Random Attractors for the Kirchhoff-Type Suspension Bridge Equations with Strong Damping and White Noises

In this paper, we investigate the existence of random attractor for the random dynamical system generated by the Kirchhoff-type suspension bridge equations with strong damping and white noises. We first prove the existence and uniqueness of solutions to the initial boundary value conditions, and then we study the existence of the global attractors of the equation.


Introduction
In this paper, we consider the following stochastic Kirchhoff-type suspension bridge equations ( ) q x H ∈ Ω is not identically equal to zero, f is a nonlinear function satisfying certain conditions.W  is the derivative of a one-dimensional two-valued Wiener process ( ) W t and ( ) q x W  formally describes white noise.
We assume that the nonlinear function satisfies the following assumptions: (a) Growth conditions: ( ) ( ) where 0 C is a positive constant.For example, obviously, ( ) (b) Dissipation conditions: ( ) ( ) ( ) where 1 2 , C C are positive constants.
When ( ) 0 f u = and ( ) 0 q x = , Equation (1.1) is regarded as a model of naval structures,which is originally in [1] introduced by Lazer and McKenna.To the best of our knowledge, Qin [2] [3] proved random attractor for stochastic Kirchhoff equation with white noise, Ma [4] investigated the asymptotic behavior of the solution for the floating beam, that is, the "noise" is absent in (1.1).No one else has studied the long-time behavior of the solutions about these problems, it is just our interest in this paper.As far as the other related problems are concerned, we refer the reader to [2]- [7] and the references therein.
It is well known that Crauel and Flandoli originally introduced the random attractor for the infinite-dimensional RDS [8] [9].A random attractor of RDS is a measurable and compact invariant random set attracting all orbits.It is the appropriate generalization of the now classical attractor exists, it is the smallest attracting compact set and the largest invariant set [10].Zhou et al. [11] studied random attractor for damped nonlinear wave equation with white noise.Fan [12] proved random attractor for a damped stochastic wave equation with multiplicative noise.These abstract results have been successfully applied to many stochastic dissipative partial differential equations.The existence of a random attractors for the wave equations has been investigated by several authors [8] [9] [10].
The outline of this paper is as follows: In Section 2, we recall many basic concepts related to a random attractor for genneral random dynamical system.In Section 3, We prove the existence and uniqueness of the solution corresponding to system (1.1) which determines RDS.In Section 4, we prove the existence of random attractor of the random dynamical system.

Random Dynamical System
In this section, we recall some basic concepts related to RDS and a random attractor for RDS in [8] [9] [10], which are important for getting our main results.

Let ( )
, X X ⋅ be a separable Hilbert space with Borel σ-algebra ( ) and ω ∈Ω .Then φ is called a random dynamical system (RDS).Moreover, φ is called a continuous RDS if φ is continuous with respect to x for 0 t ≥ and ω ∈Ω We denote , H V with the following inner products and norms,respectively: And we introduce the space ( ) , which is used throughout the paper and endow the space E with the following usual scalar product and norm: where T denotes the transposition.
More generally, define

which turns out to be a
Hilbert space with the inner product ( ) , we denote by .
It is convenient to reduce (1.1) to an evolution of the first order in time whose equivalent Itó equation is . Without loss of generality,we can assume that where P is a Wiener measure.We can define a family of measure preserving and ergodic transformations ( ) Apparently, there is no stochastic differential in (3.4) by comparing with From [13] we know that L is the infinitesimal generators of 0 C -semigroup e Lt on E. It is not difficult to check that the functions ( ) → is locally Lipschitz continuous with respect to ϕ and bounded for every ω ∈Ω .
By the classical semigroup theory of existence and uniqueness of solutions of evolution differential equations [13], so we have the following theorem: Theorem 3.1.Consider (3.5).For each ω ∈Ω and initial value ∈ , there exists a unique function ( ) , t ϕ ω such that satisfies the integral equation , d ,  , .
∫ By theorem 3.1, we can prove that for . .P a s − every ω ∈Ω the following statements hold for all 0 T > : 3) The solution mapping of (3.5) satisfies the properties of RDS.Equation (3.5) has a unique solution for every ω ∈Ω .Hence the solution mapping ( ) ( ) ( ) generates a random dynamical system, so the transformation also determines a random dynamical system corresponding to Equation (3.2).

Existence of a Random Attractor
In this section,we prove the existence of a random attractor for RDS (3.7) in E. where , .
To show the conjugation of the solution of the stochastic partial differential Equation (1) and the random partial differential Equation (4.2), we introduce the homeomorphism ( ) ( ) also determines RDS corresponding to Equation (1).Therefore, for RDS (7) we only need consider the equivalent random dynamical system where Next, we prove a positivity property of the operator Q in E that plays a vital role throughout the paper.Lemma 4.1.For any , by using the Poincaré inequality and the Young inequality, we conclude that with initial value ( ) and for all 0 t τ ≤ ≤ ( ψ ω ψ τ ω τ ω ≤ where ( ) ( ) ( ) ( ) ( ) Besides it is easy to deduce a similar absorption result for Proof.We take the inner product in E of (4.2) with ( ) where We deal with the terms in (4.7) one by one as follows: , . 4 By using (1.2)-( 1.3) and the Hölder inequality, we get Inequality (4.12) together with (1.4) yields Collecting with (4.6)-(4.15)and Lemma 4.1, we get that ( ) By the Gronwall lemma, we conclude that ( ) ( ) r ω and ( ) r ω are finite . .P a s − , we get a bounded set B of E, we choose ( ) for all ( ) for all ( ) , and for all ( ) u t be a solution of problem (1.1) with initial value ( ) we make the decomposition ( ) ( ) ( )   ( ) , B be a bounded non-random subset of E , ( ) using the Hölder inequality and the Young inequality, we get that we have that , .
Taking the scalar product in E of (4.25) with is an unknown function,which represents the downward deflection of the road bed in the vertical plane, constant of the ties, the real constant p represents the axial force acting at the end of the road bed of the bridge in the reference configuration.Namely, p is negative when the bridge is stretched, positive when compressed.Ω is an open bounded subset of 2  with sufficiently smooth boundary ∂Ω .( ) ( ) 3 Lemma 4.2.Let (1.2)-(1.4)hold, there exist a random variable ( ) 1 0 r ω > , and a bounded ball 0 B of E centered at 0 with random radius ( ) 0 0 r ω > such that for any bounded non-random set B of E , there exists a deterministic ( ) 1 T B ≤ − such that the solution