynamical system (RDS) on X over a metric dynamical system empty subsets of X is called tempered with respect to ( ) ) Definition 2.4.Let  be the collection of all tempered random sets in X.A set ∈ Ω ∈  is called a random absorbing set for RDS φ in  , if for every B ∈  and P-a.e. ω ∈ Ω , there exists ( ) , , .
Let  be the collection of all tempered random subsets of X.Then φ is said to be asymptotically compact in X if for P-a.e. ω ∈ Ω , the se- quence has a convergent subsequence in X whenever n t → ∞ , and ( ) [30], [31], [32]) Let  be the collection of all tempered random subsets of X and ( ) attractor for φ if the following conditions are satisfied, for P-a.e. ω ∈ Ω , 1) ( ) A ω is compact, and 2) ( ) where d is the Hausdorff semi-metric given by ( ) for any Z X ⊆ and Y X ⊆ .
Theorem 2.1.Let φ be a continuous random dynamical system with state space X over ( ) ( ) . If there is a closed random absorbing set ( ) B ω of φ and φ is asymptotically compact in X, then ( )

{ }
A ω is the unique random attractor of φ .
As mentioned in [23], we can define a new variable to reflect the memory kernel of (1.1) Hence, , 0.

The Random Attractor
In this section, we prove that the stochastic nonclassical diffusion problem (2.4) has a -random attractor.First, We convert system (2.4) with a random pertur- where ∆ is the Laplacian with domain ( ) ( ) . Using the change of variable ( ) ( ) ( ) , ( ) t υ satisfies the equation (which depends on the random parameter ω ) By the Galerkin method as in [34], under assumptions (1.2)-(1.8),for P-a.e. ω ∈ Ω , and for all ( ) 3) has a unique solution ( ) Throughout this article, we always write , , , , .
If u is the solution of problem (1.1) in some sense, we can define a continuous dynamical system ; We first show that the random dynamical system ϒ has a closed random absorbing set in  , and then prove that ϒ is asymptotically compact.Lemma 3.2.Assume that ( ) ( )  Proof.Taking the inner product of the first equation of (3.3) with ( ) Hence, we can rewrite (3.8) as follows ( From the first term on the right hand side of (3.8) f f f = + , First we esti- mate 1 f .By (1.4)-(1.5)and using a similar arguments as (4.2) in [35], we have ) By using (1.6)-(1.7),we arrive at By the young inequality, and using assumption (1.6), we see that ( ) ( ) Recalling that ( ) ( ) Note that ( ) s Y θ ω is the tempered, and ( ) ( ) Then ( ) r ω is the tempered since ( ) s Y θ ω has at most linear growth rate at infinity, now the proof is completed.

H g g h h
, , u t ω satisfies (3.26), and , , u t ω is the solution of (3.27).Then for ( ) The same of the problem (3.3), we also have the corresponding existence and , 0 where the positive random function ( ) r ω is defined in Lemma 3.2.Proof.From (3.10) we substituting ( ) , , ,   such that for every given 0 T ≥ , the solution of (3.31) has the following uniform estimates From (2.2) and (3.31), we obtain , , f f f and the mean value theorem, we have Using the embedding theorem, we have where we have used inequality ( )( ) ( with the inner product and norm respectively,  , f is the Borel σ-algebra on Ω , and  is the corresponding Wiener measure.Define

.
= satisfies that there exist two positive constant C and k, such that Moreover, for three Banach space 0 1 , B B and 2 B , 0 B and 1 B

5 )
In order to prove the asymptotic compactness and the existence of global attractor, we give the following results.

1 r
Then for P-a.e. ω ∈ Ω , there is a positive random function ( ) ω and a constant ( ) , 0T T B ω = > such that for all t T ≥ , uniqueness of solutions for (3.30) and(3.31).For the convenience, we obtain the solution operators of (3.30) and (3.31) by give some Lemmas to prove the asymptotic compactness.Lemma 3.3.Assume that the condition on Then for P-a.e. ω ∈ Ω , there is a constant solution of (3.30) satisfies the following uniform estimate

Y
θ ω are tempered, we can choose 2 0 T > , 2 t T ∀ > , such that (3.32) is satisfied.Lemma 3.4.Assume that the condition on Then for P-a.e. ω ∈ Ω , there is a positive random function