Random attractors for stochastic plate equations with memory in unbounded domains

Definition 2.1

Let θ : R × Ω → Ω be a ( ℬ ( R ) × ℱ , ℱ ) -measurable mapping. We say ( Ω , ℱ , P , θ ) is a parametric dynamical system if θ ( 0 , ⋅ ) is the identity on Ω , θ ( s + t , ⋅ ) = θ ( t , ⋅ ) ∘ θ ( s , ⋅ ) for all t , s ∈ R , and P θ ( t , ⋅ ) = P for all t ∈ R .

Definition 2.2

Let K : R × Ω → 2 X be a set-valued mapping with closed nonempty images. We say K is measurable with respect to ℱ in Ω if the mapping ω ∈ Ω → d ( x , K ( τ , ω ) ) is ( ℱ , ℬ ( R ) ) -measurable for every fixed x ∈ X and τ ∈ R .

Definition 2.3

A mapping Φ : R + × R × Ω × X → X is called a continuous cocycle on X over R and ( Ω , ℱ , P , { θ t } t ∈ R ) if for all τ ∈ R , ω ∈ Ω , and t , s ∈ R + , the following conditions (1)–(4) are satisfied:

1. Φ ( ⋅ , τ , ⋅ , ⋅ ) : R + × Ω × X → X is ( ℬ ( R + ) × ℱ × ℬ ( X ) , ℬ ( X ) ) -measurable;

2. Φ ( 0 , τ , ω , ⋅ ) is the identity on X ;

3. Φ ( t + s , τ , ω , ⋅ ) = Φ ( t , τ + s , θ s ω , ⋅ ) ∘ Φ ( s , τ , ω , ⋅ ) ;

4. Φ ( t , τ , ω , ⋅ ) : X → X is continuous.

Hereafter, we assume Φ is a continuous cocycle on X over R and ( Ω , ℱ , P , { θ t } t ∈ R ) , and D is the collection of some families of nonempty bounded subsets of X parameterized by τ ∈ R and ω ∈ Ω :

Definition 2.4

Let B = { B ( τ , ω ) : τ ∈ R , ω ∈ Ω } be a family of nonempty subsets of X . For every τ ∈ R , ω ∈ Ω , let

Then the family { Ω ( B , τ , ω ) : τ ∈ R , ω ∈ Ω } is called the Ω -limit set of B and is denoted by Ω ( B ) .

Definition 2.5

Let D be a collection of some families of nonempty subsets of X and K = { K ( τ , ω ) : τ ∈ R , ω ∈ Ω } ∈ D . Then K is called a D -pullback absorbing set for Φ if for all τ ∈ R and ω ∈ Ω and for every B ∈ D , there exists T = T ( B , τ , ω ) > 0 such that

If, in addition, K ( τ , ω ) is closed in X and is measurable in ω with respect to ℱ , then K is called a closed measurable D -pullback absorbing set for Φ .

Definition 2.6

Let D be a collection of some families of nonempty subsets of X . Then Φ is said to be D -pullback asymptotically compact in X if for all τ ∈ R and ω ∈ Ω , the sequence

whenever t n → ∞ , and x n ∈ B ( τ − t n , θ − t n ω ) with { B ( τ , ω ) : τ ∈ R , ω ∈ Ω } ∈ D .

Definition 2.7

Let D be a collection of some families of nonempty subsets of X and A = { A ( τ , ω ) : τ ∈ R , ω ∈ Ω } ∈ D . Then A is called a D -pullback attractor for Φ if the following conditions (1)–(3) are fulfilled: for all t ∈ R + , τ ∈ R and ω ∈ Ω ,

1. A ( τ , ω ) is compact in X and is measurable in ω with respect to ℱ ;

2. A is invariant, that is,

   Φ ( t , τ , ω , A ( τ , ω ) ) = A ( τ + t , θ t ω ) ;

3. For every B = { B ( τ , ω ) : τ ∈ R , ω ∈ Ω } ∈ D ,

   lim t → ∞ d ( Φ ( t , τ − t , θ − t ω , B ( τ − t , θ − t ω ) ) , A ( τ , ω ) ) = 0 .

Proposition 2.8

Let D be an inclusion-closed collection of some families of nonempty subsets of X , and Φ be a continuous cocycle on X over R and ( Ω , ℱ , P , { θ t } t ∈ R ) . If Φ is D -pullback asymptotically compact in X and Φ has a closed measurable D -pullback absorbing set K in D , then Φ has a unique D -pullback attractor A in D which is given by, for each τ ∈ R and ω ∈ Ω ,

Assumption I

Assume that the memory kernel function μ ∈ C 1 ( R + ) ∩ L 1 ( R + ) , nonlinear functions h ∈ C 1 ( R ) and f ∈ C 1 ( R ) satisfy the following conditions:

1. ∀ s ∈ R + and some ϖ > 0 .

   (3.4) μ ( s ) ≥ 0 , μ ′ ( s ) + ϖ μ ≤ 0 ,

   note that (3.4) implies m 0 = def ‖ μ ‖ L 1 ( R + ) = ∫ 0 ∞ μ ( s ) d s > 0 .

2. Let F ( x , u ) = ∫ 0 u f ( x , s ) d s for x ∈ R n and u ∈ R , there exist positive constants c i ( i = 1 , 2 , 3 , 4 ), such that

   (3.5) ∣ f ( x , u ) ∣ ≤ c 1 ∣ u ∣ p + ϕ 1 ( x ) , ϕ 1 ∈ L 2 ( R n ) ,

   (3.6) f ( x , u ) u − c 2 F ( x , u ) ≥ ϕ 2 ( x ) , ϕ 2 ∈ L 1 ( R n ) ,

   (3.7) F ( x , u ) ≥ c 3 ∣ u ∣ p + 1 − ϕ 3 ( x ) , ϕ 3 ∈ L 1 ( R n ) ,

   (3.8) ∂ f ∂ u ( x , u ) ≤ β , ∂ f ∂ x ( x , u ) ≤ ϕ 4 ( x ) , ϕ 4 ∈ L 2 ( R n ) ,

   where β > 0 , 1 ≤ p ≤ n + 4 n − 4 . Note that (3.5) and (3.6) imply

   (3.9) F ( x , u ) ≤ c ( ∣ u ∣ 2 + ∣ u ∣ p + 1 + ϕ 1 2 + ϕ 2 ) .

3. There exist two constants β 1 , β 2 such that

   (3.10) h ( 0 ) = 0 , 0 < β 1 ≤ h ′ ( v ) ≤ β 2 < ∞ .

Lemma 3.1

Put φ ( t + τ , τ , θ − τ ω , φ 0 ) = ( u ( t + τ , τ , θ − τ ω , u 0 ) , v ( t + τ , τ , θ − τ ω , v 0 ) , η ( t + τ , τ , θ − τ ω , η 0 , s ) ) ⊤ , where φ 0 = ( u 0 , v 0 , η 0 ) ⊤ , and let conditions (3.4)–(3.8) and (3.10) hold. Then for every ω ∈ Ω , τ ∈ R and φ 0 ∈ E ( R n ) , problem (3.12)–(3.13) has a unique ( ℱ , ℬ ( H 2 ( R n ) ) × ℬ ( L 2 ( R n ) ) × ℬ ( R μ , 2 ) ) -measurable solution φ ( ⋅ , τ , ω , φ 0 ) ∈ C ( [ τ , ∞ ) , E ( R n ) ) with φ ( τ , τ , ω , φ 0 ) = φ 0 , φ ( t , τ , ω , φ 0 ) ∈ E ( R n ) being continuous in φ 0 with respect to the usual norm of E ( R n ) for each t > τ . Moreover, for every ( t , τ , ω , φ 0 ) ∈ R + × R × Ω × E ( R n ) , the mapping

generates a continuous cocycle from R + × R × Ω × E ( R n ) to E ( R n ) over R and ( Ω , ℱ , P , { θ t } t ∈ R ) .

Assumption II

We assume that σ , δ , ε , and g ( x , t ) satisfy the following conditions:

where γ 1 = max 1 , c 1 c 3 − 1 2 , γ 2 = 1 + 1 λ + δ 2 − β 2 δ , γ 3 = 3 2 δ + 1 2 β 2 + ( β 2 − β 1 ) δ + 1 . Moreover,

and

where ∣ ⋅ ∣ denotes the absolute value of real number in R .

Lemma 4.1

Under Assumptions I and II, for every τ ∈ R , ω ∈ Ω , D = { D ( τ , ω ) : τ ∈ R , ω ∈ Ω } ∈ D , there exists T = T ( τ , ω , D ) > 0 such that for all t ≥ T the solution of problem (3.12)–(3.13) satisfies

and R ( τ , ω ) is given by

where M is a positive constant independent of τ , ω , D , and ε .

Proof

Taking the inner product of ( 3.12 ) 2 with v in L 2 ( R n ) , we get

By ( 3.12 ) 1 , we have

It follows from (3.10) and Lagrange’s mean value theorem that

where ϑ is between 0 and v + ε u z ( θ t ω ) − δ u .

By (3.10) and (4.4), we know

Substituting (4.4) into the third and fourth terms on the left-hand side of (4.3), we find that

and

Using the Cauchy-Schwarz inequality and Young’s inequality, we have