The family of random attractors for nonautonomous stochastic higher - order Kirchho ﬀ equations with variable coe

: In this paper, the stochastic asymptotic behavior of the nonautonomous stochastic higher - order Kirchho ﬀ equation with variable coe ﬃ cients is studied. By using the Galerkin method, the solution of this kind of equation is obtained, and stochastic dynamical system under this kind of equation is obtained; by using the uniform estimation, the existence of the family of (cid:2) k - absorbing sets of the stochastic dynamical system Φ k is obtained, and the asymptotic compactness of Φ k is proved by the decomposition method. Finally, the (cid:2) k - stochastic attractor family of the stochastic dynamical system Φ k in obtained.


Introduction
Let ⊂ R Ω N be a bounded domain with smooth boundary (i.e., the derivative of the function at the boundary exists and is continuous). In this paper, we study the asymptotic behavior of nonautonomous stochastic higher-order Kirchhoff equations with variable coefficients on Ω: , is a nonlinear function that satisfies certain growth conditions and dissipation conditions. 

Preparatory knowledge
In this section, we mainly give the related theories of nonautonomous stochastic dynamical systems and random attractor (the family of random attractors). First, the relevant notation needed in this paper is introduced: Define the inner product and norm on ( ) = H L Ω  Let > μ 0 be the weight function, and the weighted space with the following norm: is a separable Hilbert space the inner product and norm are respectively: , .  Assuming that ( ) ‖⋅‖ X, X is a separable Hilbert space, and ( ) B X is the Borel σ-algebra of ( ) X P Ω , , 1 is the metric probability space. if for all ∈ ∈ ∈ + τ R w t s R , Ω , , 1 the following conditions are satisfied: i: : , Ω 1 be a family of subsets parameterized by ( : , is a closed nonempty subset of X; (2) for every fixed ∈ x X and any ∈ τ R, the mapping measurable, then the family D is measurable with to in Ω 1 .
, Ω 1 1 satisfies: : , X be the set of all random tempered sets in X. : , Ω 1 of nonempty subsets of X is called a measurable -pullback attracting(or absorbing) set for { ( , Ω 1 if (1) K is measurable with respect to the P completion of in Ω 1 ; (2) for all ∈ ∈ τ R w , Ω 1 and for every ∈ D , there exists ( )> T D τ w , , 0 such that , Ω , , Φ , , , n n t n n t n n 1 1 n n has a convergent subsequence in X whenever → ∞ t n .
(3) A attracts every member of , i.e., for every ∈ ∈ D τ R , and for every ∈ w Ω 1 , denotes the Hausdorff semi-distance between two subsets P and Q of X.
, then Φ k has the family of pullback k -attracts { } A k if and only if Φ k is pullback k -asymptotically compact in X k and k has closed, -measurable pullback k -absorbing sets K k in k and the unique pullback k -attractor w is a two-sided real-valued Winner processes on the probability space ( is an ergodic metric dynamical system. For a small positive number ε, let z be a new variable given by = + z u εu t and then, system (1.1) becomes , satisfying  To show that problem (3.1) generates a random dynamical system, we let ( ) , , , and then, (3.1) can be rewritten as the equivalent system with random coefficients but without white noise: (3.8) By the standard Galerkin method: If the assumptions ( ) ( ) ( ) , Ω , τ τ with respect to the X k norm. Thus, the solution mapping can be used to define a family of continuous cocycles for (3.8).
→ X X k k be mappings given by , Ω τ τ k 1 , then Φ k is a family of continuous cocycles over ( ) . Ω 1 and ≥ ∈ t s τ R , 0, : For any bounded nonempty subset be a family of bounded nonempty subsets of X k , and for all ∈ ∈ τ R w , Ω 1 , Remember that k is the set of the aforementioned subset family D k , that is,

Uniform estimates of solutions
To prove the existence of the family of random attractors, we conduct uniform estimates on the solutions of the problem (3.8) defined on Ω, for the purposes of showing the existence of a family of k pullback absorbing sets and the pullback k asymptotic compactness of the random dynamical system. Let > ε 0 be small enough and satisfy To obtain uniform estimates of the solutions, ( ) will be given in detail later.
Proof. Taking the inner product of (3 for each term on the right-hand side of (4.2): (4.7) Using the Cauchy-Schwarz inequality, Young's inequality and Holder's inequality, we have   (4.13) The family of random attractors for higher-order Kirchhoff equations  69 Substitute (4.14) into (4.12) to obtain According to (4.1), we get Using the Gronwall inequality to integrate (4.16) over [ ] − τ t τ , with ≥ t 0 and replacing w by − θ w τ , we obtain ,  and there exists ( By (3.4), it is easy to get to any ≥ t 0, When | | → ∞ ξ , ( ) w ξ at most polynomial growth, We get from (4.18) and (4.21) that and ( ) r τ w , 10 is bounded. □ Taking the inner product of (3.8) with ( ) For each term on the right-hand side of (4.24):    According to the interpolation inequality, we have

(4.36)
The family of random attractors for higher-order Kirchhoff equations  73  Using the Cauchy-Schwarz inequality,Young's inequality and Holder's inequality, etc. we have  Substitute (4.37)-(4.40) into (4.36) to obtain     When = k m, (4.43) is also true, which will not be detailed here. Using the Gronwall inequality to integrate (4.43) over [ ] − τ t τ , and replacing w by − θ w τ we obtain

(4.45)
The family of random attractors for higher-order Kirchhoff equations  75 When | | ( ) → ∞ ξ wξ at most polynomial growth, We get from (4.45)-(4.47), ( ( ) ) such that the solution of (3.8) satisfies for ≥ ≥ t T n N , , , n n n n n n n n ,1 , ,1 . Applying ( ) − I P n to the second equation of (3.8), we obtain , we have (4.50) Applying − I P n to the first equation of (3.8), we get For the third and fourth terms on the right-hand side of (4.49), we obtain for the fifth term on the right-hand side of (4.49), we have for the sixth term on the right-hand side of (4.49), we have for the seventh, eighth, and ninth terms on the right-hand side of (4.49), we have  By using Young's inequality and Holder's inequality, we can get  (4.63) By substituting (4.52)-(4.63) into (4.50), we have   Then, there is an appropriate positive constant σ 2 so that (4.64) can be reduced to (4.71) Then applying − I P n to the first equation of (3.8), we obtain

The existence of the family of random attractors
In this section, we shall prove the existence of the family of random pullback attractors for system (3.8