In this paper, we study the random attractor family of solutions to the strongly damped stochastic Kirchhoff equation with white noise:
with the Dirichlet boundary condition
and the initial value conditions
where is a positive integer; is a constant; is a bounded region with smooth boundary in . is the Laplacian with respect to the variable to variable . M is a general real-valued function; is a non-linear and non-local source term. is derivative of a one-dimensional two-valued Wiener process and formally describes white noise.
B.L. Guo and X.K. Pu described in detail the related concepts and theories of infinite dimensional stochastic dynamical systems, and discussed in detail the existence and uniqueness, attractor and inertial manifold of some nonlinear evolution equations and wave equation solutions in   .
D.H. Cai and X.M. Fan  , considered the dissipative KDV equation with multiplicative noise.
By transforming the equation into a stochastic KDV-type equation without white noise, the existence of stochastic attractors for dynamic systems determined by the original equation is proved by discussing the dynamic absorptivity and asymptotic property determined by the new equation.
Yin et al.  have mainly studied the dissipative Hamiltonian amplitude modulated wave instability equation with multiplicative white noise.
Stochastic dynamic system has compact random attractors in space .
Xu et al.  studied the non-autonomous stochastic wave equation with dispersion and dissipation terms.
The existence of random attractors for non-autonomous stochastic wave equations with product white noise is obtained by using the uniform estimation of solutions and the technique of decomposing solutions in a region.
Lin et al.  studied the existence of stochastic attractors for higher order nonlinear strongly damped Kirchhoff equation.
The O-U process is mainly used to deal with the stochastic terms, and the existence of stochastic attractors is obtained.
Qin et al.  studied random attractors for the Kirchhoff-type suspension bridge equations with Strong Damping and white noises.
Kirchhoff stress term and dissipation term are treated. It is assumed that the non-linear term satisfies the growth and dissipation conditions.
For more relevant studies, it can be referred to references in  -  .
On the basis of some random attractors of Kirchhoff equation with white noise studied by predecessors, the existence and uniqueness of solutions of stochastic higher-order Kirchhoff equation with strong damping of white noise, nonlinear and non-local source terms and the existence of attractors of stochastic Kirchhoff equation are discussed. This paper is organized as follows. In Section 2, some basic assumptions and basic concepts related to random attractor for general random dynamical system are presented. Section 3 deals with random term and proof the existence of random attractor family by using the isomorphism mapping method.
In this section, some symbols are made and assumption Kirchhoff Stress term satisfying condition (a) and Nonlinear term satisfies condition (b). In addition, some basic definitions of stochastic dynamical systems are also introduced.
For narrative convenience, we introduce the following symbols:
Kirchhoff Stress term satisfies condition (a):
a) is locally bounded and measurable, and where is a constant;
Nonlinear term satisfies condition (b):
b) Let be nonnegative nonlocal bounded and measurable, , , and ;
Here are some basics about random attractors.
Let be a probabilistic space and define a family of transformation preserving measures and ergodicity:
then is an ergodic metric dynamical system.
Let be a complete separable metric space and be a Borel σ-algebra on X.
Definition 2.1. (  ) Let is a metric dynamic system, suppose that the mapping
is -measurable mapping and satisfies the following properties:
1) The mapping satisfies
2) is continuous, for any .
Then S is a continuous stochastic dynamical system on .
Definition 2.2. (  ) It is said that the random set is tempered, for , we have
where , for any .
Definition 2.3. (  ) Note that is the set of all random sets on X, and random set is called the absorption set on . If for any and , there exists such that
Definition 2.4. (  ) Random set called the random attractor of continuous stochastic dynamical systems on X, if random set satisfies the following conditions:
1) is a random compact set;
2) is the invariant set , that is, for any ;
3) attracts all the set on , that is, for any and , with the following limit:
where is Hausdorff half distance. (where ).
Definition 2.5. (  ) Let random set be a random absorbing set of stochastic dynamical system , and the random set satisfy:
1) Random set is a closed set on Hilbert space X.
2) For , random set satisfies the following asymptotic compactness conditions: for any sequence , there is an convergence subsequence in space X, when , Then stochastic dynamical system has a unique global attractor.
The Ornstein-Uhlenbeck process  is given as following.
Let , where . For any , the stochastic process satisfies the Ito equation
According to the nature of O-U process, there exists a probability measure P, -invariant set, and the above stochastic process
satisfies the following properties
1) The mapping is a continuous mapping, for any given ;
2) The random variable is tempered;
3) There exist a tempered set , such that
4) ; (2.12)
5) . (2.13)
3. The Existence of Random Attractor Family
In this section, we consider the existence of random attractor family. To deal with the random term we need to transform the problem (1.1) - (1.3) into a general stochastic problem. It is proved that there exists a bounded stochastic absorption set for stochastic dynamical systems. The stochastic dynamical system exists stochastic attractor family and a slowly increasing stochastic compact set.
For convenience, Equation (1.1) - (1.3) can be transformed into
Let . Then the problem (3.1) can be simplified to:
Let , then the question (3.2) can be written as:
Lemma 3.1. Let for any , if ,
Proof: For any , according to hypothesis (a), we have
Lemma 3.1 is proved.
Lemma 3.2. Let be a solution of the problem (3.2), then there exists a bounded random compact set , so that for any random set , there exists a random variable , such that
Proof: Let be a solution of the problem (3.3), by taking the inner product of two sides of the Equation (3.3) is obtained by using ,
From Lemma 1, we have
According to the inner product defined on .
According to Holder inequality, Young inequality and Poincare inequality, we have
According to hypothesis (b), we have
Combining (3.8)-(3.13) yields, we have
By the Gronwall inequality, then
And because is tempered, and is continuous about t, so according to reference  , we can get a temper random variable , so that for any , we have
Replace in Equation (3.17) with , we can obtain that
where ( )
Because is also temper, and is also tempered, so we can let
Then is also temper, is a random absorb set, and because of
then is a random absorb set of , and .
Thus, the whole proof is proved.
Lemma 3.3. When , for any , Let is a solution of the Equation (3.2) with the be initial value , it can be decompose , where satisfy
Then , for any , there exist a temper random radius , such that
Proof: Let is a solution of Equation (3.3), then according to the Equation (3.24) and (3.25), we can see that meet separately
By taking the inner product of equation within , we have
According to lemma 1 and Gronwall inequality,
Replacing by in (3.30), because is tempered, then
Taking inner product (3.30) with in and from Lemma 1 and Lemma 2, we have
Replacing by in (3.32) and from Gronwall inequality, we have
So there is exist a temper random radius
For any ,
This completes the proof of Lemma 3.3.
Lemma 3.4. The Stochastic Dynamic System , while , determined by Equation (3.2) has a compact attracting set .
Proof: Let be a closed ball with radius in space . According to the embedding relation , then is a compact set in . for any temper random set , for any , according to Lemma 3.1, , so for any , we have
So, the whole proof is complete.
According to Lemma 3.1 - Lemma 3.4, there are the following theorems.
Theorem 3.1. Random dynamical system has a family of random attractors , and there exists a slowly increasing random set ,
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