The Family of Exponential Attractors and Random Attractors for a Class of Kirchhoff Equations

To prove the existence of the family of exponential attractors, we first define a family of compact, invariant absorbing sets B_k. Then we prove that the solution semigroup has Lipschitz property and discrete squeezing property. Finally, we obtain a family of exponential attractors and its estimation of dimension by combining them with previous theories. Next, we obtain Kirchhoff-type random equation by adding product white noise to the right-hand side of the equation. To study the existence of random attractors, firstly we transform the equation by using Ornstein-Uhlenbeck process. Then we obtain a family of bounded random absorbing sets via estimating the solution of the random differential equation. Finally, we prove the asymptotic compactness of semigroup of the stochastic dynamic system; thereby we obtain a family of random attractors.

• Family of Exponential Attractors
• Lipschitz Continuous
• Squeezing Property
• Stochastic Dynamic System
• Family Random Attractors

1. Introduction

In 1991, Eden A. et al. [1] proposed the concept of inertial fractal set and how inertial fractal set is constructed. Meanwhile, they provided some applications for people to study how to prove the existence of exponential attractors. The authors’ relevant research results can be referred to [2] [3] [4] [5].

With the advent of Kirchhoff [6] equation and the existence of its solution, scholars began to study the existence of exponential attractors of Kirchhoff equation. Recently, Jia Lan, Ma Qiaozhen [7] studied the Kirchhoff-type suspension bridge equations:

{ u t t + Δ 2 u + α u t + ( p − β ‖ ∇ m u ‖ 2 ) Δ u + k 2 u + + f ( u ) = g ( x ) , ( x , t ) ∈ Ω × R + , Δ u ( x , t ) = u ( x , t ) = 0 , x ∈ ∂ Ω , t ≥ 0.

They proved the asymptotic compactness of the semigroup and showed the existence of exponential attractors by a new method of enhanced flattening property under a weaker condition of nonlinearity.

Lin Guoguang, Wang Wei [8] discussed a class of higher-order Kirchhoff-type equation with nonlinear damped term:

{ u t t + ( − Δ ) m u + ϕ ( ‖ ∇ 2 u ‖ 2 ) ( − Δ ) m u + h ( u t ) = f ( x ) , u ( x , t ) = 0 , ∂ j u ∂ v j = 0 ,       j = 1 , 2 , ⋯ , m − 1 ,       x ∈ ∂ Ω , t > 0.

They obtained the exponential attractors via proving the Lipschitz continuity and discrete squeezing property of dynamical system.

In paper [9], we studied Kirchhoff equation:

{ u t t + M ( ‖ ∇ m u ‖ p p ) ( − Δ ) 2 m u + β ( − Δ ) 2 m u t + | u | ρ u = f ( x ) , u ( x , t ) = ∂ j u ∂ v j = 0 ,       j = 1 , 2 , ⋯ , 2 m − 1 ,       x ∈ ∂ Ω , u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) ,       x ∈ ∂ Ω ,       t > 0 , (1.1)

where m > 1 , p ≥ 2 , Ω ⊆ R n ( n ≥ 1 ) is a bounded region with a smooth boundary ∂ Ω , Q = Ω × [ 0 , ∞ ) stands for the cylinder in R x n × R t , the rigid term M ( s ) ∈ C 1 [ 0, ∞ ) is a general function, β ( − Δ ) 2 m u t ( β > 0 ) is the strong dissipative term, | u | ρ u is the nonlinear term and ρ ≥ − 1 , f ( x ) denotes the external force. We have proved the existence and uniqueness of solution, a family of global attractors and its dimension estimation. In this paper, we will discuss the family of the exponential attractors and its dimension estimation. Meanwhile, we will discuss the family of random attractors of stochastic Kirchhoff equation with product white noise:

{ u t t + M ( ‖ ∇ m u ‖ p p ) ( − Δ ) 2 m u + β ( − Δ ) 2 m u t + | u | ρ u = q ( x ) d W ( t ) d t , u ( x , t ) = ∂ j u ∂ v j = 0 ,       j = 1 , 2 , ⋯ , 2 m − 1 ,       x ∈ ∂ Ω , u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) ,       x ∈ ∂ Ω ,       t > 0 , (1.2)

where W ( t ) is independent of time denotes a two-side process in probability space ( Ω , F , P ) , Ω = { ω ∈ C ( R , R ) : ω ( 0 ) = 0 } , F denotes a Borel σ -algebra generated by compact-open topology on Ω , P denotes a probability measure.

Random attractor plays an important role in stochastic dynamic systems because of its property. Lu D. proposed the concept of stochastic process and its application in the literature [10]. Then Guo Boling, Pu Xueke introduced the knowledge of random infinite dimensional dynamical system in the literature [11]. Lin G.G., Qin C.L. [12] discussed the existence of the random attractors of weekly damped Kirchhoff equation:

{ d u t + α d u + [ − ( 1 + ( ∫ D | ∇ u | 2 d x ) ρ ) Δ u + g ( u ) ] d t = f ( x ) d t + q ( x ) d W ( t ) , t ∈ [ τ , + ∞ ) , u ( x , t ) | x ∈ ∂ D = 0 , t > τ ,

Following, Lin, G.G., et al. [13] proved the exponential attractor of Kirchhoff-type equations with strongly damped terms and source terms:

d u t + [ ( − Δ ) m u t + ϕ ( ‖ ∇ m u ‖ 2 ) ( − Δ ) m u + g ( u ) ] d t = f ( x ) d t + q ( x ) d W ( t ) .

More relevant results can be referred to [14] [15] [16] [17].

2. Preliminaries

Combine paper [9] with some new definitions and assumptions, we have:

H = L 2 ( Ω ) , H 0 m ( Ω ) = H m ( Ω ) ∩ H 0 1 ( Ω ) , E k = H 0 2 m + k ( Ω ) × H 0 k ( Ω ) ,

where k = 1 , 2 , ⋯ , 2 m , U = ( u , v ) T , v = u t + ε u , C i > 0 ( i = 0 , 1 , 2 , ⋯ ) are constants. λ j denotes the jth eigenvalue of − Δ with the homogeneous Dirichlet boundary on Ω . Define the inner of E k as following:

( ( ξ , η ) , ( ξ ¯ , η ¯ ) ) = ( ∇ 2 m + k ξ , ∇ 2 m + k ξ ¯ ) + ( ∇ k η , ∇ k η ¯ ) .

M ( s ) , ρ and p satisfy the following conditions:

(A1) For ∀ s ≥ 0 , we have ε + 1 ≤ μ 0 ≤ M ( s ) ≤ μ 1 , where μ 0 , μ 1 are constants, and

μ = { μ 0 , d d t ‖ Δ m u ‖ 2 ≥ 0 μ 1 , d d t ‖ Δ m u ‖ 2 ≤ 0 ;

(A2) { 1 ≤ ρ ≤ 8 m n − 4 m , n > 4 m ρ ≥ − 1 , n ≤ 4 m ;

(A3) { 2 ≤ p ≤ 2 n n − 2 m , n > 2 m p ≥ 2 , n ≤ 2 m ;

(A4) 0 ≤ ε ≤ min { 1 + β λ 1 2 m 4 − 1 , 2 μ β , 2 μ λ 1 2 m β λ 1 2 m + 1 , ( 2 β − 2 − 1 λ 1 2 m − k ) ( 1 + μ 1 2 ) } .

3. Exponential Attractors

Definition 3.1. Compact set M k is called a family of exponential attractors of ( { S ( t ) } t ≥ 0 , E k ) , if a family of compact attractors A k ⊆ M k ⊆ E k satisfies:

1) S ( t ) M k ⊆ M k , ∀ t ≥ 0 ;

2) M k has finite fractal dimension d M k ;

3) There exist constants a 0 , a 1 > 0 such that

∀ t ≥ 0, d i s t ( S ( t ) E k , M k ) ≤ a 0 e − a 1 t .

Definition 3.2. [1] Solution semigroup S ( t ) is Lipschitz continuity, if there is a bounded function l ( t ) , such that

‖ S ( t ) U − S ( t ) V ‖ E k ≤ l ( t ) ‖ U − V ‖ E k ,   ∀ U , V ∈ E k .

Definition 3.3. [1] Assume that solution semigroup S ( t ) : E k → E k is a map satisfies Lipschitz continuity, then we say S ∗ = S ( t ∗ ) is squeezing in E k , if

∀ δ ∈ ( 0 , 1 8 ) , ∃ t ∗ > 0 , N = N ( δ ) , there is

‖ Q N ( S ∗ U − S ∗ V ) ‖ E k ≥ ‖ P N ( S ∗ U − S ∗ V ) ‖ E k , ∀ U , V ∈ E k ,

or

‖ S ∗ U − S ∗ V ‖ E k < δ ‖ U − V ‖ E k , ∀ U , V ∈ E k ,

where P N is an orthogonal projection in E k , Q N = I − P N .

Theorem 3.1. [15] Assuming that

1) S ( t ) possesses a family of ( E k , E 0 ) -compact attractors A k ;

2) S ( t ) exists a family of positive, invariant compact sets in E k ;

3) S ( t ) is Lipschitz continuous and squeezing on E k ;

then S ( t ) exists a family of ( E k , E 0 ) -type exponential attractors M k and

M k = ∪ 0 ≤ t ≤ t ∗ S ( t ) M ∗ ,     M ∗ = A k ∪ ( ∪ j = 1 ∞ ∪ k = 1 ∞ S ( t ∗ ) j ( E ( l ) ) ) ,

moreover, the fractal dimension of M k satisfies d M k ≤ c N 0 + 1 , where N 0 is the least of N which makes squeezing found.

Theorem 3.2. [9] Suppose that (A1)-(A4) are valid. Let ( u 0 , u 1 ) ∈ E k , k = 1 , 2 , ⋯ , 2 m , f ( x ) ∈ L 2 ( Ω ) , then the initial boundary value problem (1.1) has a global solution ( u , v ) that satisfies u ∈ L ∞ ( 0 , + ∞ ; H 0 2 m + k ( Ω ) ) , v ∈ L ∞ ( 0 , + ∞ ; H 0 k ( Ω ) ) ∩ L 2 ( 0 , T ; H 0 2 m + k ( Ω ) ) , and there exists a nonnegative real number R k and T = t ( Ω ) > 0 so that

‖ ( u , v ) ‖ E k 2 = ‖ ∇ 2 m + k u ‖ 2 + ‖ ∇ k v ‖ 2 ≤ R k 2 ,       ( t > T ) .

According to paper [9], the solution semigroup S ( t ) of the initial boundary value problem (1.1) exists a family of ( E k , E 0 ) -compact attractors A k , and we can define a family of positive, invariant compact sets B k = ∪ 0 ≤ t ≤ T S ( t ) B 0 k ¯ , where B 0 k = { U ∈ E k : ‖ U ‖ E k ≤ R k 2 } .

Next, we prove the problem (1.1) exists a family of exponential attractors.

Let Equation (1.1) transform into a first-order evolution equation:

U t + F ( U ) + R ( U ) = 0 , U ∈ E k , (3.1)

where

F ( U ) = ( ε u − v , ε β ( − Δ ) 2 m u + β ( − Δ ) 2 m v + ε 2 u − ε v ) T ,

R ( U ) = ( 0 , M ( ‖ ∇ m u ‖ p p ) ( − Δ ) 2 m u + | u | ρ u − f ( x ) ) T .

Lemma 3.1. (Lipschitz property) ∀ U 0 , V 0 ∈ B k , there is

‖ S ( t ) U 0 − S ( t ) V 0 ‖ E k ≤ e α 2 t ‖ U 0 − V 0 ‖ E k ,

where α 2 = max { ε 2 β 2 + C 1 2 + C 2 2 λ 1 m λ 1 2 m + k , ε } .

Proof. Let U ¯ ( t ) = U ( t ) − V ( t ) = ( u ¯ ( t ) , v ¯ ( t ) ) T , where U ( t ) = S ( t ) U 0 = ( u 1 ( t ) , v 1 ( t ) ) T , V ( t ) = S ( t ) V 0 = ( u 2 ( t ) , v 2 ( t ) ) T , u ¯ ( t ) = u 1 ( t ) − u 2 ( t ) , then we have

U ¯ t + F ( U ¯ ) + R ( U ) − R ( V ) = 0 , (3.2)

Taking the inner product of Equation (3.2) with W in E k , and we get that

1 2 d d t ‖ U ¯ t ‖ E k 2 + ( F ( U ¯ ) , U ¯ ) E k + ( R ( U ) − R ( V ) , W ) E k = 0, (3.3)

By using Young’s inequality, Holder’s inequality, Poincare’s inequality and differential mean value theorem, we obtain

( F ( U ¯ ) , U ¯ ) E k = ε ‖ ∇ 2 m + k u ¯ ‖ 2 − ( ∇ 2 m + k u ¯ , ∇ 2 m + k v ¯ ) + β ‖ ∇ 2 m + k v ¯ ‖ 2     − ε β ( ∇ 2 m + k u ¯ , ∇ 2 m + k v ¯ ) + ε 2 ( ∇ k u ¯ , ∇ k v ¯ ) − ε ‖ ∇ k v ¯ ‖ 2 ≥ ( ε − ε 2 β 2 2 ) ‖ ∇ 2 m + k u ¯ ‖ 2 + ( β − 1 2 ε − 1 2 ) ‖ ∇ 2 m + k v ¯ ‖ 2 − ε ‖ ∇ k v ¯ ‖ 2 . (3.4)

| ( R ( U ) − R ( V ) , U ¯ ) E k | = | ( ∇ k ( | u 1 | ρ u 1 − | u 2 | ρ u 2 ) , ∇ k v ¯ ) |       + | ( M ( ‖ ∇ m u 1 ‖ p p ) ∇ 2 m + k u 1 − M ( ‖ ∇ m u 2 ‖ p p ) ∇ 2 m + k u 2 , ∇ 2 m + k v ¯ ) | ≤ ( C 1 2 + C 2 2 λ 1 m 2 λ 1 2 m + k + ε 2 ) ‖ ∇ 2 m + k u ¯ ‖ 2 + ( 1 2 λ 1 2 m − k + ε + μ 1 2 2 ε ) ‖ ∇ 2 m + k v ¯ ‖ 2 . (3.5)

Substitute (3.4) and (3.5) into (3.3), we have

d d t ‖ U ¯ ‖ E k 2 + ( 2 β − 2 − 1 + μ 1 2 ε − 1 λ 1 2 m − k ) ‖ ∇ 2 m + k v ¯ ‖ 2 ≤ ( ε 2 β 2 + C 1 2 + C 2 2 λ 1 m λ 1 2 m + k ) ‖ ∇ 2 m + k u ¯ ‖ 2 + ε ‖ ∇ k v ¯ ‖ 2 .

Let α 1 = 2 β − 2 − 1 + μ 1 2 ε − 1 λ 1 2 m − k , α 2 = max { ε 2 β 2 + C 1 2 + C 2 2 λ 1 m λ 1 2 m + k , ε } , then we have

d d t ‖ U ¯ ‖ E k 2 + α 1 ‖ ∇ 2 m + k v ¯ ‖ 2 ≤ α 2 ‖ U ¯ ‖ E k 2 . (3.6)

By using Gronwall’s inequality, we get

‖ U ¯ ( t ) ‖ E k 2 ≤ e α 2 t ‖ U ¯ ( 0 ) ‖ E k 2 . (3.7)

Meanwhile, we obtain L i p B k ( S ( t ) ) ≤ e α 2 t 2 .

Lemma 3.1 is proved.

Lemma 3.2. (discrete squeezing) ∀ U 0 , V 0 ∈ B k , if

‖ Q N ( S ∗ U 0 − S ∗ V 0 ) ‖ E k ≥ ‖ P N ( S ∗ U 0 − S ∗ V 0 ) ‖ E k ,

then we have

‖ S ∗ U 0 − S ∗ V 0 ‖ E k < 1 8 ‖ U 0 − V 0 ‖ E k .

Proof. Applying Q N to Equation (3.2), we get

Q N U ¯ t ( t ) + Q N F ( U ¯ ) + Q N ( R ( U ) − R ( V ) ) = 0. (3.8)

Taking the inner product of Equation (3.8) with Q N U ¯ in E k , we obtain

1 2 d d t ‖ Q N U ¯ t ‖ E k 2 + ( Q N F ( U ¯ ) , Q N U ¯ ) E k + ( Q N ( R ( U ) − R ( V ) ) , Q N U ¯ ) E k = 0, (3.9)

Similar to the process of Lemma 3.1, we have

d d t ‖ Q N U ¯ ‖ E k 2 + α 3 ‖ ∇ 2 m + k Q N v ¯ ‖ 2 ≤ α 4 ‖ Q N U ¯ ‖ E k 2 . (3.10)

where α 3 = 2 β − 2 − 1 + μ 1 2 ε − 1 λ N + 1 2 m − k , α 4 = max { ε 2 β 2 + C 1 2 + C 2 2 λ N + 1 m λ N + 1 2 m + k , ε } .

By using Gronwall’s inequality, we get

‖ Q N U ¯ ( t ) ‖ E k 2 ≤ e α 4 t ‖ Q N U ¯ ( 0 ) ‖ E k 2 .

Suppose ‖ Q N ( S ∗ U 0 − S ∗ V 0 ) ‖ E k ≥ ‖ P N ( S ∗ U 0 − S ∗ V 0 ) ‖ E k , then we have

‖ S ∗ U 0 − S ∗ V 0 ‖ E k 2 ≤ ‖ P N ( S ∗ U 0 − S ∗ V 0 ) ‖ E k 2 + ‖ Q N ( S ∗ U 0 − S ∗ V 0 ) ‖ E k 2 ≤ 2 ‖ Q N ( S ∗ U 0 − S ∗ V 0 ) ‖ E k 2 ≤ 2 e α 4 t ∗ ‖ Q N U ¯ ( 0 ) ‖ E k 2 ≤ 2 e α 4 t ∗ ‖ U 0 − V 0 ‖ E k 2 .

Therefore, when t ∗ ≤ − 7 ln 2 α 4 , we have

‖ S ∗ U 0 − S ∗ V 0 ‖ E k ≤ 1 8 ‖ U 0 − V 0 ‖ E k .

Lemma 3.2 is proved.

In fact, Theorem 3.1 has provided the theoretical basis to prove the existence of random attractors. Because of the value of k ( k = 1 , ⋯ , 2 m ) , furthermore, we can obtain the existence theorem of the family of exponential attractors via Lemma 3.1 and Lemma 3.2 as following:

Theorem 3.3. Assume (A1)-(A4) are valid, f ∈ H , then ∀ U ∈ E k , the solution semigroup S ( t ) of the initial boundary value problem (1.1) exists a family of ( E k , E 0 ) -type exponential attractors M k , and

M k = ∪ 0 ≤ t ≤ t ∗ S ( t ) ( A k ∪ ( ∪ j = 1 ∞ ∪ k = 1 ∞ S ( t ∗ ) j ( E ( l ) ) ) ) , d M k ≤ c N 0 + 1 , k = 1 , 2 , ⋯ , 2 m .

4. Random Attractors

Define the time-translation operator on Ω : θ s ω ( t ) = ω ( t + s ) − ω ( s ) , then ( Ω , F , P , { θ t } t ∈ R ) constitutes an ergodic, metric dynamical system. According to paper [8], we have some definitions and theorems as following:

Definition 4.1. ( Ω , F , P , { θ t } t ∈ R ) is a metric dynamical system, ( B ( R + ) × F × B ( E k ) , B ( E k ) ) is measurable. ϕ is called a continuous stochastic dynamical system, if map ϕ : R + × Ω × E k → E k satisfies

1) ϕ ( 0 , ω ) x = x , x ∈ E k , ω ∈ Ω ;

2) ϕ ( t + s , ω ) = ϕ ( t , θ s ω ) ∘ ϕ ( s , ω ) , ∀ t , s ∈ R , x ∈ E k , ω ∈ Ω ;

3) ( t , ω , x ) → ϕ ( t , ω , x ) is continuous.

Definition 4.2. [8] B k ⊂ E k is called a family of tempered random sets, if

lim | s | → ∞ inf e − β s d ( B k ( θ − s ω ) ) = 0, ω ∈ Ω ,

where β > 0 , d ( B k ) = sup x ∈ B k ‖ x ‖ E k .

Definition 4.3. [8] D ( ω ) is the set of all the random sets on E k . Random set B k ( ω ) is called a family of absorption sets on D ( ω ) , if ∀ B k ( ω ) ∈ D ( ω ) , ∃ T > 0 , such that

ϕ ( t , θ − t ω ) B k ( θ − t ω ) ⊂ B 0 k ( ω ) ,       P − a . e . ω ∈ Ω .

Definition 4.4. [8] Random set A k ( ω ) is called a family of random attractors of continuous random dynamic system { S ( t , s ; ω ) } t ≥ s on E k , if A k ( ω ) satisfies:

1) A k ( ω ) is a family of random compact sets;

2) A k ( ω ) is a family of invariant sets, which means ∀ t > 0 , S ( t , ω ) A k ( ω ) = A k ( θ t ω ) ;

3) A k ( ω ) attracts all sets on D ( ω ) , which means ∀ B k ( ω ) ∈ D ( ω ) , there is

lim t → ∞ d ( S ( t , θ − t ω ) B k ( Θ − t ω ) , A k ( ω ) ) = 0 ,

where d ( A , B ) = sup x ∈ A inf y ∈ B ‖ x − y ‖ denotes the Hausdorff half-distance.

Theorem 4.1. Assume the family of random sets B k ( ω ) ∈ D ( ω ) is a family of random absorbing sets of the stochastic dynamic system { S ( t , s ; ω ) } t ≥ s , and it satisfies:

1) random set B k ( ω ) is closed set on E k ;

2) ∀ P − a . e . ω ∈ Ω , B k ( ω ) is asymptotically compact. Namely, when t n → + ∞ , there exists a convergent subsequence in E k for ∀ x n ∈ S ( t n , θ − t n ω ) .

Then there exists a unique family of global attractors A k ( ω ) of the stochastic dynamic system { S ( t , ω ) } t ≥ 0 , and

A k ( ω ) = ∩ τ ≥ t ( ω ) ∪ t ≥ τ S ( t , θ − t ω ) B k ( θ − t ω ) ¯     .

Rewrite Equation (1.2) into a stochastic differential equation:

{ d u t + ( M ( ‖ ∇ m u ‖ p p ) ( − Δ ) 2 m u + β ( − Δ ) 2 m u t + | u | ρ u ) d t = q ( x ) d W ( t ) , u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) ,       x ∈ ∂ Ω ,       t > 0 , (4.1)

Equation (4.1) can be reduced to

{ d φ + L φ d t = F ( θ t ω , φ ) , φ 0 ( ω ) = ( u 0 , u 1 + ε u 0 ) T , (4.2)

where φ = ( u , v ) T , v = u t + ε u , F ( θ t ω , φ ) = ( 0 , − | u | ρ u d t + q ( x ) d W ( t ) ) T ,

L = ( ε I − I ( ( M ( ‖ ∇ m u ‖ p p ) − ε β ) Δ 2 m + ε 2 ) I ( β Δ 2 m − ε ) I ) .

Let δ = δ ( θ t ω ) = − ∫ − ∞ 0     e s θ t ω ( s ) d s , where δ ( θ t ω ) denotes Ornstein-Uhlenbeck process and is the solution of Ito equation:

d δ + δ d t = d W .

Let z = v − q ( x ) δ ( θ t ω ) , furthermore, rewrite Equation (4.1)

{ ψ t + L ψ = F ¯ ( θ t ω , ψ ) , ψ 0 ( ω ) = ( u 0 , u 1 + ε u 0 − q ( x ) δ ( θ t ω ) ) T , (4.3)

where φ = ( u , z ) T , F ¯ ( θ t ω , φ ) = ( θ t ω , − | u | ρ u + ( ε + 1 − β Δ 2 m ) q ( x ) δ ( θ t ω ) ) T .

Lemma 4.1. ∀ x = ( x 1 , x 2 ) T ∈ E k , there is

( L x , x ) E k ≥ α 5 ‖ x ‖ E k 2 + α 6 2 ‖ ∇ 2 m + k x 2 ‖ 2 ,

where α 5 = min { β ε + ε 2 β − ε 2 2 β λ 2 m , β λ 2 m − β ε 2 2 − ε } , α 6 = β − β ε 2 + β ε .

Proof.

( L x , x ) E k = ( ∇ 2 m + k ( ε x 1 − x 2 ) , ∇ 2 m + k x 1 ) + ( M ( ‖ D m u ‖ p p ) ∇ 2 m + k x 1 , ∇ 2 m + k x 2 )   − β ε ( ∇ 2 m + k x 1 , ∇ 2 m + k x 2 ) + ε 2 ( ∇ k x 1 , ∇ k x 2 ) + β ‖ ∇ 2 m + k x 2 ‖ 2 − ε ‖ ∇ k x 2 ‖ 2 ≥ ε ‖ ∇ 2 m + k x 1 ‖ 2 − ( β ε − ε ) ‖ ∇ 2 m + k x 2 ‖ 2 − ε 2 2 ‖ ∇ k x 1 ‖ 2 − ε 2 2 ‖ ∇ k x 2 ‖ 2   + β λ 2 m 2 ‖ ∇ k x 2 ‖ 2 + β 2 ‖ ∇ 2 m + k x 2 ‖ 2 − ε ‖ ∇ k x 2 ‖ 2 ≥ ( β ε + ε 2 β − ε 2 2 β λ 2 m ) ‖ ∇ 2 m + k x 1 ‖ 2 + ( β λ 2 m − β ε 2 2 − ε ) ‖ ∇ k x 2 ‖ 2   + ( β − β ε 2 + β ε 2 ) ‖ ∇ 2 m + k x 2 ‖ 2 .

Let α 5 = min { β ε + ε 2 β − ε 2 2 β λ 2 m , β λ 2 m − β ε 2 2 − ε } , α 6 = β − β ε 2 + β ε , we obtain

( L x , x ) E k ≥ α 5 ‖ x ‖ E k 2 + α 6 2 ‖ ∇ 2 m + k x 2 ‖ 2 .

Lemma 4.2. Assume φ is the solution of problem (4.3), then there exists a bounded random compact set B ˜ 0 k ( ω ) ∈ D ( E k ) , such that for arbitrarily random set B k ( ω ) ∈ D ( E k ) , there is a random variable T k > 0 , we have

φ ( t , θ t ω ) B k ( θ − t ω ) ⊂ B ˜ 0 k ( ω ) , ∀ t > T k , ω ∈ Ω .

Proof. Suppose ψ is the solution of problem (4.3). Taking the inner product of Equation (4.3) with ψ in E k , we obtain

1 2 d d t ‖ ψ ‖ E k 2 + ( L ψ , ψ ) E k = ( F ¯ ( θ t ω , ψ ) , ψ ) E k . (4.4)

According to Lemma 4.1, we get

( L ψ , ψ ) E k ≥ α 5 ‖ ψ ‖ E k 2 + α 6 2 ‖ ∇ 2 m + k z ‖ 2 . (4.5)

( F ¯ ( θ t ω , ψ ) , ψ ) E k = ( ∇ 2 m + k q ( x ) δ , ∇ 2 m + k u ) − ( ∇ k ( | u | ρ u ) , ∇ k z )     − β ( ∇ 2 m + k q ( x ) δ , ∇ k z ) + ( ε + 1 ) ( ∇ k q ( x ) δ , ∇ k z ) ≤ ε 2 ‖ ψ ‖ E k 2 + ( 1 + λ 1 k 2 λ 1 2 m + β 2 ) ‖ λ 2 m + k z ‖ 2 + C 3     − ( 1 2 ε + β 2 + ε + 1 2 λ 1 2 m ) ‖ ∇ 2 m + k q ( x ) ‖ 2 | δ ( θ t ω ) | 2 . (4.6)

Substitute (4.5)-(4.6) into (4.4), we have

d d t ‖ ψ ‖ E k 2 + ( 2 α 5 − ε ) ‖ ψ ‖ E k 2 + ( α 6 − β − 1 + λ 1 k 2 λ 1 2 m ) ‖ ∇ 2 m + k z ‖ 2 ≤ 2 C 3 + ( 1 ε + β + ε + 1 λ 1 2 m ) ‖ ∇ 2 m + k q ( x ) ‖ 2 | δ ( θ t ω ) | 2 .

Let C 4 = 2 α 5 − ε , C 5 = 2 C 3 , C 6 = 1 ε + β + ε + 1 λ 1 2 m ‖ ∇ 2 m + k q ( x ) ‖ 2 , then we obtain

d d t ‖ ψ ‖ E k 2 + C 4 ‖ ψ ‖ E k 2 ≤ C 5 + C 6 | δ ( θ t ω ) | 2 . (4.7)

By using Gronwall’s inequality, we get

‖ ψ ( t , ω ) ‖ E k 2 ≤ e − C 4 t ‖ ψ 0 ( ω ) ‖ E k 2 + ∫ 0 t     e − C 4 ( t − r ) ( C 5 + C 6 | δ ( θ t ω ) | 2 ) d r . (4.8)

Because δ ( θ t ω ) is tempered, δ ( θ t ω ) is continuous about t. According to paper [11], we can obtain a temper random variable r 1 : Ω → R + , so that ∀ t ∈ R , ω ∈ Ω , we have

| δ ( θ t ω ) | 2 ≤ r 1 ( θ t ω ) ≤ e C 4 2 | t | r 1 ( ω ) . (4.9)

Substitute θ − t ω for ω in inequality (4.8), and let τ = r − t , then we have

‖ ψ ( t , θ − t ω ) ‖ E k 2 ≤ e − C 4 t ‖ ψ 0 ( θ − t ω ) ‖ E k 2 + ∫ − t 0     e C 4 τ ( C 5 + C 6 | δ ( θ τ ω ) | 2 ) d τ ≤ e − C 4 t ‖ ψ 0 ( θ − t ω ) ‖ E k 2 + C 5 C 6 + 2 C 6 C 4 r 1 ( ω ) . (4.10)

Because ψ 0 ( θ − t ω ) ∈ B k ( θ − t ω ) is tempered and | δ ( θ − t ω ) | is tempered, R 0 2 ( ω ) = C 5 C 6 + 2 C 6 C 4 r 1 ( ω ) is also tempered. Then B ˜ 0 k = { ψ ∈ E k : ‖ ψ ‖ E k ≤ R 0 ( ω ) } is a random attractor set. Because of

S ˜ ( t , θ − t ω ) ψ 0 ( θ − t ω ) = φ ( t , θ − t ω ) ( ψ 0 ( θ − t ω ) ) + ( 0 , q ( x ) δ ( θ − t ω ) T ) − ( 0 , q ( x ) δ ( θ − t ω ) ) T ,

B ˜ 0 k = { φ ∈ E k : ‖ φ ‖ E k ≤ R 0 ( ω ) + ‖ ∇ k q ( x ) δ ( θ − t ω ) ‖ = R ¯ 0 ( ω ) } is a random attractor set of φ ( t , ω ) .

Lemma 4.3. When k = 1 , 2 , ⋯ , 2 m , ∀ B k ( ω ) ∈ D ( E k ) , assume φ ( t ) is the solution of problem (4.2), and we decompose φ = φ 1 + φ 2 , where

{ d φ 1 + L φ 1 d t = 0 , φ 01 ( ω ) = ( u 0 , u 1 + ε u 0 ) T , (4.11)

{ d φ 2 + L φ 2 d t = F ( θ t ω , φ ) , φ 02 ( ω ) = 0 , (4.12)

then ‖ φ 1 ( t , θ − t ω ) ‖ k 2 → 0 ( t → ∞ ) , ∀ φ 0 ( θ − t ω ) ∈ B k ( θ − t ω ) .

Also, there exists a temper random radius R 1 ( ω ) , such that ∀ ω ∈ Ω , R 1 ( ω ) satisfies ‖ φ 2 ( t , Θ − t ω ) ‖ k 2 ≤ R 1 ( ω ) .

Proof. Suppose ψ = ψ 1 + ψ 2 is the solution of Equation (4.4), then we can know from Equation (4.11) and Equation (4.12) that ψ 1 , ψ 2 satisfy

{ ψ 1 t + L ψ 1 = 0 , ψ 01 = ( u 0 , u 1 + ε u 0 − q ( x ) δ ( θ t ω ) ) T , (4.13)

{ ψ 2 t + L ψ 2 = F ¯ ( ψ 2 , θ t ω ) , ψ 02 ( ω ) = 0 , (4.14)

Taking the inner product of Equation (4.13) with ψ 1 in E k , we obtain

1 2 d d t ‖ ψ 1 ‖ E k 2 + ( L ψ 1 , ψ 1 ) = 0. (4.15)

By using Gronwall’s inequality and Lemma 4.1, we have

‖ ψ 1 ( t , ω ) ‖ E k 2 ≤ e − 2 α 5 t ‖ ψ 01 ( ω ) ‖ E k 2 . (4.16)

Substitute θ − t ω for ω in inequality (4.16), and δ ( θ t ω ) ∈ B k is tempered, thus

‖ ψ 1 ( t , θ − t ω ) ‖ E k 2 ≤ e − 2 α 5 t ‖ ψ 01 ( ω ) ‖ E k 2 → 0 ( t → ∞ ) , ∀ ψ 01 ( θ − t ω ) ∈ B k ( θ − t ω )

Taking the inner product of Equation (4.14) with ψ 2 in E k , with Lemma 4.1 and Lemma 4.2 we have

d d t ‖ ψ 2 ‖ E 2 m 2 + C 4 ‖ ψ 2 ‖ E k 2 ≤ C 7 + C 6 | δ ( θ t ω ) | 2 . (4.17)

Substitute θ − t ω for ω in inequality (4.17), and by using Gronwall’s inequality, we obtain

‖ ψ 2 ( t , θ − t ω ) ‖ E k 2 ≤ e − C 4 t ‖ ψ 02 ( t , θ − t ω ) ‖ E k 2 + ∫ 0 t     e C 4 ( τ ) ( C 7 + C 6 | δ ( θ τ ω ) | 2 ) d r ≤ C 7 C 4 + 2 C 6 C 4 r 1 ( ω ) , (4.18)

Thus, there exists a temper random radius R 1 2 ( ω ) = C 7 C 4 + 2 C 6 C 4 r 1 ( ω ) , such that ∀ ω ∈ Ω , ‖ φ 2 ( t , Θ − t ω ) ‖ k 2 ≤ R 1 2 ( ω ) .

Lemma 4.4. The stochastic dynamic system { S ( t , ω ) , t ≥ 0 } determined by problem (4.3) has a family of compact absorbing sets D k ( ω ) , while t = 0 , P − a . e . ω ∈ Ω .

Proof. Suppose D k ( ω ) be a closed ball with radius R 1 2 ( ω ) in E k . According to Rellich-Kondrachov Compact Embedding Theorem, E k → E 0 . Then D k ( ω ) is the compact set in E k . For any tempered random set ∀ B k ( ω ) , φ ( t , θ − t ω ) ∈ B k , according to Lemma 4.3, we have φ 2 = φ − φ 1 ∈ D k ( ω ) , then for all t ≥ T k > 0 , when t → ∞ , via Lemma 4.2 we have

d E k ( S ( t , θ − t ω ) B k ( θ − t ω ) , D k ( ω ) ) = inf ν ( t ) ∈ D k ( ω ) ‖ φ ( t , θ − t ω ) − ν ( t ) ‖ E k 2 ≤ ‖ φ 1 ( t , θ − t ω ) ‖ E k 2 ≤ e − 2 C 4 t ‖ φ 01 ( t , θ − t ω ) ‖ E k 2 → 0

Lemma 4.4 is proved.

According to the lemma mentioned above, we verified two conditions in Theorem 4.1. Similarly, we can obtain the theorem as following:

Theorem 4.2. The stochastic dynamic system { S ( t , ω ) , t ≥ 0 } has a family of random attractors A k ( ω ) ⊂ D k ( ω ) ⊂ E k , ω ∈ Ω , and there exists a slowly increasing family of random sets D k ( ω ) , such that

A k ( ω ) = ∩ τ ≥ t ( ω ) ∪ t ≥ τ S ( t , Θ − t ω ) D k ( Θ − t ω ) ¯ , P − a . e . ω ∈ Ω .

and

S ( t , ω ) A k ( ω ) = A k ( θ t ω ) , k = 1 , 2 , ⋯ , 2 m .

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

References

1. Eden, A., Foias, C., Nicolaenko, B. and Teman, R. (1991) Inertial Sets for Dissipative Evolution Equations Part I: Construction and Applications. University of Minnesota, Twin Cities.

2. Eden, A. and Rakotoson, J.M. (1994) Exponential Attractors for a Doubly Nonlinear Equation. Journal of Mathematical Analysis and Applications, 185, 321-339. https://doi.org/10.1006/jmaa.1994.1251

3. Eden, A., Foias, C. and Kalantarov, V. (1998) A Remark on Two Constructions of Exponential Attractors for α-Contractions. Journal of Dynamics and Differential Equations, 10, 37-45. https://doi.org/10.1023/A:1022636328133

4. Eden, A., Foias, C. and Nicolaenko, B. (1994) Exponential Attractors of Optimal Lyapunov Dimension for Navier-Stokes Equations. Journal of Dynamics and Differential Equations, 6, 301-323. https://doi.org/10.1007/BF02218532

5. Eden, A., et al. (1993) Exponential Attractors and Their Relevance to Fluid Dynamics Systems. Physica D: Nonlinear Phenomena, 63, 350-360. https://doi.org/10.1016/0167-2789(93)90116-I

6. Kirchhoff, G. (1883) Vorlesungen Fiber Mechanik. Teubner, Stuttgart.

7. Jia, L. and Ma, Q.Z. (2018) The Existence of Exponential Attractors for the Kirchhoff-type Suspension Bridge Equations. Journal of Sichuan Normal University, 41, 185-189.

8. Lin, G.G. and Wang, W. (2018) Exponential Attractors for the Higher-Order Kirchhof-Type Equation with Nonlinear Damped Term. Acta Analysis Functionalis Applicata, 20, 243-249.

9. Lin, G.G. and Zhou, C.M. (2021) The Dynamic Behavior of a Class of Kirchhoff Equations with High Order Strong Damping. Journal of Applied Mathematics and Physics, 9, 1041-1055. https://doi.org/10.4236/jamp.2021.95071

10. Lu, D. (1986) Stochastic Process and Its Application. Tsinghua University Press, Beijing.

11. Guo, B. and Pu, X. (2009) Random Infinite Dimensional Dynamical System. Beijng University of Aeronautics and Astronautics Press, Beijing.

12. Lin, G.G. and Qin, C.L. (2010) Random Attractor for Weekly Damped Stochastic Kirchhoff Equation. Journal of Yunnan University, 32, 101-108, 112.

13. Lin, G.G., Chen, L. and Wang, W. (2017) Random Attractor of the Stochastic Strongly Damped for the Higher-Order Nonlinear Kirchhoff-Type Equation. International Journal of Modern Nonlinear Theory and Application, 6, 59-69. https://doi.org/10.4236/ijmnta.2017.62005

14. Lin, G.G. (2019) Dynamical Behavior of Several Kirchhoff Equations. Chongqing University Press, Chongqing.

15. Dai, Z.D. and Guo, B.L. (1997) Inertial Fractal Sets for Dissipative Zakharov System. Acta Mathematicae Applicatae Sinica, 13, 279-288. https://doi.org/10.1007/BF02025883

16. Lin, G.G. and Liu, J. (2020) A Random Attractor Family of the High Order Beam Equations with White Noise. International Journal of Modern Nonlinear Theory and Application, 9, 51-61. https://doi.org/10.4236/ijmnta.2020.93004

17. Lin, G.G. and Xia, X.S. (2018) The Exponential Attractor for a Class of Kirchhoff-Type Equations with Strongly Damped Terms and Source Terms. Journal of Applied Mathematics and Physics, 6, 1481-1493. https://doi.org/10.4236/jamp.2018.67125