Random attractors of Kirchhoff-type reaction–diffusion equations without uniqueness driven by nonlinear colored noise

Zhang Zhang; Xiaobin Yao

doi:10.1515/math-2024-0083

Article Open Access

Random attractors of Kirchhoff-type reaction–diffusion equations without uniqueness driven by nonlinear colored noise

Zhang Zhang and Xiaobin Yao

Published/Copyright: November 15, 2024

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$Open Mathematics$

From the journal Open Mathematics Volume 22 Issue 1

Abstract

In this article, we consider the asymptotic behavior of solutions for the Kirchhoff-type reaction–diffusion equations driven by a nonlinear colored noise defined on unbounded domains. We prove the existence and uniqueness of pullback random attractors by the uniformly estimate.

Keywords: reaction–diffusion equation; the Kirchhoff-type; colored noise; unbounded domain

MSC 2010: 35B40; 35B41

1 Introduction

Ornstein and Uhlenbeck were the first to propose and name the colored noise. In [1,2], first constructed colored noise to approximate the description of velocity stochastic behavior is used to determine the position of particles. Colored noise is applied in complex systems [3,4]. Colored noise is usually used to study the physical system [3–8]. Recently, there have been new advances in the pathwise dynamics of random systems, which has been proved by smooth approximations of nonlinear noise [5,9–14].

Consider the reaction–diffusion equations defined on R n :

(1.1) u t + λ u − M ( ‖ ∇ u ‖ 2 ) Δ u = f ( x , u ) + g ( t , x ) + G ( x , u ) ζ δ ( θ t ω ) u ( μ , x ) = u μ ( x ) ,

where μ ∈ R , x ∈ R n , λ > 0 are positive constants, a non-autonomous deterministic external term g ∈ L loc 2 ( R , L 2 ( R n ) ) , nonlinear functions f and G satisfy certain conditions, M is a scale function, and the colored noise ζ δ with δ > 0 . When M = 1 , the existence of random attractors of equation (1.1) was proved in [15]. Up to now, the existence of attractors for (1.1) has not been studied by any predecessors when M is a scale function. In this article, we will study the dynamics of (1.1).

Throughout the article, we suppose that both f and g are continuous functions, but not necessarily Lipschitz continuous, this brings without uniqueness of solutions for (1.1). Many authors have studied of asymptotic behavior of multivalued dynamical systems [16–20]. Zhao had considered asymptotic behavior of reaction–diffusion equations driven by white noise (additive and linear multiplicative) on unbounded domains [8,21]. The study of random equations can become deterministic ones when the system is driven by white noise, but this transformation cannot be applied to stochastic equations driven by nonlinear white noise, so we cannot obtain the existence of random attractors in systems with nonlinear white noise [5–7,22–24]. Therefore, this article focuses on Kirchhoff-type reaction–diffusion equations driven by nonlinear colored noise.

In this article, in order to prove the uniqueness of equation (1.1), we need to prove the asymptotic compactness of the random attractors. In this article, we need to solve the following 2 questions:

the study of random attractor of multi-valued dynamical systems is more complex;
the Sobolev embeddings on unbounded domains are no longer compact;
the emergence of Kirchhoff-type term makes the estimates more complicated.

To solve the above difficulties, we discuss the measurability of the random attractor of (1.1) by the method of weak upper semicontinuity in [2]. To solve the no longer compact Sobolev embedding in unbounded domains, we use the idea of the Ball energy equation in [25,26] to derive the asymptotic compactness of the solution of problem (1.1).

This article is organized as follows: we will present some results about the pullback random attractors of multivalued non-autonomous random dynamical systems in Section 2. In Section 3, we will prove the existence of the generated random dynamical systems for (1.1) and structure weak and strong upper semicontinuity of multivalued solutions. At last, we will prove the existence of the pullback random attractors for (1.1) by uniform estimates of the solution in Section 4.

2 Preliminaries

Let X ¯ and Y ¯ be the metric space and ℬ ( X ¯ ) and ℬ ( Y ¯ ) be Borel σ -algebras. We will denote all subsets of Y ¯ as 2 Y ¯ . Let G : X ¯ → 2 Y ¯ be measurable in ℬ ( X ¯ ) .

Let ( Ω , ℱ , P ) be a probability space, where Ω = { ω ∈ C ( R , R ) : ω ( 0 ) = 0 } with the open compact topology, ℱ is the Borel σ -algebra, and P is the Wiener measure. Let W be a bilateral real valued Wiener process defined on the probability space ( Ω , ℱ , P ) . The classical transformation { θ t } t ∈ R on Ω is given by θ t ω ( ⋅ ) = ω ( t + ⋅ ) − ω ( t ) , ω ∈ Ω for ω ∈ Ω . Suppose ζ δ : Ω → R by ζ δ ( ω ) = 1 δ ∫ − ∞ 0 e s δ d W for δ > 0 . Then, the process O δ ( t , ω ) = ζ δ ( θ t ω ) is called the O-U process (i.e. the colored noise), which is a stationary Gaussian process with E ( ζ δ ) = 0 and is the unique stationary solution of the stochastic equation:

d ζ δ + 1 δ ζ δ d t = 1 δ d W .

We will often use the following theorems and lemmas.

Lemma 2.1

Suppose x ¯ 0 ∈ X ¯ and y ¯ n ∈ G ( x ¯ n ) , if x ¯ n → x ¯ 0 in X ¯ , there exists y ¯ 0 ∈ G ( x ¯ 0 ) and a subsequence { y ¯ n k } of { y ¯ n } in Y ¯ , then G is upper semicontinuous at x ¯ 0 .

Lemma 2.2

Suppose Φ ¯ ( t , μ , ⋅ , ⋅ ) : Ω × X ¯ → 2 X ¯ is a multi-valued function. A metric space Ω and a separable Banach space X ¯ . For t ∈ R + , μ ∈ R , the mapping Φ ¯ ( t , μ , ⋅ , ⋅ ) is weakly upper semicontinuous and is ℬ ( Ω ) × ℬ ( X ¯ ) -measurable.

Theorem 2.3

Suppose Φ ¯ : ( Ω , ℱ , P , { θ } t ∈ R ) be a multi-valued non-autonomous cocycle on X ¯ with D being an inclusion-closed collection of some families of nonempty subsets of X ¯ , we have the following assumptions

Φ ¯ is D -pullback asymptotically compact in X ¯ ;
For all t ∈ R + , μ ∈ R , ω ∈ Ω , then Φ ¯ ( t , μ , ω , ⋅ ) : X ¯ → 2 Y ¯ is upper semicontinuous;
K = { K ( μ , ω ) : μ ∈ R , ω ∈ Ω } ∈ D is a closed D -pullback absorbing set of Φ ¯ ;
For each m ∈ N , t ∈ R + , μ ∈ R , Φ ¯ ( t , μ , ⋅ , K ( μ , ⋅ ) ) : Ω m → 2 Y ¯ is weakly upper semicontinuous.

Let A ¯ be a unique D -pullback attractor in D for Φ ¯ that has

A ¯ ( μ , ω ) = ⋂ r ≥ 0 ⋃ t ≥ r Φ ¯ ( t , μ − t , θ − t ω , K ( μ − t , θ − t ω ) ) ¯ .

The colored noise ζ δ (see [27]), we have:

Lemma 2.4

If 0 < δ ≤ 1 and Ω is a { θ t } t ∈ R -invariant subset, for ω ∈ Ω ,

(2.1) lim t → ± ∞ ω ( t ) t = 0 ;
let ( t , ω ) ↦ ζ δ ( θ t ω ) = − 1 δ 2 ∫ − ∞ 0 e s δ θ t ω ( s ) d s is a stationary solution of d ζ δ + 1 δ ζ δ d t = 1 δ d W with continuous trajectories satisfying
(2.2) lim t → ± ∞ ∣ ζ δ ( θ t ω ) ∣ t = 0 ,

(2.3) lim t → ± ∞ 1 t ∫ 0 t ζ δ ( θ s ω ) d s = 0 ;
if T > 0 , ε > 0 , there exist δ 0 = δ 0 ( μ , ω , T , ε ) > 0 such that for all 0 < δ < δ 0 and t ∈ [ μ , μ + T ] ,
(2.4) ∫ 0 t ζ δ ( θ s ω ) d s − ω ( t ) < ε .

By (2.4), there exist δ 0 = δ 0 ( μ , ω , T ) > 0 , c = c ( μ , ω , T ) , such that for 0 < δ < δ 0 and t ∈ [ μ , μ + T ] ,

(2.5) ∫ 0 t ζ δ ( θ s ω ) d s ≤ c .

Hereafter, we denote H = L 2 ( R n ) with ( ⋅ , ⋅ ) and ‖ ⋅ ‖ and denote V = H 1 ( R n ) and V ′ = H − 1 ( R ) with ‖ ⋅ ‖ V , and L p ( R n ) ( p > 2 ) with norm ‖ ⋅ ‖ p . As usual, let c in the article represents a generic positive constant.

3 Construction of multi-valued non-autonomous cocycles

Defining a reaction–diffusion equation with the Kirchhoff-type driven by colored noise:

(3.1) u t + λ u − M ( ‖ ∇ u ‖ 2 ) Δ u = f ( x , u ) + g ( t , x ) + G ( x , u ) ζ δ ( θ t ω ) , t > μ , x ∈ R n ,

with initial condition

(3.2) u ( μ , x ) = u μ ( x ) , x ∈ R n ,

where M is a continuous function related to s and satisfies

(3.3) M ( s ) > κ ≥ 0 .

f and G : R n × R → R are nonlinear continuous functions, and for u ∈ R , y ∈ R n ,

(3.4) f ( y , u ) u ≤ − α 1 ∣ u ∣ p + ψ 1 ( y ) ,

(3.5) ∣ f ( y , u ) ∣ ≤ α 2 ∣ u ∣ p − 1 + ψ 2 ( y ) ,

(3.6) ∣ G ( y , u ) ∣ ≤ ψ 3 ( ∣ u ∣ q − 1 + 1 ) ,

where α 1 , α 2 , and p , q ( 2 ≤ q < p ) are positive constants, ψ 1 ∈ L 1 ( R n ) , ψ 2 ∈ L p 1 ( R ) , ψ 3 ∈ L p 1 ∩ L p q ( R n ) with p 1 = p p − 1 , p q = p p − q .

We will prove that the solution is meaningful.

Definition 3.1

Given μ ∈ R , ω ∈ Ω , and u μ ∈ H , let a continuous function u ( ⋅ , μ , ω , u τ ) : [ μ , ∞ ) → H with L loc p ( ( μ , ∞ ) , L p ( R n ) ) ∩ L loc 2 ( ( μ , ∞ ) , V ) , and u t ∈ L loc 2 ( ( μ , ∞ ) , V ′ ) + L loc p 1 ( ( μ , ∞ ) , L p 1 ( R ) ) . Then, u is called a solution of (3.1) and (3.2) with initial data u τ if u ( μ , μ , ω , u μ ) = u μ and for every k ∈ V ∩ L p ( R n ) ,

(3.7) ( u t , k ) + λ ( u , k ) + ( M ( ‖ ∇ u ‖ 2 ) Δ u , k ) = ( g ( t , ⋅ ) , k ) + ∫ R ( f ( x , u ) + ζ δ ( θ t ω ) G ( x , u ) ) k d x

in the sense of distribution on ( μ , ∞ ) .

Then, we will discuss the existence of the solution to problems (3.1) and (3.2) on unbounded domains. First, we will consider the situation in bounded domains. Furthermore, we assume ξ ∈ N , define O ξ = { x ∈ R : ∣ x ∣ < ξ } ,

(3.8) u ξ t + λ u ξ − M ( ‖ ∇ u ξ ‖ 2 ) Δ u ξ = f ( x , u ξ ) + g ( t , x ) + G ( x , u ξ ) ζ δ ( θ t ω ) , t > μ , x ∈ O ξ , u ξ ( t , x ) = 0 , t > τ , ∣ x ∣ = ξ , u ξ ( μ , x ) = u τ ( x ) , x ∈ O ξ .

Similarly [28], under conditions (3.3)–(3.6), one can easily obtain that for every μ ∈ R , ω ∈ Ω , and u μ ∈ H , problem (3.8) has at least one solution u ξ ∈ C ( [ μ , ∞ ) , H ) ∩ L loc p ( ( μ , ∞ ) , L p ( R n ) ) ∩ L loc 2 ( ( μ , ∞ ) , V ) . As ξ → ∞ , we can obtain the following result for (3.1) and (3.2).

Lemma 3.2

If (3.3)–(3.6) hold and g ∈ L loc 2 ( R , H ) . Then, for each μ ∈ R , ω ∈ Ω , and u μ ∈ H , problems (3.1) and (3.2) in the sense of Definition 3.1, there exists a solution u ( ⋅ , μ , ω , u μ ) and for all t ≥ μ , which satisfies the energy equation

(3.9) d d t ‖ u ( t , μ , ω , u μ ) ‖ 2 + 2 λ ‖ u ( t , μ , ω , u μ ) ‖ 2 + 2 M ( ‖ ∇ u ‖ 2 ) ‖ ∇ u ‖ 2 = 2 ∫ R 2 f ( x , u ( t , μ , ω , u μ ) ) u d x + 2 ( g , u ( t , μ , ω , u μ ) ) + 2 ζ δ ( θ t ω ) ∫ R n G ( x , u ( t , μ , ω , u μ ) ) u d x .

In order to obtain the weak semicontinuity and strong upper semicontinuity of the solutions of (3.1) and (3.2), for all m ∈ N , we suppose

(3.10) Ω ¯ m = { ω ∈ Ω ¯ : ∣ ω ( t ) ∣ ≤ ∣ t ∣ , ∣ t ∣ ≥ m } ,

which obtain properties as follows:

Lemma 3.3

[15] If the colored noise ζ δ with δ ∈ ( 0 , 1 ] , and Ω ¯ m ∈ Ω ¯ with m ∈ N from (3.10).

Let ω n → ω with ω n , ω ∈ Ω ¯ m , for t in a compact interval as n → ∞ , then ζ δ ( θ t ω n ) → ζ δ ( θ t ω ) .
If each m ∈ N , Ω ¯ m is a closed subset of Ω ¯ with Ω ¯ = ∪ m = 1 ∞ Ω ¯ m .
Taken m ∈ N and ω ∈ Ω ¯ m , for every t ≤ − m , we obtain
(3.11) ∣ ζ δ ( θ t ω ) ∣ ≤ 2 δ ∣ t ∣ + 1 .

We will discuss the uniform estimates of solutions for problems (3.1) and (3.2) as follows.

Lemma 3.4

If (3.3)–(3.6) hold. Suppose δ > 0 , μ ∈ R , L > 0 , T > 0 , and ω n → ω with ω n , ω ∈ Ω ¯ m and c = c ( δ , μ , T , L , m , ω ) > 0 , then the solutions of (3.1) and (3.2) satisfy

(3.12) ‖ u ¯ ( t , μ , ω n , u μ ) ‖ 2 + ∫ μ t e λ ( s − t ) ( 2 M ( ‖ ∇ u ( s , μ , ω n , u μ ) ‖ 2 ) ‖ ∇ u ( s , μ , ω n , u μ ) ‖ 2 + λ 2 ‖ u ( s , μ , ω n , u μ ) ‖ 2 + ‖ u ( s , μ , ω n , u μ ) ‖ p p ) d s ≤ c ,

which n ∈ N , t ∈ [ μ , μ + T ] , and u μ ∈ H with ‖ u μ ‖ ≤ L .

Proof

By (3.3), (3.4), (3.6), (3.8), and applying Young’s inequality, we obtain

(3.13) d d t ‖ u ‖ 2 + 3 2 λ ‖ u ‖ 2 + 2 M ( ‖ ∇ u ‖ 2 ) ‖ ∇ u ‖ 2 + α 1 ‖ u ‖ p p ≤ c ( 1 + ∣ ζ δ ( θ t ) ω ∣ p q + ∣ ζ δ ( θ t ) ω ∣ p 1 + ‖ g ( t , ⋅ ) ‖ 2 ) .

By Gronwall’s inequality, we have

(3.14) ‖ u ( t , μ , ω n , u μ ) ‖ 2 + ∫ μ t e λ ( s − t ) ( 2 M ( ‖ ∇ u ( s ) ‖ 2 ) ‖ ∇ u ( s ) ‖ 2 + λ 2 ‖ u ( s ) ‖ 2 + α 1 ‖ u ( s ) ‖ p p ) d s ≤ e λ ( μ − t ) ‖ u μ ‖ 2 + c 1 ∫ μ t ( 1 + ∣ ζ δ ( θ s ω n ) ∣ p q + ∣ ζ δ ( θ s ω n ) ∣ p 1 + ‖ g ( s , ⋅ ) ‖ 2 ) d s .

For s ∈ [ μ , μ + T ] as n → ∞ and ω n → ω , we know that ζ δ ( θ s ω n ) → ζ δ ( θ s ω ) . Suppose there exists N 1 = N 1 ( T , μ , ω ) ≤ 1 , for every n ≤ N 1 and s ∈ [ μ , μ + T ] , ∣ ζ δ ( θ s ω n ) ∣ ≤ 1 + ∣ ζ δ ( θ s ω ) ∣ , then ζ δ ( θ s ω ) is continuity in s and exists c ¯ 1 = c 1 ( δ , T , μ , ω ) > 0 , for every n ≥ N 1 and s ∈ [ μ , μ + T ] ,

(3.15) ∣ ζ δ ( θ s ω ) ∣ ≤ c ¯ 1 , ∣ ζ δ ( θ s ω n ) ∣ ≤ 1 + c ¯ 1 ,

along with (3.14) and (3.15) to obtain (3.12).□

Now, we discuss the weak continuity and strong continuity of solutions of (3.1) and (3.2).

Lemma 3.5

If (3.3)–(3.6) hold, for μ ∈ R , t ≥ τ , and ω n → ω with ω n , ω ∈ Ω ¯ m . Suppose a solution u ^ ( ⋅ , μ , ω n , u τ , n ) of (3.1) and (3.2) with initial data u μ , n .

If u μ , n ⇀ u μ in H, then a solution u ^ of problems (3.1) and (3.2) with u μ such that
(3.16) u ( t , μ , ω n , u μ , n ) ⇀ u ^ ( t , μ , ω , u μ ) in H
and
(3.17) u ^ ( ⋅ , μ , ω n , u μ , n ) ⇀ u ^ ( ⋅ , μ , ω , u μ ) in L 2 ( ( μ , μ + T ) , V ) .
If u μ , n → u μ strongly in H, then a solution u ^ of problem (3.1) and (3.2) with u μ such that
u ^ ( t , μ , ω n , u μ , n ) → u ^ ( t , μ , ω , u μ ) in H .

Proof

(1) By u μ , n ⇀ u μ in H , there exists c 1 > 0 such that ‖ u μ , n ‖ ≤ c 1 for every n ∈ N . Denote u n ( m ) = u ( m , μ , ω n , u μ , n ) for m ∈ [ μ , μ + T ] . If ω n → ω with ω n , ω ∈ Ω ¯ , by (3.3), (3.5), and Lemma 3.4, for all n ∈ N , we have

(3.18) { u n } n = 1 ∞ is bounded in L ∞ ( μ , μ + T ; H ) ∩ L 2 ( μ , μ + T ; V ) ∩ L p ( μ , μ + T ; L p ( R n ) ) ,

(3.19) { M ( ‖ ∇ u n ‖ 2 ) } n = 1 ∞ is bounded in L 2 ( μ , μ + T , V ) ,

(3.20) { f ( x , u n ) } n = 1 ∞ and ζ δ ( θ t ω n ) G ( x , u n ) n = 1 ∞ is bounded in L p 1 ( μ , μ + T ; L p 1 ( R n ) ) ,

(3.21) ∂ u n ∂ t n = 1 ∞ is bounded in L 2 ( μ , μ + T ; V ′ ) ∩ L p 1 ( μ , μ + T ; L p 1 ( R n ) ) .

Then, there exist u ∈ L ∞ ( μ , μ + T ; H ) ∩ L 2 ( μ , μ + T ; V ) ∩ L p ( μ , μ + T ; L p ( R n ) ) , χ ∈ L p 1 ( μ , μ + T ; L p 1 ( R n ) ) , and u ¯ ( t 0 ) ∈ H for ∀ t 0 ∈ [ μ , μ + T ] , such that, up to a subsequence

(3.22) u n ⇀ * u ^ in L ∞ ( μ , μ + T ; H ) ,

(3.23) u n ⇀ u in L p ( μ , μ + T ; L p ( R n ) ) and L 2 ( μ , μ + T ; V ) ,

(3.24) M ( ‖ ∇ u n ‖ 2 ) ⇀ M ( ‖ ∇ u ‖ 2 ) in L 2 ( μ , μ + T , V ) ,

(3.25) f ( x , u n ) + ζ δ ( θ t ω n ) G ( x , u n ) ⇀ χ in L p 1 ( μ , μ + T ; L p 1 ( R n ) ) ,

(3.26) ∂ u n ∂ t ⇀ ∂ u ∂ t in L 2 ( μ , T ; V ′ ) ∩ L p 1 ( μ , T ; L p 1 ( R n ) ) ,

(3.27) u n ( t 0 , μ , ω , u μ ) ⇀ u ^ ( t 0 ) in H .

We obtain u is a solution of (3.1) and (3.2) with initial data u ( μ ) = u μ by the standard method. Further

u n ( m ) ⇀ u ^ ( m ) in H , for m ∈ [ μ , μ + T ] .

(2) Denote u ^ ( t ) = u ^ ( t , μ , ω , u μ ) . By the weak convergence in (1), we have

(3.28) liminf n → ∞ ‖ u n ( t ) ‖ ≥ ‖ u ^ ( t ) ‖ .

Next, we prove strong convergence

(3.29) limsup n → ∞ ‖ u n ( t ) ‖ ≤ ‖ u ^ ( t ) ‖ .

Taking t ≥ μ , T > 0 , such that t ∈ [ μ , μ + T ] , by (3.9), we find any weak solution of (3.1) and (3.2) for t ∈ [ μ , μ + T ] ,

(3.30) d d t ‖ u ^ ‖ 2 + 2 λ ‖ u ^ ‖ 2 + 2 M ( ‖ ∇ u ^ ‖ 2 ) ‖ ∇ u ^ ‖ 2 = 2 ( f ( x , u ^ ) , u ^ ) + 2 ( g , u ^ ) + 2 ζ δ ( θ t ω ) ( G ( x , u ^ ) , u ^ ) .

Applying Gronwall’s inequality, we find

(3.31) ‖ u ^ ( t , μ , ω , u μ ) ‖ 2 = e − λ ( t − μ ) ‖ u μ ‖ 2 − ∫ μ t e − λ ( s − t ) ( λ ‖ u ( s ) ‖ 2 + 2 M ( ‖ ∇ u ( s ) ‖ 2 ) ‖ ∇ u ( s ) ‖ 2 ) d s + 2 ∫ μ t e − λ ( s − t ) ( g ( s ) , u ( s ) ) d s + 2 ∫ μ t e − λ ( s − t ) ( ( f ( x , u ( s ) ) , u ( s ) ) + ζ δ ( θ s ω n ) ( G ( x , u ( s ) ) , u ( s ) ) ) d s .

Replacing ω by ω n and u μ by u μ , n in (3.31), we have

(3.32) ‖ u n ( t ) ‖ 2 = e − λ ( t − μ ) ‖ u μ , n ‖ 2 − ∫ μ t e − λ ( s − t ) ( λ ‖ u n ( s ) ‖ 2 + 2 M ( ‖ ∇ u n ( s ) ‖ 2 ) ‖ ∇ u n ( s ) ‖ 2 ) d s + 2 ∫ μ t e − λ ( s − t ) ( g ( s ) , u n ( s ) ) d s + 2 ∫ μ t e − λ ( s − t ) ( ( f ( x , u n ( s ) ) , u n ( s ) ) + ζ δ ( θ s ω n ) ( G ( x , u n ( s ) ) , u n ( s ) ) ) d s ≜ ∑ i = 1 4 I i ( n ) .

Next, we will estimate each term in (3.32). First, since u μ , n → u μ in H , we find

(3.33) lim n → ∞ I 1 ( n ) = e − λ ( t − μ ) ‖ u μ ‖ 2 .

By (3.16) and (3.17), we have

(3.34) limsup n → ∞ I 2 ( n ) ≤ − λ ∫ μ t e − λ ( s − t ) ‖ u ^ ( s ) ‖ 2 − 2 ∫ μ t e − λ ( s − t ) M ( ‖ ∇ u ^ ( s ) ‖ 2 ) ( ‖ ∇ u ^ ( s ) ‖ 2 ) d s .

By (3.17), we obtain

(3.35) lim n → ∞ I 3 ( n ) = 2 ∫ μ t e − λ ( s − t ) ( g ( s ) , u ^ ( s ) ) d s .

Next, we estimate I 4 ( n ) , for k ∈ N , we obtain

(3.36) I 4 ( n ) = 2 ∫ μ t e − λ ( s − t ) ∫ ∣ x ∣ ≥ k ( f ( x , u n ( s ) ) u n ( s ) + ζ δ ( θ s ω n ) G ( x , u n ( s ) ) u n ( s ) ) d x d s + 2 ∫ μ t e − λ ( s − t ) ∫ O k ( f ( x , u n ( s ) ) u n ( s ) + ζ δ ( θ s ω n ) G ( x , u n ( s ) ) u n ( s ) ) d x d s ≜ I 41 ( n ) + I 42 ( n ) .

By (3.4), (3.6), and Young’s inequality, such that for k ≥ N , N ( ε ) > 0 , we have

(3.37) I 41 ( n ) = c 1 ∫ μ t e − λ ( s − t ) ∫ ∣ x ∣ ≥ k ( ∣ ψ 1 ( x ) ∣ + ∣ ψ 3 ( x ) ∣ p 1 + ∣ ψ 3 ( x ) ∣ p q ) d x d s ≤ ε .

By (3.22), (3.23), and (3.26), a diagonal process and a compactness argument, we can obtain a subsequence, for k ∈ N ,

u n → u ^ in L 2 ( μ , μ + T ; L 2 ( O k ) ) .

For any ( t , x ) ∈ ( μ , μ + T ) × O k , u ^ n ( t , x ) → u ^ ( t , x ) . Since f ( x , u ^ ) + ζ δ ( θ t ω ) G ( x , u ^ ) is continuous in u ^ , we have

f ( x , u n ) + ζ δ ( θ t ω n ) G ( x , u n ) → f ( x , u ^ ) + ζ δ ( θ t ω ) G ( x , u ^ ) .

By (3.4) and (3.6), we obtain

(3.38) ψ 1 + c 1 ( ∣ ψ 3 ∣ p 1 + ∣ ψ 3 ∣ p q ) − f ( x , u ^ ) u ^ − ζ δ ( θ t ω ) G ( x , u ^ ) u ^ ≥ 0 .

By applying Fatou’s lemma and by Lemma 3.3 (1), we find

(3.39) liminf n → ∞ ( − I 42 ( n ) ) ≥ − 2 ∫ μ t e λ ( s − t ) ∫ O k ( ( f ( x , u ¯ ( s ) ) , u ¯ ( s ) + G ( x , u ¯ ( s ) ) , u ¯ ( s ) ) ) d x d s ,

together with (3.37) to obtain

(3.40) limsup n → ∞ I 4 ( n ) ≤ 2 ∫ μ t e λ ( s − t ) ∫ O k ( ( f ( x , u ^ ( s ) ) , u ^ ( s ) ) + ζ δ ( θ s ω ) ( G ( x , u ^ ( s ) ) , u ^ ( s ) ) ) d x d s + ε .

Let k → ∞ , by (3.33), (3.34), and (3.31), we have

limsup n → ∞ ‖ u n ( t ) ‖ 2 ≤ ‖ u ^ ( t ) ‖ 2 + ε .

If ε → 0 , we obtain (3.29).□

By the consequence Lemma 3.5, we will prove the weak continuity of solution of (3.1) and (3.2) as follows.

Lemma 3.6

If (3.3)–(3.6) hold, μ ∈ R and ω ∈ Ω . Let a solution u ¯ ( ⋅ , μ , ω ) of (3.1) and (3.2) with initial date u μ , n . If u μ , n ⇀ u μ in H, then a solution of problems (3.1) and (3.2) u ^ ( ⋅ , μ , ω , u μ ) with initial condition u μ such that

u ^ ( t , μ , ω , u μ , n ) ⇀ u ^ ( t , μ , ω , u μ ) in H f o r t ≥ μ

and

u ^ ( ⋅ , μ , ω , u μ , n ) ⇀ u ^ ( ⋅ , μ , ω , u μ ) in L 2 ( ( μ , μ + T ) , V ) for T > 0 .

Denote U ( t + μ , μ , ω , u μ ) be the collection of all solutions of (3.1) and (3.2) at time t + μ with initial condition u μ and initial time μ ; that is

U ( t + μ , μ , ω , u μ ) = { ξ ( t + μ , μ , ω , u μ ) : ξ is a solution of (3.1 ) and ( 3.2) } .

By Lemmas 2.4, 3.3, and 3.5, we have the following result.

Lemma 3.7

If (3.3)–(3.6) hold, ∀ μ ∈ R , the mapping

U ( t , μ , ⋅ , ⋅ ) : Ω ¯ × H → 2 H

is measurable in ℬ ( Ω ¯ ) × ℬ ( H ) .

By Lemmas 3.3 and 3.5, we obtain the following lemma.

Lemma 3.8

Let (3.3)–(3.6) hold. If ∀ t ∈ R + , μ ∈ R , ω ∈ Ω ¯ , and u μ ∈ H , then the U ( t + μ , μ , ω , u μ ) is a set closed in H.

Proof

If ϕ n ∈ U ( t + μ , μ , ω , u μ ) and ϕ ∈ H , we will prove ϕ n → ϕ , then ϕ ∈ U ( t + μ , μ , ω , u μ ) . Since ϕ n ∈ U ( t + μ , μ , ω , u μ ) , there exists u n ( t + μ , μ , ω , u μ ) ∈ U ( t + μ , μ , ω , u μ ) such that

(3.41) ϕ n = u n ( t + μ , μ , ω , u μ ) .

We know that there exists m ∈ N such that ω ∈ Ω ¯ m . We have that problems (3.1) and (3.2) have a solution u ^ ( t , μ , ω , u μ ) with initial data u μ by Lemma 3.5 (1) such that

u n ( t + μ , μ , ω , u μ ) ⇀ u ^ ( t + μ , μ , ω , u μ ) in H ,

from (3.41), we obtain that

(3.42) ϕ n ⇀ u ^ ( t + μ , μ , ω , u μ ) .

By (3.42), since ξ n → ξ in H , we have

ϕ = u ^ ( t + μ , μ , ω , u μ ) ∈ U ( t + μ , μ , ω , u μ ) .

This proof is complete.□

Next, we define a multi-valued non-autonomous random dynamical system for (3.1) and (3.2). Denote multi-valued mapping Φ ¯ : R + × R × Ω × H → 2 H

(3.43) Φ ¯ ( t , μ , ω , u μ ) = U ( t + μ , μ , θ − μ ω , u μ ) = { u ( t + μ , μ , θ − μ ω , u μ ) : u is a solution of (3.1) and (3.2) } ,

where ( t , μ , ω , u μ ) ∈ R + × R × Ω × H . Since θ − μ : Ω → Ω is measurable, from Lemma 3.7 we obtain that for ∀ μ ∈ R , Φ ¯ ( ⋅ , μ , ⋅ , ⋅ ) : R + × Ω × H → 2 h is measurable. Then, Φ ¯ is a multi-valued non-autonomous cocycle in H by Lemma 3.8.

From the result of Lemma 3.5(2) and Lemma 2.1, we can obtain the upper semicontinuity of Φ ¯ as follows.

Lemma 3.9

If (3.3)–(3.6) hold. The mapping Φ ¯ ( t , μ , ω , ⋅ ) : H → 2 H is upper semicontinuous for ∀ t ∈ R + , μ ∈ R , ω ∈ Ω .

4 Existence and uniqueness of D -pullback random attractors

In this section, we discuss the existence of the D -pullback random attractor of Φ ¯ , where D = { D = { D ( μ , ω ) : μ ∈ R , ω ∈ Ω ¯ } } satisfying for ∀ μ ∈ R , ω ∈ Ω ¯ , and γ > 0 ,

(4.1) lim t → − ∞ e γ t ‖ D ( τ + t , θ t ω ) ‖ 2 = 0 .

For this purpose, we consider that g satisfies

(4.2) ∫ − ∞ 0 e λ r ‖ g ( r + μ , ⋅ ) ‖ 2 d r ≤ ∞ , μ ∈ R ,

and for ∀ c > 0 ,

(4.3) lim t → − ∞ e c t ∫ − ∞ 0 e λ r ‖ g ( r + t , ⋅ ) ‖ 2 d r = 0 .

Then, we will obtain uniform estimates of the solutions in H .

Lemma 4.1

If (3.3)–(3.6) hold. Suppose μ ∈ R , ω ∈ Ω , D = D ( μ , ω ) : μ ∈ R , ω ∈ Ω ∈ D . For every t ≥ T and s ∈ [ − t , 0 ] , there exists T = T ( μ , ω , D , δ ) > 0 such that the solution of problems (3.1) and (3.2) satisfies

(4.4) ‖ u ( μ + s , μ − t , θ − μ ω , u μ − t ) ‖ 2 + ∫ μ t μ + s e λ ( r − μ − s ) ( 2 M ( ‖ ∇ u ‖ 2 ) ‖ ∇ u ( r ) ‖ 2 + λ 2 ‖ u ( r ) ‖ 2 + α 1 ‖ u ( r ) ‖ p p ) d r ≤ e − λ s ℒ ( μ , ω ) ,

where u μ − t ∈ D ( μ − t , θ − t ω ) and

(4.5) ℒ ( μ , ω ) = ℛ ∫ − ∞ 0 e λ r ( 1 + ∣ ζ δ ( θ r ω ) ‖ p q + ‖ ζ δ ( θ r ω ) ‖ p 1 + ‖ g ( r + μ , ⋅ ) ‖ 2 ) d r ,

where ℛ is a positive constant and independent of s , μ , ω , D , and δ .

Proof

By multiplying e − λ t to (3.13) and integrating from μ − t to μ + s with t ≥ 0 and s ∈ [ − t , 0 ] , then there exists T 1 = T ( μ , ω , D , δ ) > 0 for every t ≥ T 1 and s ∈ [ − t , 0 ] ,

‖ u ( μ + s , μ − t , θ − μ ω , u μ − t ) ‖ 2 + ∫ μ − t μ + s e λ ( r − μ − s ) ( 2 M ( ‖ ∇ u ‖ 2 ) ‖ ∇ u ( r ) ‖ 2 + λ 2 ‖ u ( r ) ‖ 2 + α 1 ‖ u ( r ) ‖ p p ) d r ≤ e − λ ( t + s ) ‖ u μ − t ‖ 2 + c 1 ∫ μ − t μ + s e λ ( r − μ − s ) ( 1 + ∥ ζ δ ( θ r ω ) ‖ p q + ‖ ζ δ ( θ r ω ) ‖ p 1 + ‖ g ( r + μ , ⋅ ) ‖ 2 ) d r ≤ e − λ ( t + s ) ‖ D ( μ t , θ − t ω ) ‖ 2 + c 1 ∫ − t s e λ ( r − s ) ( 1 + ∥ ζ δ ( θ r ω ) ‖ p q + ‖ ζ δ ( θ r ω ) ‖ p 1 + ‖ g ( r + μ , ⋅ ) ‖ 2 ) d r = e − λ s ( e − λ t ‖ D ( μ − t , θ − t ω ) ‖ 2 + c 1 ∫ − t s e λ r ( 1 + ∥ ζ δ ( θ r ω ) ‖ p q + ‖ ζ δ ( θ r ω ) ‖ p 1 + ‖ g ( r + μ , ⋅ ) ‖ 2 ) d r ) ≤ c 2 e − λ s ∫ − ∞ 0 e λ r ( 1 + ∥ ζ δ ( θ r ω ) ‖ p q + ‖ ζ δ ( θ r ω ) ‖ p 1 + ‖ g ( r + μ , ⋅ ) ‖ 2 ) d r .

Due to u μ − t ∈ D ( μ − t , θ − t ω ) , we obtain

(4.6) e − λ t ‖ D ( μ − t , θ − t ω ) ‖ 2 → 0 as t → ∞ .

From the consequence of Lemma 4.1, we will prove the existence of the D -pullback absorbing set for Φ ¯ .

Lemma 4.2

If (3.3)–(3.6) and (4.2)–(4.3) hold, then K ∈ D is the D -pullback absorbing set of Φ ¯ :

(4.7) K ( μ , ω ) = { u ∈ H : ‖ u ‖ 2 ≤ ℒ ( μ , ω ) , μ ∈ L , ω ∈ Ω ¯ } ,

where ℒ ( μ , ω ) from (4.5).

Lemma 4.3

If (3.3)–(3.6) and (4.2)–(4.3) hold and Φ ¯ has a closed D -pullback absorbing set K . For m ∈ N , t ∈ R + , and μ ∈ R , the mapping Φ ¯ ( t , μ , ⋅ , K ( ⋅ ) ) : Ω ¯ m → 2 H is weakly upper semicontinuous.

Proof

Suppose ω n → ω in Ω ¯ m and u ˜ n ∈ Φ ¯ ( t , μ , ω n , K ( μ , ω n ) ) . We have u ˜ ∈ Φ ¯ ( t , μ , ω , K ( μ , ω ) ) such that

(4.8) u ˜ n ⇀ u ˜ in H .

Due to u ˜ n ∈ Φ ¯ ( t , μ , ω n , K ( μ , ω n ) ) and n ∈ N , there exist u μ , n ∈ K ( μ , ω n ) and a solution u n ( ⋅ , μ , θ − μ ω n , u μ , n ) with u ˜ μ , n such that

(4.9) ‖ u ˜ μ , n ‖ 2 ≤ ℛ ( μ , ω n ) = ℒ ∫ − ∞ 0 e λ r ( 1 + ∣ ζ δ ( θ r ω n ) ∣ p q + ∣ ζ δ ( θ r ω n ) ∣ p 1 + ‖ g ( r + μ , ⋅ ) ‖ 2 ) d r

and

(4.10) u ˜ n = u ˜ n ( t + μ , μ , θ − μ ω n , u μ , n ) .

By (3.11), for ∀ n ∈ R , r ≤ − m and take δ ∈ ( 0 , 1 ] , then there have c 1 > 0 such that

(4.11) ∣ ζ δ ( θ r ω n ) ∣ p q ≤ c 1 ( ∣ r ∣ p q + 1 ) ∣ ζ δ ( θ r ω n ) ∣ p 1 ≤ c 1 ( ∣ r ∣ p 1 + 1 ) .

Applying the dominated convergence theorem, by Lemma 3.1 and (4.11), which obtain

lim n → ∞ ∫ − ∞ − m e λ r ( 1 + ∣ ζ δ ( θ r ω n ) ∣ p q + ∣ ζ δ ( θ r ω n ) ∣ p 1 + ‖ g ( r + μ , ⋅ ) ‖ 2 ) d r = ∫ − ∞ − m e λ r ( 1 + ∣ ζ δ ( θ r ω ) ∣ p q + ∣ ζ δ ( θ r ω ) ∣ p 1 + ‖ g ( r + μ , ⋅ ) ‖ 2 ) d r ,

together with Lemma 3.1 (1), we obtain

(4.12) lim n → ∞ ∫ − ∞ 0 e λ r ( 1 + ∣ ζ δ ( θ r ω n ) ∣ p q + ∣ ζ δ ( θ r ω n ) ∣ p 1 + ‖ g ( r + μ , ⋅ ) ‖ 2 ) d r = ∫ − ∞ 0 e λ r ( 1 + ∣ ζ δ ( θ r ω ) ∣ p q + ∣ ζ δ ( θ r ω ) ∣ p 1 + ‖ g ( r + μ , ⋅ ) ‖ 2 ) d r .

By (4.9) and (4.12), we have

(4.13) limsup n → ∞ ‖ u μ , n ‖ 2 ≤ L ∫ − ∞ 0 e λ r ( 1 + ∣ ζ δ ( θ r ω ) ∣ p q + ∣ ζ δ ( θ r ω ) ∣ p 1 + ‖ g ( r + μ , ⋅ ) ‖ 2 ) d r ,

then we know that the bounded subsequence { u μ , n } n = 1 ∞ in H , thus there exists u μ ∈ H

(4.14) u μ , n ⇀ u μ in H

and

(4.15) ‖ u μ ‖ 2 ≤ ℒ ∫ − ∞ 0 e λ r ( 1 + ∣ ζ δ ( θ r ω ) ∣ p q + ∣ ζ δ ( θ r ω ) ∣ p 1 + ‖ g ( r + μ , ⋅ ) ‖ 2 ) d r .

By (4.14) and Lemma 3.5(1), we find that problems (3.1) and (3.2) with initial data u μ have a solution u ¯ = u ¯ ( ⋅ , μ , θ − μ ω , u μ ) for s ≥ μ , such that

(4.16) u ¯ n ( s , μ , θ − μ ω n , u μ , n ) ⇀ u ¯ ( s , μ , θ − μ ω , u μ ) in H ,

according to (4.10) for t ∈ R + , we have

(4.17) u ˜ n ⇀ u ¯ ( t + μ , μ , θ − μ ω , u μ ) in H .

From (4.15) and (4.16), we obtain (4.8) with u ˜ = u ¯ ( t + μ , μ , θ − μ ω , u μ ) , which completes the proof.□

Next, we will discuss the asymptotic compactness of solutions of (3.1) and (3.2).

Lemma 4.4

If (3.3)–(3.6) and (4.2) and (4.3) hold, then for every μ ∈ R and ω ∈ Ω , the sequence

{ u ¯ ( μ , μ − t n , θ − μ ω , u μ − t n ) } n = 1 ∞

has a convergent subsequence in H of solutions of (3.1) and (3.2), if t n → ∞ and u μ − t n ∈ D ( μ − t n , θ − t n ω ) with D ∈ D .

Proof

When s = 0 , we have that u ¯ ( μ , μ − t n , θ − μ ω , u μ − t n ) is bounded in H by Lemma 4.1. Then, there exist u ˜ ∈ H such that a subsequence

(4.18) u ¯ ( μ , μ − t n , θ − μ ω , u μ − t n ) ⇀ u ˜ .

Hence,

(4.19) liminf n → ∞ ‖ u ¯ ( μ , μ − t n , θ − μ ω , u μ − t n ) ≥ ‖ u ˜ ‖ .

In addition, we need to prove

(4.20) limsup n → ∞ ‖ u ¯ ( μ , μ − t n , θ − μ ω , u μ − t n ) ≤ ‖ u ˜ ‖ ,

which together with (4.19) proves the strong convergence of u ( μ , μ − t n , θ − μ ω , u μ − t n ) in H . Next, we show (4.20) by energy equation (3.30). By Lemma 4.1, there exists N 1 = N 1 ( μ , ω , D ) ≥ 1 , such that for all n ≥ N 1 , s ∈ [ − t n , 0 ] ,

(4.21) ‖ u ¯ ( μ + s , μ − t n , θ − μ ω , u μ − t n ) ‖ 2 ≤ e − λ s ℒ ( μ , ω ) .

Suppose i be a positive integer and N 2 = N 2 ( μ , ω , D , i ) ≥ N 1 and t n ≥ i for every n ≥ N 2 . By (4.21), we find for n ≥ N 2 ,

(4.22) ‖ u ¯ ( μ − i , μ − t n , θ − μ ω , u μ t n ) ‖ 2 ≤ e λ i ℒ ( μ , ω ) .

Therefore, the sequence { u ¯ ( μ − i , μ − t n , θ − μ ω , u μ − t n ) } n = 1 ∞ is bounded in H , we obtain u ˜ i ∈ H by a diagonal process for all i ∈ N such that

(4.23) u ¯ ( μ − i , μ − t n , θ − μ ω , u μ t n ) ⇀ u ˜ i in H , as n → ∞ .

As r ∈ [ − i , 0 ] ,

(4.24) u ¯ ( μ + r , μ − t n , θ − μ ω , u μ − t n ) = u ¯ ( μ + r , μ − i , θ − μ ω , u ( μ − i , μ − t − n , θ − μ ω , u μ t n ) ) .

A subsequence ( t n ′ , u μ − t n ′ ) (may depend on i ) and a solution u ¯ i ( ⋅ , μ − i , θ − μ ω , u ˜ i ) of problems (3.1) and (3.2) with initial data u i ˜ for i ∈ N , for every s ∈ [ − i , 0 ] , as n ′ → ∞ , we obtain

u ¯ ( μ + s , μ − t n ′ , θ − μ ω , u μ − t n ′ ) ⇀ u ¯ i ( μ + s , μ − i , θ − μ ω , u ˜ i ) in H

and

u ¯ ( ⋅ , μ − i , θ − μ ω , u ¯ ( μ − i , μ − t n ′ , θ − μ ω , u μ − t n ′ ) ) ⇀ u ¯ i ( ⋅ , μ − i , θ − μ ω , u ˜ i ) in L 2 ( ( μ − i , μ ) , V )

Nevertheless, applying a diagonal process, for each one i ∈ N , s ∈ [ − i , 0 ] , as n → ∞ , we can obtain a common subsequence

(4.25) u ¯ ( μ + s , − t n , θ − μ ω , u μ − t n ) ⇀ u ¯ i ( μ + s , μ − i , θ − μ ω , ) in H

and

(4.26) u ¯ ( ⋅ , μ − i , θ − μ ω , u ¯ ( μ − i , μ − t n , θ − μ ω , u μ − t n ) ) ⇀ u ¯ i ( ⋅ , μ − i , θ − μ ω , u ¯ i ) in L 2 ( ( μ − i , μ ) , V ) .

Combining (4.18) with (4.24) and (4.25) when s = 0 , we obtain

(4.27) u ˜ = u ¯ i ( μ , μ − i , θ − μ ω , u ˜ i ) .

Denote u n , μ − i = u ¯ ( μ − i , μ − t n , θ − μ ω , u μ − t n ) , u n ( r ) = u ¯ ( r , μ − i , θ − μ ω , u n , μ − i ) , and u i ( μ + s ) by u i ( μ + s , μ − i , θ − μ ω , u i ˜ ) . For u ¯ ( μ , μ − i , θ − μ ω , u n , μ − i ) with initial data u n , μ − i , by (4.24) and (3.31), we have

(4.28) ‖ u ( μ , μ − t n , θ − μ ω , u μ − t n ) ‖ 2 = e λ i ‖ u n , μ − i ‖ 2 − 2 ∫ μ − i μ e λ ( s − μ ) M ( ‖ ∇ u n ‖ 2 ) ‖ ∇ u n ( s ) ‖ 2 d s + 2 ∫ μ − i μ e λ ( s − μ ) ( f ( x , u n ( s ) , u n ( s ) ) d s ) + 2 ∫ μ − i μ e λ ( s − μ ) ( g ( s ) , u n ( s ) ) d s + 2 ∫ μ − i μ e λ ( s − μ ) ζ δ ( θ s − μ ω ) ( G ( x , u n ( s ) ) , u n ( s ) ) d s − λ ∫ μ − i μ e λ ( s − μ ) ‖ u n ( s ) ‖ 2 d s = e λ i ‖ u n , μ − i ‖ 2 − ∫ − i 0 e λ s ( 2 M ( ‖ ∇ u n ( s + μ ) ‖ 2 ) ‖ ∇ u n ( s + μ ) ‖ 2 + λ ‖ u n ( s + μ ) ‖ 2 ) d s + 2 ∫ − i 0 e λ s ( g ( s + μ ) , u n ( s + μ ) ) d s + 2 ∫ − i 0 e λ s ( f ( x , u n ( s + μ ) ) + ζ δ ( θ s ω ) G ( x , u n ( s + μ ) ) , u n ( s + μ ) ) d s ≜ ∑ j = 1 4 J i ( i , n ) .

Next, we will estimate each term in (4.28). First, by (4.6) for n ≥ N 2 , we obtain that

J 1 ( i , n ) ≤ e − λ t n ‖ d ( μ − t n , θ − t n , θ − t n ω ) ‖ 2 + c 1 ∫ − ∞ − i e λ r ( 1 + ∣ ζ δ ( θ r ω ) ∣ p q + ∣ ζ δ ( θ r ω ) ∣ p 1 + ‖ g ( r + μ , ⋅ ) ‖ 2 ) d r .

Due to D ∈ D , we obtain that

(4.29) limsup n → ∞ J 1 ( i , n ) ≤ c 1 ∫ − ∞ − i ( 1 + ∣ ζ δ ( θ r ω ) ∣ p q + ∣ ζ δ ( θ r ω ) ∣ p 1 + ‖ g ( r + τ , ⋅ ) ‖ 2 ) d r .

By (4.25) and (4.26), we have

(4.30) limsup n → ∞ J 2 ( i , n ) ≤ − 2 ∫ − i 0 e λ s ( 2 M ( ‖ ∇ u i ( s + μ ) ‖ 2 ) ‖ ∇ u i ( s + μ ) ‖ 2 + λ ‖ u i ( s + μ ) ‖ 2 ) d s .

By (4.26), we obtain that

(4.31) lim n → ∞ J 3 ( i , n ) = 2 ∫ − i 0 e λ s ( g ( s + μ ) , u i ( s + μ ) ) d s .

Furthermore, for k ∈ N , we have

(4.32) J 4 ( i , n ) = 2 ∫ − i 0 ∫ ∣ x ∣ ≥ k e λ s ( f ( x , u n ( s + μ ) ) u n ( s + μ ) + ζ δ ( θ s ω ) G ( x , u n ( s + μ ) ) u n ( s + μ ) ) d x d s + 2 ∫ − i 0 ∫ O k e λ s ( f ( x , u n ( s + μ ) ) u n ( s + μ ) + ζ δ ( θ s ω ) G ( x , u n ( s + μ ) ) u n ( s + μ ) ) d x d s ≜ J 41 ( i , n ) + J 42 ( i , n ) .

By (3.3)–(3.6), ψ 1 ∈ L 1 ( R n ) and ψ 3 ∈ L p 1 ( R n ) ∩ L p q ( R n ) , for η > 0 , there exist X 1 = X 1 ( η ) ≥ 1 , k ≥ X 1 , such that

(4.33) limsup n → ∞ J 41 ( i , n ) ≤ η .

By a diagonal process, we prove that, for k ∈ N , such that

(4.34) limsup n → ∞ J 42 ( i , n ) ≤ 2 ∫ − i 0 ∫ O k e λ s ( f ( x , u i ( s + μ ) ) u i ( s + μ ) + ζ δ ( θ s ω ) G ( x , u i ( s + μ ) ) u i ( s + μ ) ) d x d s ,

which together with (4.33) implies that for k ≥ X 1 ,

limsup n → ∞ J 4 ( i , n ) ≤ 2 ∫ − i 0 ∫ O k e λ s ( f ( x , u i ( s + μ ) ) u i ( s + μ ) + ζ δ ( θ s ω ) G ( x , u i ( s + μ ) ) u i ( s + μ ) ) d x d s + η .

As k → ∞ and η → 0 , we have

(4.35) limsup n → ∞ J 4 ( i , n ) ≤ 2 ∫ − i 0 ∫ R n e λ s ( f ( x , u i ( s + μ ) ) u i ( s + μ ) + ζ δ ( θ s ω ) G ( x , u i ( s + μ ) ) u i ( s + μ ) ) d x d s .

From (4.28)–(4.32) and (4.35), we obtain

(4.36) limsup n → ∞ ‖ u ( μ , μ − t n , θ − μ ω , u μ − t n ) ‖ 2 ≤ c 1 ∫ − ∞ − i e λ r ( 1 + ∣ ζ δ ( θ r ω ) ∣ p q + ∣ ζ δ ( θ r ω ) ∣ p 1 + ‖ g ( r + μ , ⋅ ) ‖ 2 ) d r − ∫ − ∞ 0 e λ s ( 2 M ( ‖ ∇ u i ( s + μ ) ‖ 2 ) ‖ ∇ u i ( s + μ ) ‖ 2 + λ ‖ u i ( s + μ ) ‖ 2 ) d s + 2 ∫ − i 0 e λ s ( g ( s + ) , u i ( s + μ ) ) d s + 2 ∫ − i 0 e λ s ( f ( x , u i ( s + μ ) ) + ζ δ ( θ s ω ) G ( x , u i ( s + μ ) ) , u i ( s + μ ) ) d s .

Applying energy equation (3.31) to u ¯ , by (3.30), we have

(4.37) ‖ u ¯ ‖ 2 = ‖ u i ( μ , μ − i , θ − μ ω , u i ¯ ) ‖ 2 = e − λ i ‖ u ¯ i ‖ 2 − 2 ∫ μ − i μ e λ ( s − μ ) M ( ‖ ∇ u i ( s ) ‖ 2 ) ‖ ∇ u i ( s ) ‖ 2 d s + 2 ∫ μ − i μ e λ ( s − μ ) ( f ( x , u i ( s ) ) , u i ( s ) ) d s + 2 ∫ μ − i μ e λ ( s − μ ) ( g ( s ) , u i ( s ) ) d s + 2 ∫ μ − i μ e λ ( s − μ ) ζ δ ( θ s − μ ω ) ( G ( x , u i ( s ) ) , u i ( s ) ) d s − λ ∫ μ − i μ e λ ( s − μ ) ‖ u i ( s ) ‖ 2 d s = e − λ i ‖ u ¯ i ‖ 2 − ∫ − i 0 e λ s ( 2 M ( ‖ ∇ u i ( s + μ ) ‖ 2 ) ‖ ∇ u i ( s + μ ) ‖ 2 + λ ‖ u i ( s + μ ) ‖ 2 ) d s + 2 ∫ − i 0 e λ s ( g ( s + μ ) , u i ( s + μ ) ) d s + 2 ∫ − i 0 e λ s ( f ( x , u i ( s + μ ) ) + ζ δ ( θ s ω ) G ( x , u i ( s + μ ) ) , u i ( s + μ ) ) d s .

By (4.36) and (4.37), we obtain

limsup n → ∞ ‖ u ( μ , μ − t n , θ − μ ω , u μ − t n ) ‖ 2 ≤ ‖ u ¯ ‖ 2 + c 1 ∫ − ∞ − i e λ r ( 1 + ∣ ζ δ ( θ r ω ) ∣ p q + ∣ ζ δ ( θ r ω ) ‖ p 1 + ‖ g ( r + μ , ⋅ ) ‖ 2 ) d r .

Letting i → ∞ , we obtain (4.20), which completes the proof.□

Conversely, we will prove Φ ¯ is the D -pullback compactness.

Lemma 4.5

If (3.3)–(3.6) and (4.2) and (4.3) hold, then Φ ¯ is D -pullback asymptotically compact in H.

Proof

Suppose μ ∈ R , ω ∈ Ω , D ∈ D , and u ¯ n ∈ Φ ( t n , μ − t n , θ − t n ω , D ( μ − t n , θ − t n ω ) ) with t n → ∞ , we will show a convergent subsequence exists of u ¯ n . There exist u μ − t n ∈ D ( μ − t n , θ − t n ω ) and u n ( ⋅ , μ − t n , θ − μ ω , u μ − t n ) with initial data u μ − t n by assumption such that

(4.38) u ¯ n = u n ( μ , μ − t n , θ − μ , u μ − t n ) .

The sequence { u n ( μ , μ − t n , θ − μ ω , u μ − t n ) } n = 1 ∞ has convergent subsequence { u ˜ } n = 1 ∞ in H , which completes the proof.□

Finally, by Lemmas 3.9, 4.2, 4.3, 4.5, and Theorem 2.3, we obtain the main result of the article as follows.

Theorem 4.6

If (3.3)–(3.6) and (4.2) and (4.3) hold, then problems (3.1) and (3.2) have a unique D -pullback random attractor A ¯ ∈ D in H .

Acknowledgements

The authors would like to thank the referees for their valuable comments and suggestions.

Funding information: The research was supported by the Natural Science Foundation of Qinghai Province (No. 2024-ZJ-931) and the Natural Science Foundation of China (No. 12161071).
Author contributions: The whole manuscript was written by Zhang Zhang. All authors reviewed and approved the manuscript.
Conflict of interest: The authors declare that there are no conflicts of interest regarding the publication of this article.

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Received: 2023-10-17

Revised: 2024-09-16

Accepted: 2024-10-01

Published Online: 2024-11-15

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/math-2024-0083

Keywords for this article

reaction–diffusion equation; the Kirchhoff-type; colored noise; unbounded domain

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