In this paper, we consider the following stochastic Kirchhoff-type suspension bridge equations

{ u t t + Δ 2 u + Δ 2 u t + ( p − | ∇ u | 2 ) Δ u + b u + + f ( u ) = q ( x ) W ˙ ,   Ω × [ τ , + ∞ ) , τ ∈ ℝ ,   u ( x , t ) = ∇ u ( x , t ) = 0 ,   x ∈ ∂ Ω , t ≥ τ , u ( x , τ ) = u 0 ( x ) ,   u t ( x , τ ) = u 1 ( x )   x ∈ Ω ,   (1.1)

where u ( x , t ) is an unknown function,which represents the downward deflection of the road bed in the vertical plane, u + = u for u ≥ 0 and u + = 0 for u < 0 . b > 0 denotes the spring 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 ℝ 2 with sufficiently smooth boundary ∂ Ω . q ( x ) ∈ H 3 ( 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 f ∈ C 2 ( ℝ , ℝ ) with f ( 0 ) = 0 , which satisfies the following assumptions:

(a) Growth conditions:

| f ( s ) | ≤ C 0 ( 1 + | s | k ) ,     k ≥ 1, ∀   s ∈ ℝ , (1.2)

where C 0 is a positive constant. For example, obviously, f ( s ) = | s | k − 1 s satisfies (1.2).

(b) Dissipation conditions:

F ( s ) : = ∫ 0 s   f ( r ) d r ≥ C 1 ( | s | k + 1 − 1 ) , ∀ s ∈ ℝ , (1.3)

and

s f ( s ) ≥ C 2 ( F ( s ) − 1 ) , ∀ s ∈ ℝ , (1.4)

where C 1 , C 2 are positive constants.

When f ( u ) = 0 and q ( x ) = 0 , 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.

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 s-algebra B ( X ) , and let ( Ω , F , P ) be a probability space. θ t : Ω → Ω , t ∈ ℝ is a family of measure preserving transformations such that ( t , ω ) ↦ θ t ω is measurable, θ 0 = i d and θ t + s = θ t θ s for all t , s ∈ ℝ . The flow θ t together with the probability space ( Ω , F , P , ( θ t ) t ∈ ℝ ) is called a metric dynamical system.

Definition 2.1. Let ( Ω , F , P , ( θ t ) t ∈ ℝ ) be a metric dynamical system. Suppose that the mapping ϕ : ℝ + × Ω × X → X is ( B ( ℝ + ) × F × B ( X ) ) -measurable and satisfies the following properties:

1) ϕ ( 0 , ω ) x = x , x ∈ X and ω ∈ Ω ;

2) ϕ ( t + s , ω ) = ϕ ( t , θ s ω ) ∘ ϕ ( s , ω ) , for all t , s ∈ ℝ + , x ∈ X 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 t ≥ 0 and ω ∈ Ω .

Definition 2.2. A set-valued map D : Ω → 2 X is said to be a closed (compact)random set if D ( ω ) is closed (compact) for P − a . s . ω ∈ Ω , and ω ↦ d ( x , D ( ω ) ) is p − a . s . measurable for all x ∈ X .

Definition 2.3. If K and B are random sets such that for P − a . s . ω there exists a time t B ( ω ) such that for all t ≥ t B ( ω ) ,

ϕ ( t , θ − t ω ) B ( θ − t ω ) ⊂ K ( ω ) ,

then K is said to absorb B, and t B ( ω ) is called the absorption time.

Definition 2.4. A random set A = A ( ω ) ω ∈ Ω ⊂ X is called a random attractor associated to the RDS ϕ if P − a . s . :

1) A is a random compact set, i.e., A ( ω ) is compact for P − a . s . ω ∈ Ω , and the map ω ↦ d ( x , A ( ω ) ) is measurable for every x ∈ X ;

2) A is ϕ -invariant, i.e., ϕ ( t , ω ) A ( ω ) = A ( θ t ω ) for all t ≥ 0 and P − a . s . ω ∈ Ω ;

3) A attracts every set B in X, i.e., for all bounded (and non-random) B ⊂ X ,

l i m t → ∞ d ( ϕ ( t , θ − t ω ) B ( θ − t ω ) , A ( ω ) ) = 0,

where d ( ⋅ , ⋅ ) denotes the Hausdorff semi-distance:

d ( A , B ) = sup x ∈ A inf y ∈ B d ( x , y ) , A , B ∈ X .

Note that ϕ ( t , θ − t ω ) x can be interpreted as the position of the trajectory which was in x at time − t . Thus, the attraction property holds from t = − ∞ .

Theorem 2.1. [8] (Existence of a random attractor) Let ϕ be a continuous random dynamical system on X over ( Ω , F , P , ( θ t ) t ∈ ℝ ) . Suppose that there exists a random compact set K ( ω ) absorbing every bounded non-random set B ⊂ X . Then the set

A = A ( ω ) ω ∈ Ω = ∪ B ⊂ X ∧ B ( ω ) ¯

is a global random attractor for ϕ , where the union is taken over all bounded B ⊂ X , and ∧ B ( ω ) is the w-limitsset of B given by

∧ B ( ω ) = ∩ s ≥ 0 ∪ t ≥ s     ϕ ( t , θ − t ω ) B ( θ − t ω ) ¯

With the usual notation, we denote

H = L 2 ( Ω ) ,     V = H 2 ( Ω ) D ( A ) = H 2 ( Ω ) ∩ H 0 1 ( Ω ) ,     D ( A 2 ) = { u ∈ H 4 ( Ω ) : A 2 u ∈ L 2 ( Ω ) } ,

where A = − Δ , A 2 = Δ 2 . We denote H , V with the following inner products and norms,respectively:

( u , v ) = ∫ Ω   u v d x ,     ‖ u ‖ 2 = ( u , u ) , ∀ u , v ∈ H , ( ( u , v ) ) = ∫ Ω   Δ u Δ v d x ,     ‖ u ‖ 2 2 = ( ( u , u ) ) , ∀ u , v ∈ V .

And we introduce the space E = D ( A ) × H , which is used throughout the paper and endow the space E with the following usual scalar product and norm:

( y 1 , y 2 ) E = ( ( u 1 , u 2 ) ) + ( v 1 , v 2 ) , ∀ y i = ( u i , v i ) T ∈ E , i = 1 , 2 , ‖ y ‖ E 2 = ( y , y ) E ,     ∀ y = ( u , v ) T ∈ E

where T denotes the transposition.

More generally, define H r = D ( A r 2 ) for r ∈ ℝ , which turns out to be a Hilbert space with the inner product ( u , v ) r = ( A r 2 u , A r 2 v ) , we denote by ‖   ⋅   ‖ r = ‖ A r 2 ‖ the norm on H r induced by the above inner product. Let λ > 0

be the first eigenvalue of A 2 u = λ u , u ( x , t ) = ∇ u ( x , t ) = 0 , x ∈ ∂ Ω , by the compact embeddings H r + 1 → H r along with the generalized Poincaré inequality, we have

‖ u ‖ r + 1 2 ≥ λ ‖ u ‖ r 2 . (3.1)

It is convenient to reduce (1.1) to an evolution of the first order in time

{ u t = v , v t = − A 2 u − A 2 v + ( p − | ∇ u | 2 ) A u − b u + − f ( u ) + q ( x ) W ˙ , u ( x , τ ) = u 0 ( x ) , u t ( x , τ ) = u 1 ( x ) , x ∈ Ω , (3.2)

whose equivalent Itó equation is

{ d u = v d t , d v = − A 2 u d t − A 2 v d t + ( p − | ∇ u | 2 ) A u d t − b u + d t − f ( u ) d t + q ( x ) d W , u ( x , τ ) = u 0 ( x ) ,   u t ( x , τ ) = u 1 ( x ) ,   x ∈ Ω , (3.3)

where W ( t ) is a one-dimensional two-sided real-valued Wiener process on ( Ω , F , P , ( θ t ) t ∈ ℝ ) . Without loss of generality,we can assume that

Ω = { ω ( t ) = W ( t ) ∈ C ( ℝ , ℝ ) : ω ( 0 ) = 0 } ,

where P is a Wiener measure. We can define a family of measure preserving and ergodic transformations ( θ t ) t ∈ ℝ by

θ t ω ( ⋅ ) = ω ( ⋅ + t ) ω ( t ) ,   t ∈ ℝ , ω ∈ Ω   .

Let z = v − q ( x ) W , we consider the random partial differential equation equivalent to (3.3)

{ d u d t = z + q ( x ) W , d z d t = − A 2 u − A 2 z + ( p − | ∇ u | 2 ) A u − b u + − f ( u ) − A 2 q ( x ) W , u ( x , τ ) = u 0 ( x ) ,   z ( τ , ω ) = z ( x , τ , ω ) = u 1 ( x ) − q ( x ) W ( τ ) ,   x ∈ Ω   . (3.4)

Apparently, there is no stochastic differential in (3.4) by comparing with stochastic differential Equation (3.3). Let

φ = ( u z ) , L = ( 0 I − A 2 − A 2 ) , F ( φ , ω ) = ( q ( x ) W ( p − | ∇ u | 2 ) A u − b u + − f ( u ) − A 2 q ( x ) W ) ,

then (3.4) can be written as

φ ˙ = L φ + F ( φ , ω ) ,   φ ( τ , ω ) = ( u 0 , z ( τ , ω ) ) T . (3.5)

From [13] we know that L is the infinitesimal generators of C 0 -semigroup e L t on E. It is not difficult to check that the functions F ( φ , ω ) : E → E 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 φ ( τ , ω ) = ( u 0 , z ( τ , ω ) ) T = ( u 0 , u 1 ( x ) − q ( x ) W ( τ ) ) T ∈ E , there exists a unique function φ ( t , ω ) such that satisfies the integral equation

φ ( t , ω ) = e L ( t − τ ) φ ( τ , ω ) + ∫ τ t   F ( φ ( s ) , ω ) d s ,     ∀ φ ( τ , ω ) ∈ E .

By theorem 3.1, we can prove that for P − a . s . every ω ∈ Ω the following statements hold for all T > 0 :

1) If φ ( τ , ω ) ∈ E , then φ ( t , ω ) ∈ C ( [ τ , τ + T ) ; D ( A ) ) × C ( [ τ , τ + T ) ; H ) .

2) φ ( t , φ ( τ , ω ) ) is continuous in t and φ ( τ , ω ) .

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

S ˜ ( t , ω ) : φ ( τ , ω ) ↦ φ ( t , ω ) (3.6)

generates a random dynamical system, so the transformation

S ( t , ω ) : φ ( τ , ω ) + ( 0 , q ( x ) W ( τ ) ) T ↦ φ ( t , ω ) + ( 0 , q ( x ) W ( t ) ) T (3.7)

also determines a random dynamical system corresponding to Equation (3.2).

In this section,we prove the existence of a random attractor for RDS (3.7) in E. Let z ˜ = z + ε u , ψ = ( u , z ˜ ) T , where

ε = λ 2 4 λ 2 + 3 λ + 4 . (4.1)

So Equation (3.4) can be written as

ψ ˙ + Q ψ = F ˜ ( ψ , ω ) ,   ψ ( τ , ω ) = ( u 0 , z ( τ , ω ) + ε u 0 ) T ,   t ≥ τ . (4.2)

where

Q = ( ε I − I ( 1 − ε ) A 2 + ε 2 I A 2 − ε I ) ,

F ˜ ( ψ , ω ) = ( q ( x ) W ( p − | ∇ u | 2 ) A u − b u + − f ( u ) + ( ε − A 2 ) q ( x ) W ) .

The mapping

S ˜ ε ( t , ω ) : ( u 0 , z ( τ , ω ) + ε u 0 ) T ↦ ( u ( t ) , z ( t ) + ε u ( t ) ) T ,     E → E ,   t ≥ τ

is defined by (4.2).

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

R ε : ( u , z ) T ↦ ( u , z + ε u ) T

with the inverse homeomorphism R − ε . Then the transformation

S ˜ ε ( t , ω ) = R ε S ( t , ω ) R − ε (4.3)

also determines RDS corresponding to Equation (1). Therefore, for RDS (7) we only need consider the equivalent random dynamical system S ε ( t , ω ) = R ε S ( t , ω ) R − ε , where S ε ( t , ω ) is decided by

ξ ˙ + Q ξ = G ( ξ , ω ) ,   ξ ( τ , ω ) = ( u 0 , u 1 + ε u 0 ) T ,   t ≥ τ , (4.4)

where

ξ = ( u ( t ) u t ( t ) + ε u ( t ) ) ,   G ( ξ , ω ) = ( 0 ( p − | ∇ u | 2 ) A u − b u + − f ( u ) + q ( x ) W ˙ ) .

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 ϕ = ( u , z ) T ∈ E , there holds

( Q ϕ , ϕ ) E ≥ ε 2 ‖ ϕ ‖ E 2 + ε 4 ‖ u ‖ 2 2 + λ 2 ‖ z ‖ 2 .

Proof. Since Q ϕ = ( ε u − z , ( 1 − ε ) A 2 u + ε 2 u + A 2 z − ε z ) T , by using the Poincaré inequality and the Young inequality, we conclude that

( Q ϕ , ϕ ) E = ε ‖ u ‖ 2 2 − ε ( A u , A z ) + ε 2 ( u , z ) + ‖ A z ‖ 2 − ε ‖ z ‖ 2 ≥ ε ‖ u ‖ 2 2 − ε 8 ‖ u ‖ 2 2 − 2 ε ‖ A z ‖ 2 − ε 8 ‖ u ‖ 2 2 − 2 ε 3 λ ‖ z ‖ 2 + ‖ A z ‖ 2 − ε ‖ z ‖ 2 ≥ ε 2 ‖ ϕ ‖ E 2 + ε 4 ‖ u ‖ 2 2 + ( 1 − 2 ε ) λ ‖ z ‖ 2 − ( 2 ε λ + 3 ε 2 ) ‖ z ‖ 2 = ε 2 ‖ ϕ ‖ E 2 + ε 4 ‖ u ‖ 2 2 + λ 2 ‖ z ‖ 2 .

where ε = λ 2 4 λ 2 + 3 λ + 4 . ,

Lemma 4.2. Let (1.2)-(1.4) hold, there exist a random variable r 1 ( ω ) > 0 , and a bounded ball B 0 of E centered at 0 with random radius r 0 ( ω ) > 0 such that for any bounded non-random set B of E , there exists a deterministic T ( B ) ≤ − 1 such that the solution ψ ( t , ω , ψ ( τ , ω ) ) = ( u ( t , ω ) , z ˜ ( t , ω ) ) T of (4.2) with initial value ( u 0 , u 1 + ε u 0 ) T ∈ B satisfies for P − a . s . ω ∈ Ω ,

‖ ψ ( − 1, ω , ψ ( τ , ω ) ) ‖ E ≤ r 0 ( ω ) ,     τ ≤ T ( B ) ,

and for all τ ≤ t ≤ 0

‖ ψ ( t , ω , ψ ( τ , ω ) ) ‖ E 2 ≤ R ( τ , ω ) , (4.5)

where z ˜ ( t ω ) = u t ( t ) + ε u ( t ) − q ( x ) W ( t ) , and R ( τ , ω ) is given by

R ( τ , ω ) = 2 e − ε 1 ( t − τ ) ( ‖ u 0 ‖ 2 2 + ‖ u 1 + ε u 0 ‖ 2 + ‖ q ‖ 2 | W | 2 + ∫ Ω F ( u 0 ) d x + | ∇ u 0 | 2 − p ) 2 )                       + r 1 2 ( ω ) .

Besides it is easy to deduce a similar absorption result for

ϕ ( − 1 ) = ( u ( − 1 ) , u t ( − 1 ) + ε u ( − 1 ) ) T

instead of ψ ( − 1 ) .

Proof. We take the inner product in E of (4.2) with ψ = ( u , z ˜ ) T , where z ˜ = u t + ε u − q ( x ) W , we get

1 2 d d t ‖ ψ ‖ E 2 + ( Q ψ , ψ ) E = ( F ˜ ( ψ , ω ) , ψ ) E ,     ∀ t ≥ τ , (4.6)

where

( F ˜ ( ψ , ω ) , ψ ) E = ( ( u , q ( x ) W ) ) − b ( u + , z ˜ ) − ( f ( u ) , z ˜ )     − ( A 2 q ( x ) W , z ˜ ) + ε ( q ( x ) W , z ˜ ) + ( ( p − | ∇ u | 2 ) A u , z ˜ ) . (4.7)

We deal with the terms in (4.7) one by one as follows:

( ( u , q ( x ) W ) ) ≤ ε 4 ‖ u ‖ 2 2 + 1 ε ‖ q ‖ 2 2 | W ( t ) | 2 ; (4.8)

− b ( u + , z ˜ ) = − b ( u + , u t + ε u − q ( x ) W ) = − 1 2 d d t b ‖ u + ‖ 2 − ε b ‖ u + ‖ 2 + b ( u + , q ( x ) W ) ≤ − 1 2 d d t b ‖ u + ‖ 2 − ε b 2 ‖ u + ‖ 2 + b ‖ q ‖ 2 2 ε | W | 2 ; (4.9)

ε ( q ( x ) W , z ˜ ) ≤ λ 4 ‖ z ˜ ‖ 2 + ε 2 λ ‖ q ‖ 2 | W | 2 ; (4.10)

| − ( A 2 q ( x ) W , z ˜ ) | ≤ ‖ q ‖ 4 2 λ | W | 2 + λ 4 ‖ z ˜ ‖ 2 . (4.11)

By using (1.2)-(1.3) and the Hölder inequality, we get

( f ( u ) , q ( x ) W ) ≤ C 0 ∫ Ω ( 1 + | u | k ) q ( x ) W ( t ) d x ≤ C 0 ‖ q ‖ | W | + C 0 ( ∫ Ω | u | k + 1 d x ) k k + 1 ‖ q ‖ k + 1 | W | ≤ C 0 ‖ q ‖ | W | + C 0 C 1 − k k + 1 ( ∫ Ω ( F ( u ) + C 1 ) d x ) k k + 1 ‖ q ‖ k + 1 | W | ≤ C 0 ‖ q ‖ | W | + ε C 0 C 1 − 1 2 ∫ Ω F ( u ) d x + C 0 2 ε ‖ q ‖ k + 1 k + 1 | W | k + 1 + ε C 0 | Ω | 2 . (4.12)

Inequality (4.12) together with (1.4) yields

− ( f ( u ) , z ˜ ) = − ( f ( u ) , u t + ε u − q ( x ) W ) ≤ − d d t ∫ Ω F ( u ) d x − ε C 2 ∫ Ω F ( u ) d x + ε C 2 | Ω | + ( f ( u ) , q ( x ) W ) ≤ − d d t ∫ Ω F ( u ) d x − ε ( 2 C 2 − C 0 C 1 − 1 ) 2 ∫ Ω F ( u ) d x + C 0 ‖ q ‖ | W |     + C 0 2 ε ‖ q ‖ k + 1 k + 1 | W | k + 1 + ε ( C 0 + 2 C 2 ) 2 | Ω | . (4.13)

( ( p − | ∇ u | 2 ) A u , z ˜ ) = ( p − | ∇ u | 2 ) ( A u , u t + ε u − q ( x ) W ) = ( p − | ∇ u | 2 ) [ 1 2 d | ∇ u | 2 d t + ε | ∇ u | 2 − ( A u , q ( x ) W ) ] = − 1 4 d d t ( | ∇ u | 2 − p ) 2 − ε ( | ∇ u | 2 − p ) 2 − ε p ( | ∇ u | 2 − p )     + ( | ∇ u | 2 − p ) ( A u , q ( x ) W ) . (4.14)

( | ∇ u | 2 − p ) ( A u , q ( x ) W ) ≤ ε 4 ( | ∇ u | 2 − p ) 2 + ε 2 | ∇ u | 4 + 1 2 ε 3 | ∇ q | 4 | W | 4 . (4.15)

Collecting with (4.6)-(4.15) and Lemma 4.1, we get that

d d t [ ‖ ψ ‖ E 2 + b ‖ u + ‖ 2 + 2 ∫ Ω F ( u ) d x + 1 2 ( | ∇ u | 2 − p ) 2 + 2 C 1 | Ω | ] +   ε 1 [ ‖ ψ ‖ E 2 + b ‖ u + ‖ 2 + 2 ∫ Ω F ( u ) d x + 1 2 ( | ∇ u | 2 − p ) 2 + 2 C 1 | Ω | ] ≤ M ( 1 + | W ( t ) | + | W ( t ) | 2 + | W ( t ) | 4 + | W ( t ) | k + 1 ) ,

where ε 1 = min { ε , 2 C 2 − C 0 C 1 − 1 2 ε } , C 2 > C 0 C 1 − 1 2 , and

M = max { ε p 2 + ε ( C 0 + 2 C 2 + 2 C 1 ) | Ω | , C 0 ‖ q ‖ ,                                 ‖ q ‖ 2 2 ε + ( ε 2 λ + b 2 ε ) ‖ q ‖ 2 + ‖ q ‖ 4 2 λ , C 0 ‖ q ‖ k + 1 k + 1 2 ε , ‖ ∇ q ‖ 4 2 ε 3 } .

By the Gronwall lemma, we conclude that

‖ ψ ( t , ω , ψ ( τ , ω ) ) ‖ E 2 ≤ e − ε 1 ( t − τ ) ( ‖ ψ ( τ , ω ) ‖ E 2 + b ‖ u 0 + ‖ 2 + 2 ∫ Ω F ( u 0 ) d x + 1 2 ( | ∇ u 0 | 2 − p ) 2 + 2 C 1 | Ω | )     + M ∫ τ t   e − ε 1 ( t − s ) ( 1 + | W ( s ) | + | W ( s ) | 2 + | W ( s ) | 4 + | W ( s ) | k + 1 ) d s

≤ 2 e − ε 1 ( t − τ ) [ ‖ u 0 ‖ 2 2 + ‖ u 1 + ε u 0 ‖ 2 + ‖ q ‖ 2 | W ( τ ) | 2 + b ‖ u 0 + ‖ 2     + ∫ Ω F ( u 0 ) d x + 1 4 ( | ∇ u 0 | 2 − p ) 2 ] + M ∫ τ t   e − ε 1 ( t − s ) ( 1 + | W ( s ) | + | W ( s ) | 2 + | W ( s ) | 4 + | W ( s ) | k + 1 ) d s . (4.16)

Let

r 0 2 ( ω ) = 2 ( 1 + s u p τ ≤ − 1 e ε 1 τ ‖ q ‖ 2 | W ( τ ) | 2 ) + M ε 1     + M ∫ − ∞ − 1   e − ε 1 ( − 1 − s ) ( | W ( s ) | + | W ( s ) | 2 + | W ( s ) | 4 + | W ( s ) | k + 1 ) d s ,

r 1 2 ( ω ) = M ε 1 + M ∫ − ∞ 0   e ε 1 s ( | W ( s ) | + | W ( s ) | 2 + | W ( s ) | 4 + | W ( s ) | k + 1 ) d s ,

where r 0 2 ( ω ) and r 1 2 ( ω ) are finite P − a . s . , we get a bounded set B of E, we choose T ( B ) ≤ − 1 such that

e − ε 1 ( − 1 − τ ) ( ‖ u 0 ‖ 2 2 + ‖ u 1 + ε u 0 ‖ 2 + b ‖ u 0 + ‖ 2 + ∫ Ω F ( u 0 ) d x + C 1 | Ω | + 1 4 ( | ∇ u 0 | 2 − p ) 2 ) ≤ 1 (4.17)

for all ∀ ( u 0 , u 1 + ε u 0 ) T ∈ B , and

e ε 1 τ ( ‖ u 0 ‖ 2 2 + ‖ u 1 + ε u 0 ‖ 2 + b ‖ u 0 + ‖ 2 + ∫ Ω F ( u 0 ) d x + C 1 | Ω | + 1 4 ( | ∇ u 0 | 2 − p ) 2 ) ≤ 1 (4.18)

for all ∀ ( u 0 , u 1 + ε u 0 ) T ∈ B , and for all τ ≤ T ( B ) . ,

Let u ( t ) be a solution of problem (1.1) with initial value ( u 0 , u t + ε u 0 ) T ∈ B . we make the decomposition u ( t ) = y 1 ( t ) + y 2 ( t ) , where y 1 ( t ) and y 2 ( t ) satisfy

{ y 1 t t + Δ 2 y 1 + Δ 2 y 1 t + ( p − | ∇ u | 2 ) Δ y 1 = 0 ,   Ω × [ τ , + ∞ ) , τ ∈ ℝ ,   y 1 ( x , t ) = ∇ y 1 ( x , t ) = 0 ,   x ∈ ∂ Ω , t ≥ τ ,   y 1 ( x , τ ) = u 0 ( x ) ,   y 1 t ( x , τ ) = u 1 ( x ) ,   x ∈ Ω   .   (4.19)

and

{ y 2 t t + Δ 2 y 2 + Δ 2 y t + ( p − | ∇ u | 2 ) Δ y 2 + b u + + f ( u ) = q ( x ) W ˙ ,   Ω × [ τ , + ∞ ) , τ ∈ ℝ ,   y 2 ( x , t ) = ∇ y 2 ( x , t ) = 0 ,   x ∈ ∂ Ω , t ≥ τ ,   y 2 ( x , τ ) = 0 ,   y 2 t ( x , τ ) = 0 ,   x ∈   Ω   .   (4.20)

Lemma 4.3. Let p < λ 4 , B be a bounded non-random subset of E , ∀ ( u 0 , u 1 + ε u 0 ) T ∈ B

‖ Y 1 ( 0 ) ‖ E 2 = ‖ y 1 ( 0 ) ‖ 2 2 + ‖ y 1 t ( 0 ) + ε y 1 ( 0 ) ‖ 2 ≤ e ε τ ( 1 − ε ) C ( p ) ( ‖ u 0 ‖ 2 2 − ( p − | ∇ u 0 | 2 ) | ∇ u 0 | 2 + ‖ u 1 + ε u 0 ‖ 2 ) , (4.21)

where Y 1 = ( y 1 , y 1 t + ε y 1 ) T satisfies (4.19), C ( p ) = { 0 , p ≤ 0 , 1 − p λ , p < λ 4 .

Proof. Let v = y 1 t + ε y 1 , taking the scalar product in H of (4.19), we get that

1 2 d d t [ ‖ v ‖ 2 + ( 1 − ε ) ‖ y 1 ‖ 2 2 + ( | ∇ u 0 | 2 − p ) | ∇ y 1 | 2 ] + ‖ v ‖ 2 2 − ε ‖ v ‖ 2 + ε ( 1 − ε ) ‖ y 1 ‖ 2 2 + ε ( | ∇ u 0 | 2 − p ) | ∇ y 1 | 2 + ε 2 ( y 1 , v ) = 0, (4.22)

using the Hölder inequality and the Young inequality, we get that

‖ v ‖ 2 2 − ε ‖ v ‖ 2 + ε ( 1 − ε ) ‖ y 1 ‖ 2 2 + ε 2 ( y 1 , v ) ≥ ε ( 1 − ε ) ‖ y 1 ‖ 2 2 + ‖ v ‖ 2 2 − ε ‖ v ‖ 2 − ε ( 1 − ε ) 2 ‖ y 1 ‖ 2 2 − ε 3 2 ( 1 − ε ) λ ≥ ε ( 1 − ε ) 2 ‖ y 1 ‖ 2 2 + ( λ − ε − ε 2 λ ) ‖ v ‖ 2 ≥ ε ( 1 − ε ) 2 ‖ y 1 ‖ 2 2 + ε 2 ‖ v ‖ 2 , (4.23)

we have that

d d t [ ‖ v ‖ 2 + ( 1 − ε ) ‖ y 1 ‖ 2 2 + ( | ∇ u 0 | 2 − p ) | ∇ y 1 | 2 ] +   ε [ ‖ v ‖ 2 + ( 1 − ε ) ‖ y 1 ‖ 2 2 + 2 ( | ∇ u 0 | 2 − p ) | ∇ y 1 | 2 ] ≤ 0.

Let ϑ ( t ) = ‖ v ‖ 2 + ( 1 − ε ) ‖ y 1 ‖ 2 2 + ( | ∇ u 0 | 2 − p ) | ∇ y 1 | 2 , we can get that

ϑ ( t ) ≥ ( 1 − ε ) C ( p ) ‖ y 1 ‖ 2 2 + ‖ v ‖ 2 + | ∇ u | 2 | ∇ y 1 | 2 > ( 1 − ε ) C ( p ) ‖ Y 1 ‖ E 2 > 0

and for p < λ 4 we get

ε [ ‖ v ‖ 2 + ( 1 − ε ) ‖ y 1 ‖ 2 2 + 2 ( | ∇ u 0 | 2 − p ) | ∇ y 1 | 2 ] ≥ ε 2 ϑ ( t ) ,

so

d d t ϑ ( t ) + ε 2 ϑ ( t ) ≤ 0,

By the Gronwall lemma, getting that (4.21). ,

Lemma 4.4. Assume that (1.2) holds, there exists a random radius r 2 ( ω ) , such that for P − a . s . ω ∈ Ω ,

‖ A 1 2 Y 2 ( 0, ω ; Y 2 ( τ , ω ) ) ‖ E 2 ≤ r 2 2 ( ω ) , (4.24)

where Y 2 = ( y 2 , y 2 t + ε y 2 − q ( x ) W ) T satisfies (4.20).

Proof. Let Y 2 = ( y 2 , y 2 t + ε y 2 − q ( x ) W ) T , Equation (4.20) can be written to

Y 2 t + Q Y 2 = H ( Y 2 , ω ) ,   Y 2 ( τ ) = ( 0 , − q ( x ) W ( τ ) ) T , (4.25)

where

H ( Y 2 , ω ) = ( q ( x ) W ( p − | ∇ u | 2 ) A y 2 − b u + − f ( u ) + ( ε − A 2 ) q ( x ) W ) .

Taking the scalar product in E of (4.25) with A Y 2 , we get that

1 2 d d t ‖ A 1 2 Y 2 ‖ E 2 + ( Q Y 2 , A Y 2 ) E = ( H ( Y 2 , ω ) , A Y 2 ) E , (4.26)

where

( H ( Y 2 , ω ) , A Y 2 ) E = ( ( A y 2 , q ( x ) W ) ) + ( ( p − | ∇ u | 2 ) A y 2 , A ( y 2 t + ε y 2 − q ( x ) W ) )     + ( − b u + − f ( u ) + ( ε − A 2 ) q ( x ) W , A ( y 2 t + ε y 2 − q ( x ) W ) ) . (4.27)

Due to Lemma 4.1, we get that

( Q Y 2 , A Y 2 ) E ≥ ε 2 ‖ A 1 2 Y 2 ‖ E 2 + ε 4 ‖ A 1 2 y 2 ‖ 2 2 + λ 2 ‖ A 1 2 ( y 2 t + ε y 2 − q ( x ) W ) ‖ 2 . (4.28)

Using the Young inequality, we have that

( ( A y 2 , q ( x ) W ) ) ≤ ε 4 ‖ A 1 2 y 2 ‖ 2 2 + 1 ε ‖ A 1 2 q ‖ 2 2 | W | 2 ; (4.29)

| − ( b u + , A ( y 2 t + ε y 2 − q ( x ) W ) ) | ≤ 2 b 2 λ ‖ A 1 2 u + ‖ 2 2 + λ 8 ‖ A 1 2 ( y 2 t + ε y 2 − q ( x ) W ) ‖ 2 ; (4.30)

| − ( A 2 q ( x ) W , A ( y 2 t + ε y 2 − q ( x ) W ) ) | ≤ 2 λ ‖ A 5 2 q ‖ 2 | W | 2 + λ 8 ‖ A 1 2 ( y 2 t + ε y 2 − q ( x ) W ) ‖ 2 ; (4.31)

( ε q ( x ) W , A ( y 2 t + ε y 2 − q ( x ) W ) ) ≤ 2 ε 2 λ ‖ A 1 2 q ‖ 2 | W | 2 + λ 8 ‖ A 1 2 ( y 2 t + ε y 2 − q ( x ) W ) ‖ 2 . (4.32)

By (1.2), (4.5) and Sobolev embedding theorem, we obtain that f ( u ) is uniformly bounded in L ∞ , that is, there exists a constant M > 0 such that

| f ′ ( s ) | L ∞ ≤ M . (4.33)

Combining with (4.33), the Sobolev embedding theorem and the Young inequality, we have that

| − ( f ( u ) , A ( y 2 t + ε y 2 − q ( x ) W ) ) | ≤ ‖ A 1 2 f ( u ) ‖ ‖ A 1 2 ( y 2 t + ε y 2 − q ( x ) W ) ‖ ≤ 2 μ M 2 λ ‖ u ‖ 2 2 + λ 8 ‖ A 1 2 ( y 2 t + ε y 2 − q ( x ) W ) ‖ 2 , (4.34)

where μ is a positive constant.

− ( ( p − | ∇ u | 2 ) A y 2 , A ( y 2 t + ε y 2 − q ( x ) W ) ) = 1 2 d d t ( | ∇ u | 2 − p ) ‖ y 2 ‖ 2 2 + ε ( | ∇ u | 2 − p ) ‖ y 2 ‖ 2 2 + ( p − | ∇ u | 2 ) ( A y 2 , q ( x ) W ) ≥ 1 2 d d t ( | ∇ u | 2 − p ) ‖ y 2 ‖ 2 2 + ε ( | ∇ u | 2 − p ) ‖ y 2 ‖ 2 2 − ( | p | + | ∇ u | 2 ) ‖ y 2 ‖ 2 ‖ A q ‖ | W |

≥ 1 2 d d t ( | ∇ u | 2 − p ) ‖ y 2 ‖ 2 2 + ε 2 ( | ∇ u | 2 − p ) ‖ y 2 ‖ 2 2 − ε 2 | p | ‖ y 2 ‖ 2 2       − 1 2 ‖ A q ‖ 2 | W | 2 − 1 2 ( | p | + | ∇ u | 2 ) 2 ‖ y 2 ‖ 2 2 = 1 2 d d t ( | ∇ u | 2 − p ) ‖ y 2 ‖ 2 2 + ε 2 ( | ∇ u | 2 − p ) ‖ y 2 ‖ 2 2       − 1 2 ( ε | p | + ( | p | + | ∇ u | 2 ) 2 ) ‖ y 2 ‖ 2 2 − 1 2 ‖ A q ‖ 2 | W | 2 (4.35)

Let E ( t ) = ‖ A 1 2 Y 2 ‖ E 2 + ( | ∇ u | 2 − p ) ‖ y 2 ‖ 2 2 , by the Poincaré and C ( p ) , we get

that, E ( t ) ≥ C ( p ) ‖ A 1 2 Y 2 ‖ E 2 > 0 . Using (4.27)-(4.35) and (4.5), for τ ≤ T ( ω ) , from (4.27) we get that

d d t E ( t ) + ε E ( t ) ≤ C ( ε | p | + ( | p | + R ( s , ω ) ) 2 + 4 b 2 + 4 μ M 2 λ ) R ( s , ω )     + ( 2 ε ‖ A 1 2 q ‖ 2 2 + 4 λ ‖ A 5 2 q ‖ 2 + 4 ε 2 λ ‖ A 1 2 q ‖ 2 + 1 2 ‖ A q ‖ 2 ) | W ( t ) | 2 ,   τ ≤ t ≤ 0.

Using the Gronwall lemma, we get that

E ( 0 ) ≤ e ε τ ‖ A 1 2 q ‖ 2 + C ∫ τ 0   e ε s ( ε | p | + ( | p | + R ( s , ω ) ) 2 + 4 b 2 + 4 μ M 2 λ ) R ( s , ω ) d s     + ( 2 ε ‖ A 1 2 q ‖ 2 2 + 4 λ ‖ A 5 2 q ‖ 2 + 4 ε 2 λ ‖ A 1 2 q ‖ 2 + 1 2 ‖ A q ‖ 2 ) ∫ τ 0   e ε s | W ( s ) | 2 d s . (4.36)

Set

r 2 2 ( ω ) = ( 2 ε ‖ A 1 2 q ‖ 2 2 + 4 λ ‖ A 5 2 q ‖ 2 + 4 ε 2 λ ‖ A 1 2 q ‖ 2 + 1 2 ‖ A q ‖ 2 ) ∫ − ∞ 0   e ε s | W ( s ) | 2 d s     + ‖ A 1 2 q ‖ 2 s u p τ ≤ 0 e ε τ | W ( τ ) | 2     + C ∫ − ∞ 0   e ε s ( ε | p | + ( | p | + R ( s , ω ) ) 2 + 4 b 2 + 4 μ M 2 λ ) R ( s , ω ) d s .

Since l i m t → ∞ W ( t ) t = 0 , r 2 2 ( ω ) is finite P-a.s., together with 4.18 and 4.36, we get that

‖ A 1 2 Y 2 ( 0, ω ; Y 2 ( τ , ω ) ) ‖ E 2 ≤ r 2 2 ( ω ) C ( p ) .

This completes the proof of Lemma 4.4. ,

Theorem 4.5. Let p < λ 4 , (1.2)-(1.4) hold, q ( x ) ∈ H 3 ( Ω ) ∩ H 0 2 ( Ω ) , then the

random dynamical system S ε ( t , ω ) possesses a nonempty compact random attractor A .

Proof. Let B 1 ( ω ) be the ball of H 3 ( Ω ) × ( H 2 ( Ω ) ∩ H 0 1 ( Ω ) ) of radius

r 2 ( ω ) C ( p ) , by the compact embedding H 3 ( Ω ) , it follows that B 1 ( ω ) is

compact in E. for every bounded non-random set B of E and any ψ ( 0 ) ∈ S ˜ ε ( t , θ − t ω ) B , by Lemma 4.4, we know that Y 2 ( 0 ) = ψ ( 0 ) − Y 1 ( 0 ) ∈ B 1 ( ω ) . Therefore, for τ ≤ 0 ,

inf l ( 0 ) ∈ B 1 ( ω ) ‖ ψ ( 0 ) − l ( 0 ) ‖ E 2 ≤ ‖ Y 1 ( 0 ) ‖ E 2 ≤ e ε τ ( 1 − ε ) C ( p ) ( ‖ u 0 ‖ 2 2 − ( p − | ∇ u 0 | 2 ) | ∇ u 0 | 2 + ‖ u 1 + ε u 0 ‖ 2 ) .

So, for all t ≥ 0 ,

d ( S ˜ ε ( t , θ − t ω ) B , B 1 ( ω ) ) ≤ e − ε τ ( 1 − ε ) C ( p ) ( ‖ u 0 ‖ 2 2 − ( p − | ∇ u 0 | 2 ) | ∇ u 0 | 2 + ‖ u 1 + ε u 0 ‖ 2 ) .

From relation (4.3) between S ε ( t , ω ) and S ˜ ε ( t , ω ) , w can obtain that for any non-random bounded B ⊂ E P − a . s . ,

d ( S ˜ ε ( t , θ − t ω ) B , B 1 ( ω ) ) → 0,   as   t → + ∞ .

Hence, the RDS S ε ( t , ω ) associated with (3.7) possesses a uniformly attracting compact set B 1 ( ω ) ⊂ E . Using Theorem 2.1, we complete the proof.

This work is supported by the National Natural Sciences Foundation of People’s Republic of China under Grant 11561076.

Qin, C.L., Du, J.J. and Lin, G.G. (2017) Random Attractors for the Kirchhoff-Type Suspension Bridge Equations with Strong Damping and White Noises. International Journal of Modern Nonlinear Theory and Application, 6, 134-147. https://doi.org/10.4236/ijmnta.2017.64012