Random pullback attractor of a non-autonomous local modified stochastic Swift-Hohenberg with multiplicative noise

Abstract: In this paper, we study the existence of the random -pullback attractor of a non-autonomous local modified stochastic Swift-Hohenberg equation with multiplicative noise in stratonovich sense. It is shown that a random -pullback attractor exists in ) D ( 2 0 H when its external force has exponential growth. Due to the stochastic term, the estimate are delicate, we overcome this difficulty by using the Ornstein-Uhlenbeck(O-U) transformation and its properties.


Introduction
The Swift-Hohenberg(S-H) type equations arise in the study of convective hydrodynamical, plasma confinement in toroidal and viscous film flow, was introduced by authors in [1]. After that, Doelman and Standstede [2] proposed the following modified Swift-Hohenberg equation for a pattern formation system near the onset to instability ,  is antisymmetry and  2 is symmetry. We will use the symmetry principle study S-H equation.
The dynamical properties of the S-H equation are important for the studies pattern formation system have been extensively investigated by many authors; see [3][4][5][6][7][8]. Polat [8] establish the existence of global attractor for the system (1.1), and then Song et al. [7] improved the result in H. k Recently for non-autonomous modified S-H equation: it has also attracted the interest of many authors. If 0 = l , equation (1.2) becomes a non-autonomous modified S-H equaiton. Park [9] proved the existence of -pullback attractor when the external force has exponential growth, Xu et al. [10] established the existence of uniform attractor when the external force is a constant, Guo et al. [11] investigated the equation when 0 = ) t , x ( g and proved the existence of random attractor which need the spatial variable in one dimension. For ( , ) 0 g x t = ， to the best of our knowledge, the existence of random -pullback attractor for equation (1. 2) has not yet considered.
In this paper, we consider the following one dimensional non-autonomous local modified stochastic S-H equation with multiplicative noise: l are arbitrary constants, ) t ( W is a two-sided real-valued Wiener process on a probability space which will be specified later. For the external force )) The assumption is same as [8,12], through simple calculation, for all R t∈ , we have (1.7) An outline of this paper is as follows: In section 2, we recall some basic concepts about random -pullback attractors. In Section 3, we prove that the stochastic dynamical system generated by (1.3) exists a random -pullback attractor in ) D ( H 2 0 .

Preliminaries
There are many research results on random attractors and related issues. The reader is referred to [13][14][15][16][17][18][19] for more details, we only list the definitions and abstract result Let ( , and ( , , )  be a probability space. In this paper, the term -a.s.(the abbreviation for almost surely) denotes that an event happens with probability one. In other words, the set of possible exception may be non-empty, but it has probability zero.
In the sequel, we use to denote a collection of some families of nonempty bounded subsets of X:

Random pullback attractor for modified Swift-Hohenberg
In this section, we will use abstract theory in section 2 to obtain the random -pullback attractor for equation (1.3)-(1.5). First we introduce an Ornstein-Uhlenbeck process, We known from [6], it is the solution of Langevin equation is an ergodic metric dynamical system.
From [15,16,20], it is known that the random variable () z  is tempered and there exists a  [2,8,22] We now apply abstract theory in Section 2 to obtain the random -pullback attractors for non-autonomous modified Swift-Hohenberg equation, by the equivalent, we only consider the random -pullback attractor of equation ( For our purpose that the following Gagliardo-Nirenberg inequality will be used.
Using Young inequality, we get By integration by parts, we obtain Proof. Taking inner product of equation ( By the H older inequality, Young inequality and Gagliardo-Nirenberg inequality, we get Then we have ) .