The family of random attractors for nonautonomous stochastic higher-order Kirchhoff equations with variable coefficients

Lemma 2.1

[26] Suppose that μ ∈ L N 2 ( Ω ) ∩ C 0 ∞ ( Ω ) , then for all u ∈ C 0 ∞ ( Ω ) , there exists α > 0 such that

where α = β − 2 ‖ μ ‖ N / 2 − 1 .

Let μ > 0 be the weight function, and the weighted space L μ p = L μ p ( Ω ) with the following norm:

for 1 ≤ p < + ∞ . Clearly L μ 2 = L μ 2 ( Ω ) is a separable Hilbert space the inner product and norm are respectively:

For p : 1 ≤ p < ∞ , the Banach space L μ p is uniformly convex, reflexive space, and ( L μ p ) ′ = L μ p ′ , where p ′ is the conjugate number of p .

Lemma 2.2

[26] Suppose that μ ∈ L N 2 ( Ω ) ∩ C 0 ∞ ( Ω ) , then D 1 , 2 is compactly embedded in L μ 2 . Let

and the corresponding inner product and norm are, respectively,

At the same time, a general form of Poincare inequality: λ 1 ‖ ∇ r u ‖ 2 ≤ ‖ ∇ r + 1 u ‖ 2 , where λ 1 is the first eigenvalue of − Δ . In the text, C i is a positive constant, C ( ⋅ ) represents a positive constant that depends on the parameters in parentheses, and C m n is the corresponding number of combinations.

Definition 2.3

[12] Let θ t : R × Ω 1 → Ω 1 be a family of ( B ( X ) × ℱ , ℱ ) -measurable mappings such that θ 0 ( ⋅ ) is the identity on Ω 1 ∀ t , s ∈ R , θ t + s ( ⋅ ) = θ t ( ⋅ ) ∘ θ s ( ⋅ ) , P θ t ( ⋅ ) = P . A mapping Φ : R + × R × Ω 1 × X → X is called a continuous cocycle or continuous random dynamical system (RDS) on X over R and ( Ω 1 , ℱ , P , ( θ t ) t ∈ R ) if for all τ ∈ R , w ∈ Ω 1 , t , s ∈ R + the following conditions are satisfied:

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

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

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

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

Definition 2.4

[13] The family D = { D ( τ , w ) ⊆ X : τ ∈ R , w ∈ Ω 1 } satisfies:

1. for all ( τ , w ) ∈ R × Ω 1 D ( τ , w ) is a closed nonempty subset of X ;

2. for every fixed x ∈ X and any τ ∈ R , the mapping w ∈ Ω 1 → dist X ( x , B ( τ , w ) ) is ( ℱ , B ( R + ) ) measurable, then the family D is measurable with to ℱ in Ω 1 .

Definition 2.5

[15] For all σ > 0 , w ∈ Ω 1 D = { D ( τ , w ) ⊆ X : τ ∈ R , w ∈ Ω 1 } satisfies:

then D = { D ( τ , w ) ⊆ X : τ ∈ R , w ∈ Ω 1 } is called tempered.

Let D = D ( X ) be the set of all random tempered sets in X .

Definition 2.6

[12] A family K = { K ( τ , w ) ⊆ X : τ ∈ R , w ∈ Ω 1 } ∈ D of nonempty subsets of X is called a measurable D -pullback attracting(or absorbing) set for { Φ ( t , τ , w ) } t ≥ 0 , τ ∈ R , w ∈ Ω 1 if

1. K is measurable with respect to the P completion of ℱ in Ω 1 ;

2. for all τ ∈ R , w ∈ Ω 1 and for every D ∈ D , there exists T ( D , τ , w ) > 0 such that

   Φ ( t , τ − t , θ − t w , D ( τ − t , θ − t w ) ) ⊆ K ( τ , w ) , ∀ t ≥ T ( D , τ , w ) .

Definition 2.7

[15] Φ is said to be asymptotically compact in X if for τ ∈ R , w ∈ Ω 1 , D = { D ( τ , w ) ⊆ X : τ ∈ R , w ∈ Ω 1 } ∈ D , x n ∈ B ( τ − t n , θ − t n w ) { Φ ( t n , τ − t n , θ − t n w , x n ) } n = 1 ∞ has a convergent subsequence in X whenever t n → ∞ .

Definition 2.8

[13] A family A = { A ( τ , w ) ⊆ X : τ ∈ R , w ∈ Ω 1 } ∈ D is called a D -pullback random attractor for { Φ ( t , τ , w ) } t ≥ 0 , τ ∈ R , w ∈ Ω 1 if

1. A ( τ , w ) is measurable in Ω 1 with respect to ℱ and compact in X for ∀ τ ∈ R , w ∈ Ω 1 ,

2. A is invariant, i.e., for ∀ τ ∈ R and w ∈ Ω 1 , ∀ t ≥ 0 ,

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

3. A attracts every member of D , i.e., for every D ∈ D , τ ∈ R and for every w ∈ Ω 1 ,

   lim t → + ∞ dist X ( Φ ( t , τ − t , θ − t w , B ( τ − t , θ − t w ) ) , A ( τ , w ) ) = 0 ,

   where dist X ( P , Q ) denotes the Hausdorff semi-distance between two subsets P and Q of X .

Lemma 2.9

[12] Let D be a neighborhood-closed collection of ( τ , w ) -parametrized families of nonempty subsets of X and Φ be a continuous cocycle on X over R and ( Ω 1 , ℱ , P , { θ t } t ∈ R ) , then Φ has a pullback D -attract A if and only if Φ is pullback D asymptotically compact in X and Φ has a closed, ℱ -measurable pullback D -absorbing set K in D and the unique pullback D -attractor A = { A ( τ , w ) } is given by

Similarly, Lemma 2.9 can be extended to Lemma 2.10 of the family of pullback attractors.

Lemma 2.10

Let D k be neighborhood-closed collections of ( τ , w ) -parametrized families of nonempty subsets of X k , k = 1 , 2 , … , m , and Φ k be the family of continuous cocycles on X k , k = 1 , 2 , … , m over R and ( Ω 1 , ℱ , P , { θ t } t ∈ R ) , then Φ k has the family of pullback D k -attracts { A k } if and only if Φ k is pullback D k -asymptotically compact in X k and D k has closed, ℱ -measurable pullback D k -absorbing sets K k in D k and the unique pullback D k -attractor A k = { A k ( τ , w ) } is given by

Lemma 4.1

Suppose M satisfies ( M ) , h ( x ) ∈ V m + k ( Ω ) , k = 0 , 1 , … , m , (3.2)–(3.6) hold, f ( x , t ) satisfies ( F 1 ) , and B k = { B k ( τ , w ) : τ ∈ R , w ∈ Ω 1 } ∈ D k for P − a . e . w ∈ Ω 1 , τ ∈ R initial value satisfies ( u τ − t , v τ − t ) ∈ B k ( τ − t , θ − τ w ) , there exists T k = T k ( τ , w , B k ) > 0 such that for all t ≥ T k , the solution ( u ( τ , τ , w , u τ − t ) , v ( τ , τ , w , v τ − t ) ) = ( u τ − t , v τ − t ) of problem (3.8) satisfies

where r 1 k ( τ , w ) will be given in detail later.

Proof

Taking the inner product of (3.8) with v in L μ 2 ( Ω ) , we find that

for each term on the right-hand side of (4.2):