2010 Mathematics Classification: 35K10, 35K25, 35K35
In this paper, we consider the following stochastic strongly damped higher- order nonlinear Kirchhoff-type equation with white noise:
with the Dirichlet boundary condition
and the initial value conditions
where is a bounded domain of, with a smooth boundary, is the Laplacian with respect to the variable, is a real function of and, is the damping coefficient, f is a given external force, v is the outer norm vector, is a nonlinear forcing, their respectively satis- fies the following conditions:
where are positive constants.
As well as we known, the study of stochastic dynamical is more and more widely the attention of scholars, and the study of random attractor has become an important goal. In a sense, the random attractor is popularized for classic determine dynamical system of the global attractor. Global attractor of Kirchhoff- type equations have been investigated by many authors, see, e.g.,     , however, the existence random attractor has also been studied by many authors, in  , Zhaojuan Wang, Shengfan Zhou and Anhui Gu, they study the asymp- totic dynamics for a stochastic damped wave equation with multiplicative noise defined on unbounded domains, and investigate the existence of a random attractor, they overcome the difficulty of lacking the compactness of Sobolev embedding in unbounded domains by the energy equation. In  , Guigui Xu, Libo Wang and Guoguang Lin study the long time behavior of solution to the stochastic strongly damped wave equation with white noise, in this paper, they use the method introduced in  , so that they needn’t divide the equation into two parts. In  , Zhaojuan Wang, Shengfan Zhou and Anhui Gu study the asymptotic dynamics of the stochastic strongly damped wave equation with homogeneous Neuman boundary condition, and prove the existence of a ran- dom attractor. The other long time behavior of solution of evolution equations, we can see  -  .
In this work, we deal with random term by using Ornstein-Uhlenbeck process, the key is to handle the nonlinear terms and strongly damped, and is also difficult to be conducted. So far as we know, there were no result on random attractor for the stochastic higher-order Kirchhoff-type equ- ation with nonlinear strongly dissipation and white noise. It is therefore im- portant to investigate the existence of random attractor on (1.1)-(1.3).
This paper is organized as follows: In Section 2, we recall many basic concepts related to a random attractor for genneral random dynamical system. In Section 3, we introduce O-U process and deal with random term. In Section 4, we prove the existence of random attractor of the random dynamical system.
In this section, we collect some basic knowledge about general random dy- namical system (    ).
Let be a separable Hilbert space with Borel s-algebra. Let be the metric dynamical system on the probability space.
Definition 2.1. (see   ). A continuous random dynamical system on X over is a -measurable mapping . Such that the following properties hold (1)
1) is the identity on X;
2) for all;
3) is continuous for all.
Definition 2.2. (see  )
1) A set-valued mapping, is said to be a random set if the mapping is measurable for any. If is also closed (compact) for each, is called a random closed (com- pact) set. A random set is said to be bounded if there exist and a random variable such that
2) A random set is called tempered provided for,
Let Y be the set of all random tempered sets in X.
3) A random set is said to be a random absorbing set if for any tempered random set, and, there exists such that
4) A random set is said to be a random attracting set if for any tempered random set, and, we have
where is the Hausdorff semi-distance given by for any.
5) is said to be asymptotically compact in X if for has a convergent subsequence in X whenever, and with.
6) A random compact set is said to be a random attractor if it is a random attracting set and for and all.
Theorem 2.1. (  ) Let be a continuous random dynamical system with state space X over. If there is a closed random absorbing set of and is asymptotically compact in X, then is a random attractor of, where
Moreover, is the unique random attractor of.
3. O-U Process and Stochastic Dynamical System
, , ,
Let, and define a weighted inner product and norm in E
3.1. O-U Process
O-U process is given by Wiener process on the metric system, we can see (    ).
Let, where, for, meet Itô
equation:. And there is a probability measure P, -in- variant set; so that stochastic process meet the following properties:
1) For, mapping for continuous mapping;
2) Random variable is called tempered;
3) Exist temper set, such that
3.2. Stochastic Dynamical System
For convenience, we rewrite the Question (1.1)-(1.3):
Let, and (defined in  ), then (3.2.1) has the following simple matrix form
Let, then (3.2.1) can be rewritten as the equivalent system:
In   they have proven that the operator L of (3.2.3) is the infinitesimal generation operator of semigroup in Hilbert space E, is continuous in t and globally Lipschitz continuous in for each. By the classical theory concerning the existence and uniqueness of the solutions    , so we have the following theorem.
Theorem 3.2.1. Consider (3.2.3). For each and initial value , there exists a unique function such that satisfies the integral equation
For, let the solution mapping of
generates a random dynamical system.
Define two isomorphic mapping:
And inverse isomorphic mapping:
Then the mapping generates a random dynamical system associated with (1.1)-(1.3); and mapping generates a random dynamical system associated with (3.2.2).
Notice that all of the above random dynamical system, are equivalent. Hence we only need to consider the random dynamical system.
4. The Existence of Random Attractor
First, we prove the random dynamical system exists a bounded random absorb set, hence we let be all temper subsets in E.
Lemma 4.1. (Lemma 3.1 of  ) Let, for any , we have
where are determined in  , , is first eigenvalues of (1.1).
Lemma 4.2. Let is a solve of (3.2.2), then there is a bounded random com- pact set, such that for arbitrarily random set, existence a random variable, so that
Proof. Let is a solve of (3.2.3), applying the inner product of the equation (3.2.3) with, we discover that
According to (4.1) and (4.4)-(4.10), we have
According to Gronwall inequation, , we have
Because is tempered, and is continuous about t, according to  , we can get a temper random variables, such that , we have
Substituting by in (4.12), we know
Because is tempered, and is also tempered, hence we let
then is also tempered, is called a random absorb set, and because of
then is a random absorb set of, and.
Next, we will prove the random dynamical system has a compact absorb set
Lemma 4.3. For, let be a solve of (3.2.2), initial value, we decompose, where satisfy
and exist a temper random radius, such that, satisfy
Proof. Let be a solve of (3.2.3), according to (4.17) and (4.18), we know meet separately
Taking inner product (4.21) with, we have
according to Lemma 4.1 and Gronwall inequality, we have
substituting by, and is tempered, then
So, (4.19) is hold. Taking inner product (4.22) with
, we have
according to Lemma 4.1, Lemma 4.2, (4.24) and Young inequality, we have
where are given by Lemma 4.2, and
Due to Gronwall inequality, and substituting by
, we have
According to (4.14) and (4.16), then
Then is tempered, and because
hence, we set
then, for, we have
and is tempered.
Lemma 4.4. (3.2.2) the identified stochastic dynamical system, while exist a compact attracting set.
Proof. Let be a closed ball, radius in space
, because, so is a compact
set in E, for arbitrarily temper random set, for, ac- cording to Lemma 4.3, , so for, we have
Theorem 4.1. The random dynamical system has a unique random attractor in E, where
in which is a tempered random compact attracting for.