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In this paper, we studied a class of damped high order Beam equation stochas-tic dynamical systems with white noise. First, the Ornstein-Uhlenbeck process is used to transform the equation into a noiseless random equation with random variables as parameters. Secondly, by estimating the solution of the equation, we can obtain the bounded random absorption set. Finally, the isomorphism mapping method and compact embedding theorem are used to obtain the system. It is progressively compact, then we can prove the existence of ran-dom attractors.

We studied the random high order Beam equation with strong damping and white noise in this paper.

u t t + β ( − Δ ) 2 m u t + M ( ‖ D m u ‖ p p ) u t + N ( ‖ D m u ‖ p p ) ( − Δ ) m u + α Δ 2 m u = q ( x ) W ˙ , (1)

u ( x , t ) = 0 , ∂ i u ∂ v i = 0 , i = 1 , 2 , ⋯ , 2 m , x ∈ D , t ∈ [ 0 , + ∞ ) , (2)

u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) . (3)

where M ( ‖ D m u ‖ p p ) u t , N ( ‖ D m u ‖ p p ) ( − Δ ) m u are given functions, m ≥ 1 , q d W describe a addable white noise, Ω denotes a area which bounded with smooth homogeneous Dirichlet boundary, ∂ Ω denotes the boundary of Ω . α , β are constants, α Δ 2 m u , β ( − Δ ) 2 m u t are strong damping terms, W ( t ) denotes a one-dimensional two-sided Wiener process on a 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 is a random collection, which is measurable, compact, constant, and attract all the orbits. If it exists, it will be the smallest absorbing sets of system solution concentration, it’s also the biggest invariant set. In a sense, random attractor is reminded as a reasonable extension of the global attractor to the classical dynamical system. Recently, more and more scholars have focused on the random dynamic system.

Guo [

Lin [

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

Qin [

u t t − Δ u − α Δ u t − β u t t + h ( u ) u t + λ u + f ( x , u ) = g ( x , t ) u + ε u ⋅ d W d t .

Crauel and Flandoli [

d u = ( a u x x x x + u x x + β u u x x + r u ) d t = f ( x ) d t + b u d W ( t ) , x ∈ D , t > 0.

For more relevant studies, it can be referred to references in [

In this section, some symbols and assumptions are introduced for convenience.

Let the operator A = Δ with Dirichlet boundary condition be selfadjoint, positive definite and linear. Set the eigenvalue of A is { λ i } i ∈ N , and satisfies

0 < λ 1 ≤ λ 2 ≤ ⋯ ≤ λ m ≤ ⋯ , when m → + ∞ , λ m → + ∞ . (4)

Among them

2 n n + 2 m ≤ p { < 2 n n − 2 m , n > 2 m < ∞ , n ≤ 2 m

Set E k = H 0 2 m + k ( Ω ) × H 0 k ( Ω ) , k = 1 , 2 , ⋯ , 2 m and define a weighted inner product and norm in E k

( y 1 , y 2 ) E k = ( D 2 m + k u 1 , D 2 m + k u 2 ) + ( D k v 1 , D k v 2 ) , (5)

∀ y i = ( u i , v i ) T , y = ( u , v ) T ∈ E k , i = 1 , 2 (6)

and

0 < ε < min { α − 1 β , ( α − 2 ) β λ 1 2 m ( β 2 − 1 ) λ 1 2 m − 4 M 2 ( ‖ D m u ‖ p p ) } , β > α − 2 ε . (7)

Definition 2.1 [

S : R + × Ω × X → X , ( t , w , x ) ↦ S ( t , w , x ) , (8)

satisfy

1) For all s , t ≥ 0 and ω ∈ Ω , mapping S ( t , w ) : = S ( t , w , ⋅ ) satisfy

S ( 0 , w ) = i d , S ( t + s , w ) = S ( t , θ s w ) ∘ S ( s , w ) ; (9)

2) For every w ∈ Ω , mapping ( t , w ) ↦ S ( t , w , x ) continuous.

It is said that S is a continuous random dynamic system on ( Ω , F , P , ( θ t ) t ∈ R ) .

Definition 2.2 [

lim | s | → ∞ inf e − β s d ( B ( θ − s w ) ) = 0.

and d ( B ) = sup x ∈ B ‖ x ‖ X , for all x ∈ X .

Definition 2.3 [

S ( t , θ − t ω ) ( B ( θ − t ω ) ) ⊂ B 0 ( ω ) . (10)

Definition 2.4 [

1) A ( w ) is a random compact set;

2) A ( w ) is a invariant set, for every t > 0 , S ( t , w ) A ( w ) = A ( θ t w ) ;

3) A ( ω ) attracts all sets on D ( w ) , for any B ( w ) ∈ D ( w ) and P − a ⋅ e ⋅ ω ∈ Ω , we have the limit formula:

lim t → ∞ d ( S ( t , θ − t w ) ( B ( θ − t w ) ) , A ( w ) ) = 0 (11)

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

Definition 2.5 [

1) Random set B k ( w ) is a closed set on Hilbert space X;

2) For P − a ⋅ e ⋅ w ∈ Ω , random set B k ( w ) meet the following progressive compactness conditions: For any sequence x n ∈ s ( t n , θ − t n w ) B k ( θ − t n w ) in t n → + ∞ , there is a convergent subsequence in space X. Then the stochastic dynamic system ( s ( t , w ) ) t ≥ 0 has a unique global attractor

A k ( w ) = ∩ τ ≥ t k ( w ) ∪ t ≥ τ s ( t , θ − t w ) B k ( θ − t w ) ¯ (12)

For convenience, Equations (1)-(3) can be reduced to

{ d u = u t d t d u t + ( β ( − Δ ) 2 m u t + M ( ‖ D m u ‖ p p ) u t + N ( ‖ D m u ‖ p p ) ( − Δ ) m u + α Δ 2 m u ) d t = q ( x ) d W ( t ) t ∈ [ 0 , + ∞ ) , u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) , x ∈ D . (13)

Set ϕ = ( u , y ) T , y = u t + ε u , then Equation (1) is equivalent to the following stochastic differential equation

{ d ϕ + L ϕ d t = F ( θ t ω , ϕ ) , ϕ 0 ( ω ) = ( u 0 , u 1 + ε u 0 ) T . (14)

and

ϕ = ( u y ) , F ( θ t ω , ϕ ) = ( 0 q ( x ) d W ( t ) ) ,

L = ( ε I − I ( ( α − β ε ) A 2 m + N ( ‖ D m u ‖ p p ) A m + ε 2 − ε M ( ‖ D m u ‖ p p ) ) I ( β A 2 m + M ( ‖ D m u ‖ p p ) − ε ) I )

δ = δ ( θ t ω ) = − ∫ − ∞ 0 e s θ t ω ( s ) d s ,

δ ( θ t ω ) denotes a Ornstein-Uhlenbeck process, it is a stationary solution of Itô equation

d δ + δ d t = d W . (15)

v = y − q ( x ) δ ( θ t ω ) , then Equation (14) 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 − q ( x ) δ ( θ t ω ) ) T . (16)

And

φ = ( u v ) ,

F ¯ ( θ t ω , φ ) = ( q ( x ) δ ( θ t ω ) q ( x ) d W ( t ) + ( ε − β A 2 m − 1 − M ( ‖ D m u ‖ p p ) ) q ( x ) δ ( θ t ω ) ) ,

L = ( ε I − I ( ( α − β ε ) A 2 m + N ( ‖ D m u ‖ p p ) A m + ε 2 − ε M ( ‖ D m u ‖ p p ) ) I ( β A 2 m + M ( ‖ D m u ‖ p p ) − ε ) I ) .

This section mainly considers existence of the random attractor of problem (1). First, we can prove that the random dynamic system S ( t , ω ) has a bounded random absorption set. For this reason, all slowly increasing subsets in the space E are denoted as D ( E ) .

Lemma 1 E k = H 0 2 m + k ( Ω ) × H 0 k ( Ω ) , for every y = ( y 1 , y 2 ) T ∈ E k , ( k = 1 , 2 , ⋯ , 2 m ) , When

k 1 = min { ε + β ε + 1 − α 2 β + ε ( ε 2 − ε M ( ‖ D m u ‖ p p ) ) 2 λ 1 2 m , ε − M ( ‖ D m u ‖ p p ) 2 + β λ 1 2 m 2 − ε + M ( ‖ D m u ‖ p p ) } , k 2 = β ( β ε + 2 − α ) 2 .

Then

( L y , y ) E k ≥ k 1 ‖ y ‖ E k 2 + k 2 ‖ D 2 m + k y 2 ‖ 2 . (17)

Proof

( L y , y ) E k = ( D 2 m + k ( ε y 1 − y 2 ) , D 2 m + k y 1 ) + ( D k ( ( α − β ε ) A 2 m y 1 + N ( ‖ D m u ‖ p p ) A m y 1 + β A 2 m y 2 ( ε 2 − ε M ( ‖ D m u ‖ p p ) ) y 1 + M ( ‖ D m u ‖ p p ) y 2 ) , D k y 2 ) = ε ‖ D 2 m + k y 1 ‖ 2 + ( α − β ε − 1 ) ( D 2 m + k y 1 , D 2 m + k y 2 ) + N ( ‖ D m u ‖ p p ) ( D 2 m + k y 1 , D k y 2 ) + β ‖ D 2 m + k y 2 ‖ 2

+ ( ε 2 − ε M ( ‖ D m u ‖ p p ) ) ( D k y 1 , D k y 2 ) + ( M ( ‖ D m u ‖ p p ) − ε ) ‖ D k y 2 ‖ 2 ≥ ε ‖ D 2 m + k y 1 ‖ 2 + β ε + 1 − α 2 β ‖ D 2 m + k y 1 ‖ 2 + β ( β ε + 1 − α ) 2 ‖ D 2 m + k y 2 ‖ 2 + ε 2 − ε M ( ‖ D m u ‖ p p ) 2 ‖ D k y 1 ‖ 2 + ε 2 − ε M ( ‖ D m u ‖ p p ) 2 ‖ D k y 2 ‖ 2 + β 2 ‖ D 2 m + k y 2 ‖ 2 + β λ 1 2 m 2 ‖ D k y 2 ‖ 2 + ( M ( ‖ D m u ‖ p p ) − ε ) ‖ D k y 2 ‖ 2

= ( ε + β ε + 1 − α 2 β + ε ( ε 2 − ε M ( ‖ D m u ‖ p p ) ) 2 λ 1 2 m ) ‖ D 2 m + k y 1 ‖ 2 + ( β 2 + β ( β ε + 1 − α ) 2 ) ‖ D 2 m + k y 2 ‖ 2 + ( ε − M ( ‖ D m u ‖ p p ) 2 + β λ 1 2 m 2 + M ( ‖ D m u ‖ p p ) − ε ) ‖ D k y 2 ‖ 2

≥ k 1 ‖ y ‖ E k 2 + k 2 ‖ D 2 m + k y 2 ‖ 2 (18)

According to the Formula (7), we can get

β 2 + β ( β ε + 1 − α ) 2 ≥ 0 ,

ε − M ( ‖ D m u ‖ p p ) 2 + β λ 1 2 m 2 + M ( ‖ D m u ‖ p p ) − ε = β λ 1 2 m 2 − ε + ε + M ( ‖ D m u ‖ p p ) 2 ≥ 0.

So set

k = min { ε + β ε + 1 − α 2 β + ε ( ε 2 − ε M ( ‖ D m u ‖ p p ) ) 2 λ 1 2 m , ε − M ( ‖ D m u ‖ p p ) 2 + β λ 1 2 m 2 − ε + M ( ‖ D m u ‖ p p ) ε } , k 2 = β ( β ε + 2 − α ) 2 . (19)

Lemma 2 ϕ denotes a solution of problem (14), then there is a bounded random compact set B ˜ 0 k ( ω ) ∈ D ( E k ) , so that there is a random variable T B k ( ω ) > 0 for any slowly increasing random set B ( ω ) ∈ D ( E k ) , such that

ϕ ( t , θ − t ω ) B ( θ − t ω ) ⊂ B ˜ 0 k ( ω ) , ∀ t ≥ T B k ( ω ) , ω ∈ Ω . (20)

Proof φ denotes a solution of problem (16), use φ = ( u , v ) T ∈ E k to take the inner product with the Equation (16), we obtain

1 2 d d t ‖ φ ‖ E k 2 + ( L φ , φ ) E k = ( F ¯ ( θ t ω , φ ) , φ ) . (21)

From Lemma 1

( L φ , φ ) E k ≥ k 1 ‖ φ ‖ E k 2 + k 2 ‖ D 2 m + k v ‖ 2 , (22)

and

( F ¯ ( θ t ω , φ ) , φ ) = ( D 2 m + k q ( x ) δ ( θ t ω ) , D 2 m + k u ) + ( ε − β A 2 m − 1 − M ( ‖ D m u ‖ p p ) ) ( D k q ( x ) δ ( θ t ω ) , D k v ) ≤ ( ε 2 + ( 1 + M ( ‖ D m u ‖ p p ) ) 2 2 ε ) ‖ D k q ( x ) ‖ 2 | δ ( θ t ω ) | 2 + ( ε 2 + 2 ε M 2 ( ‖ D m u ‖ p p ) λ 1 2 m ) ‖ D 2 m + k v ‖ 2 + ( β 2 2 ε + 1 2 )

‖ D 2 m + k q ( x ) ‖ 2 | δ ( θ t ω ) | 2 + 1 2 ‖ D 2 m + k u ‖ 2 . (23)

From Equations (22) and (23), Equation (21) can be written as

d d t ‖ φ ‖ E k 2 + 2 k 1 ‖ φ ‖ E k 2 + ( 2 k 2 − ε − 4 ε M 2 ( ‖ D m u ‖ p p ) λ 1 2 m ) ‖ D 2 m + k v ‖ 2 ≤ ( ε + ( 1 + M ( ‖ D m u ‖ p p ) ) 2 ε ) ‖ D k q ( x ) ‖ 2 | δ ( θ t ω ) | 2 + ( β 2 ε + 1 ) ‖ D 2 m + k q ( x ) ‖ 2 | δ ( θ t ω ) | 2 + ‖ D 2 m + k u ‖ 2 (24)

Set

C 1 = ‖ D 2 m + k u ‖ 2 , η = 2 k 1 ,

P 1 = ( ε + ( 1 + M ( ‖ D m u ‖ p p ) ) 2 ε ) ‖ D k q ( x ) ‖ 2 | δ ( θ t ω ) | 2 + ( β 2 ε + 1 ) ‖ D 2 m + k q ( x ) ‖ 2 | δ ( θ t ω ) | 2 ,

then

d d t ‖ φ ‖ E k 2 + η ‖ φ ‖ E k 2 ≤ P 1 + C 1 . (25)

According to the Formula (7), we can get

2 k 2 − ε − 4 ε M 2 ( ‖ D m u ‖ p p ) λ 1 2 m = β ( β ε + 2 − α ) − ε − 4 ε M 2 ( ‖ D m u ‖ p p ) λ 1 2 m > 0. (26)

From the Gronwall’s inequality, P − a . e . ω ∈ Ω , we have

‖ φ ( t , ω ) ‖ E k 2 ≤ e − η t ‖ φ 0 ( ω ) ‖ E k 2 + ∫ 0 t e − η ( t − r ) ( C 1 + P 1 ‖ δ ( θ t ω ) ‖ 2 ) d r , (27)

because δ ( θ t ω ) is slowly increasing, and δ ( θ t ω ) is continuous with respect to t, according to the literature [

| δ ( θ t ω ) | 2 ≤ r 1 ( θ t ω ) ≤ e n 2 t r 1 ( ω ) . (28)

Substituting θ − t ω for ω in (27), we get

‖ φ ( t , θ − t ω ) ‖ E k 2 ≤ e − η t ‖ φ 0 ( θ − t ω ) ‖ E k 2 + ∫ 0 t e − η ( t − r ) ( C 1 + P 1 ‖ δ ( θ − t ω ) ‖ 2 ) d r , (29)

and

∫ 0 t e − η ( t − r ) ( C 1 + p 1 | δ ( θ r − t ω ) | 2 ) d r = ∫ 0 t e η τ ( C 1 + p 1 | δ ( θ τ ω ) | 2 ) d r ≤ C 1 η + 2 η p 1 r 1 ( ω ) . (30)

Because ϕ 0 ( θ − t ω ) ∈ B k ( θ − t ω ) is slowly increasing, and | δ ( θ − t ω ) | is also slowly increasing, so let

R 0 2 ( ω ) = C 1 η + 2 η p 1 r 1 ( ω ) . (31)

Then R 0 2 ( ω ) also slowly increasing, B ^ 0 k = { φ ∈ E k : ‖ φ ‖ E k ≤ R 0 ( ω ) } denotes a random absorption set, because

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

so

B ˜ 0 k ( ω ) = { ϕ ∈ E k : ‖ ϕ ‖ E k ≤ R 0 ( ω ) + ‖ D k q ( x ) δ ( ω ) ‖ = R ¯ 0 ( ω ) } . (32)

Then B ˜ 0 k ( ω ) is the random absorption set of ϕ ( t , ω ) , and B ˜ 0 k ( ω ) ∈ D ( E k ) .

So the lemma is proved.

Lemma 3 When k = m , for any B m ( ω ) ∈ D ( E m ) , ϕ ( t ) is the solution of Equation (14) under the initial value condition ϕ 0 = ( u 0 , u 1 + ε u 0 ) T ∈ B m . It can be decomposed into ϕ = ϕ 1 + ϕ 2 , where ϕ 1 and ϕ 2 satisfy

{ d ϕ 1 + L ϕ 1 d t = 0 ϕ 10 ( ω ) = ( u 0 , u 1 + ε u 0 ) T (33)

{ d ϕ 2 + L ϕ 2 d t = F ( ω , ϕ ) ϕ 20 ( ω ) = 0 (34)

Then

‖ ϕ 1 ( t , θ − t ω ) ‖ E m 2 → 0 ( t → ∞ ) , ∀ ϕ 0 ( θ − t ω ) ∈ B ( θ − t ω ) , (35)

and there is a slowly increasing random radius R 1 ( ω ) , so that for every ω ∈ Ω , satisfy

‖ ϕ 2 ( t , θ − t ω ) ‖ E m 2 ≤ R 1 ( ω ) . (36)

Proof φ = φ 1 + φ 2 = ( u 1 , u 1 t + ε u 1 ) T + ( u 2 , u 2 t + ε u 2 − q ( x ) δ ( θ t ω ) ) T is a solution of Equation (16), from Equations (33) and (34), it can be seen that φ 1 and φ 2 satisfy

{ φ 1 t + L φ 1 d t = 0 , φ 10 = φ 0 = ( u 0 , u 1 + ε u 0 ) T , (37)

{ φ 2 t + L φ 2 d t = F ( ω , φ ) φ 20 = 0 (38)

Using φ 1 = ( u 1 , u 1 t + ε u 1 ) T and Equation (37) to take the inner product, we get

1 2 d d t ‖ φ 1 ‖ E m 2 + ( L φ 1 , φ 1 ) E m = 0 , (39)

according to lemma 2 and the Gronwall’s inequality, we obtain

‖ φ 1 ( t , ω ) ‖ E m 2 ≤ e − 2 k 1 t ‖ φ 0 ( ω ) ‖ E m 2 , (40)

replace ω in (40) with θ − t ω , and because δ ( θ − t ω ) ∈ B is slowly increasing, then

‖ φ 1 ( t , θ − t ω ) ‖ E m 2 ≤ e − 2 k 1 t ‖ φ 0 ( θ − t ω ) ‖ E m 2 → 0 ( t → ∞ ) , ∀ φ 0 ( θ − t ω ) ∈ B m (41)

Using φ 2 = ( u 2 , u 2 t + ε u 2 − q ( x ) δ ( θ t ω ) ) T and Equation (38) to take the inner product, we get

d d t ‖ φ 2 ‖ E m 2 + η ‖ φ 2 ‖ E m 2 ≤ C 2 + P 2 | δ ( θ t ω ) | 2 , (42)

and P 2 = P 1 | δ ( θ t ω ) | 2 .

According to lemma 1, lemma 2, Equation (29), the Gronwall’s inequality, and replace ω with θ − t ω , we obtain

‖ φ 2 ( t , θ − t ω ) ‖ E m 2 ≤ e − η t ‖ ϕ 20 ( θ − t ω ) ‖ E m 2 + ∫ 0 t e − η ( t − r ) ( C 2 + P 2 | δ ( θ r − t ω ) | 2 ) d r ≤ C 3 η + 2 η P 2 r 1 ( ω ) , (43)

set

R 1 2 ( ω ) = C 3 η + 2 η p 2 r 1 ( ω ) , (44)

for every ω ∈ Ω ,

‖ ϕ 2 ( t , θ − t ω ) ‖ E m ≤ R 1 ( ω ) , (45)

and R 1 ( ω ) is slowly increasing. The lemma is proved.

Lemma 4 The stochastic dynamic system { S ( t , ω ) , t ≥ 0 } determined by Equation (17) has a compact absorption set K ( ω ) ⊂ E k under condition t = 0 , P − a . e . ω ∈ Ω

Proof Suppose K ( ω ) is a closed sphere with R 1 ( ω ) as the radius in space D ( A 3 4 ) × D ( A 1 4 ) . According to the embedding relationship E k ⊂ E 0 , K ( ω )

is a compact set in E k . For any slowly increasing random set B k ( ω ) in E, for ∀ ϕ ( t , θ − t ω ) ∈ B k , according to Lemma 3.1, there is φ 2 = φ − φ 1 ∈ K ( ω ) , so for every t ≥ T B k ( ω ) > 0 ,

d E k ( S ( t , θ − t ω ) B k ( θ − t ω ) , K ( ω ) ) = inf ϑ ( t ) ∈ K ( ω ) ‖ φ ( t , θ − t ω ) − ϑ ( t ) ‖ E k 2 ≤ ‖ φ ( t , θ − t ω ) ‖ E k 2 ≤ e − η t ‖ φ 0 ( θ − t ω ) ‖ E k 2 → 0 , ( t → ∞ ) (46)

Therefore, for any slowly increasing random set B k ( ω ) in E k , there is

d E k ( S ( t , θ − t ω ) B ( θ − t ω ) , K ( ω ) ) → 0 , t → ∞ , ω ∈ Ω . (47)

According to Lemma 1 to Lemma 4, there is the following theorem.

Theorem 1 Random dynamic system { S ( t , ω ) , t ≥ 0 } has a random attractor A k ( ω ) ⊂ K ( ω ) ⊂ E k , ω ∈ Ω , and there is a slowly increasing random set K ( ω ) , P a . e . ω ∈ Ω ,

A k ( w ) = ∩ t ≥ 0 ∪ τ ≥ t S ( t , θ − τ w ) K ( θ − τ w ) ¯ , (48)

and

S ( t , w ) A k ( ω ) = A k ( θ t ω ) . (49)

We studied a class of damped high order Beam equation stochastic dynamical systems with white noise, by using the Ornstein-Uhlenbeck process, estimating the solution of the equation and the isomorphism mapping method, then we can get the existence of the random attractor family, I wish there will be some more convenient methods can be shown off. Further we can make the inertial manifolds of the model.

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

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