We studied the random high order Beam equation with strong damping and white noise in this paper.
where are given functions, , describe a addable white noise, denotes a area which bounded with smooth homogeneous Dirichlet boundary, denotes the boundary of . are constants, are strong damping terms, denotes a one-dimensional two-sided Wiener process on a probability space , , F denotes a Borel -algebra generated by compact-open topology on , P denotes a probability measure.
Random attractor is a random collection, which is measurable, compact, constant, and attract all the orbits. If it exists, it will be the smallest absorbing sets of system solution concentration, it’s also the biggest invariant set. In a sense, random attractor is reminded as a reasonable extension of the global attractor to the classical dynamical system. Recently, more and more scholars have focused on the random dynamic system.
Guo  wrote a book about the random infinite dimensional dynamical system, it’s the first book at home. It includes any experience of the author with random dynamic study and some research results, the latest development and results also be introduced.
Lin  studied the existence of stochastic attractors of high order nonlinear Beam equation.
Qin  proved the random attractor for stochastic Beam equations with addable white noise, Xu  studied the non-autonomous stochastic wave equation with dispersion and dissipation terms.
Crauel and Flandoli  studied the random attractor of the infinite dimensional equation. Cai and Fan  considered the dissipative KDV equation with multiplicative noise.
For more relevant studies, it can be referred to references in  - .
In this section, some symbols and assumptions are introduced for convenience.
Let the operator with Dirichlet boundary condition be selfadjoint, positive definite and linear. Set the eigenvalue of A is , and satisfies
, when , . (4)
Set and define a weighted inner product and norm in
Definition 2.1  Set is a metric dynamic system, if , measurable mapping
1) For all and , mapping satisfy
2) For every , mapping continuous.
It is said that S is a continuous random dynamic system on .
Definition 2.2  It is said that the random set is slowly increasing, if , , there is
and , for all .
Definition 2.3  denotes a collection of all random sets on X, random set denotes a absorption set on , if for every and , there is make
Definition 2.4  The random set becomes the random attractor of the continuous random dynamic system on X. If the random set satisfies:
1) is a random compact set;
2) is a invariant set, for every , ;
3) attracts all sets on , for any and , we have the limit formula:
denotes the Hausdorff half distance. ( ).
Definition 2.5  Random set is the random absorption set of the random dynamic system , and the random set satisfies
1) Random set is a closed set on Hilbert space X;
2) For , random set meet the following progressive compactness conditions: For any sequence in , there is a convergent subsequence in space X. Then the stochastic dynamic system has a unique global attractor
3. Existence of Random Attractors
3.1. Existence and Uniqueness of Solution
For convenience, Equations (1)-(3) can be reduced to
Set , , then Equation (1) is equivalent to the following stochastic differential equation
denotes a Ornstein-Uhlenbeck process, it is a stationary solution of Itô equation
, then Equation (14) can be reduced to
3.2. The Existence of Random Attractors
This section mainly considers existence of the random attractor of problem (1). First, we can prove that the random dynamic system has a bounded random absorption set. For this reason, all slowly increasing subsets in the space E are denoted as .
Lemma 1 , for every , , When
According to the Formula (7), we can get
Lemma 2 denotes a solution of problem (14), then there is a bounded random compact set , so that there is a random variable for any slowly increasing random set , such that
Proof denotes a solution of problem (16), use to take the inner product with the Equation (16), we obtain
From Lemma 1
From Equations (22) and (23), Equation (21) can be written as
According to the Formula (7), we can get
From the Gronwall’s inequality, , we have
because is slowly increasing, and is continuous with respect to t, according to the literature , a slowly increasing random variable can be obtained, so for there is
Substituting for in (27), we get
Because is slowly increasing, and is also slowly increasing, so let
Then also slowly increasing, denotes a random absorption set, because
Then is the random absorption set of , and .
So the lemma is proved.
Lemma 3 When , for any , is the solution of Equation (14) under the initial value condition . It can be decomposed into , where and satisfy
and there is a slowly increasing random radius , so that for every , satisfy
Proof is a solution of Equation (16), from Equations (33) and (34), it can be seen that and satisfy
Using and Equation (37) to take the inner product, we get
according to lemma 2 and the Gronwall’s inequality, we obtain
replace in (40) with , and because is slowly increasing, then
Using and Equation (38) to take the inner product, we get
According to lemma 1, lemma 2, Equation (29), the Gronwall’s inequality, and replace with , we obtain
for every ,
and is slowly increasing. The lemma is proved.
Lemma 4 The stochastic dynamic system determined by Equation (17) has a compact absorption set under condition ,
Proof Suppose is a closed sphere with as the radius in space . According to the embedding relationship ,
is a compact set in . For any slowly increasing random set in E, for , according to Lemma 3.1, there is , so for every ,
Therefore, for any slowly increasing random set in , there is
According to Lemma 1 to Lemma 4, there is the following theorem.
Theorem 1 Random dynamic system has a random attractor , , and there is a slowly increasing random set , ,
We studied a class of damped high order Beam equation stochastic dynamical systems with white noise, by using the Ornstein-Uhlenbeck process, estimating the solution of the equation and the isomorphism mapping method, then we can get the existence of the random attractor family, I wish there will be some more convenient methods can be shown off. Further we can make the inertial manifolds of the model.
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