Random Attractors for Non-Autonomous Benjamin–Bona–Mahony Equations Driven by Laplace Multiplier Noise in High-Dimensional Unbounded Channel

Abstract

We investigate the long-time dynamics of a non-autonomous stochastic Benjamin–Bona–Mahony equation on a high-dimensional unbounded channel domain $O \subseteq R^{d}$ , driven by an operator-type Laplace-multiplier noise. We construct a non-autonomous random dynamical system on $H_{0}^{1} (O)$ and prove the existence of a compact pullback random attractor, which is cocycle-invariant and pullback attracts every tempered random set in $H_{0}^{1} (O)$ . The lack of compactness of Sobolev embeddings on unbounded domains is overcome by combining spectral decomposition on bounded truncations with uniform tail estimates outside large bounded sets.

Keywords

Benjamin-Bona-Mahony equation non-autonomous random dynamical system asymptotic compactness unbounded channels Laplace-multiplier noise

1. Introduction

This paper studies the long-time dynamics of the following non-autonomous stochastic Benjamin–Bona–Mahony (BBM) equation defined on an unbounded channel domain $O \subseteq R^{d}$ ( $d \in N$ ) with smooth boundary:

\begin{aligned} {\begin{aligned} ψ_{t} - Δ ψ_{t} - ν Δ ψ + \nabla \cdot \vec{F} (ψ) = g (t, x) + S ψ \circ d W, t > τ \in R, \\ ψ (t, x) = 0, x \in \partial O, t > τ, \\ ψ (τ, x) = ψ_{τ} (x), x \in O, \end{aligned} \end{aligned}

(1.1)

where

ν > 0

S = I - Δ

, and

g \in L_{loc}^{2} (R; L^{2} (O))

is a time-dependent forcing. The nonlinear flux

\vec{F} (s) = (F_{1} (s), \dots, F_{d} (s))

satisfies

\begin{aligned} F_{i} (0) = 0, | F_{i}^{'} (s) | ⩽ γ_{1} + γ_{2} | s |^{q - 1}, \end{aligned}

(1.2)

with

q \in [1, \frac{d}{d - 2})

d ⩾ 3

and

q \in [1, \infty)

d = 1, 2

, where

γ_{1}, γ_{2} > 0

. Let

f_{i} (s) := \int_{0}^{s} F_{i} (t) d t

for

i = 1, \dots, d

. Then

\begin{aligned} | F_{i} (s) | ⩽ γ_{1} | s | + γ_{2} | s |^{q}, | f_{i} (s) | ⩽ γ_{1} | s |^{2} + γ_{2} | s |^{q + 1}, i = 1, \dots, d . \end{aligned}

(1.3)

Attractors provide a basic framework for describing the asymptotic behavior of dissipative PDEs and the associated infinite-dimensional dynamical systems. For the research of attractors of deterministic problems, we refer to Babin and Vishik (1992), Hale (1988), Hao and Caraballo (2025), Robinson (2001), Wang, Li et al. (2025). For stochastic problems, random attractors have been developed to capture the long-time dynamics of stochastic PDEs and random dynamical systems. For background on existence, measurability, stability, and dimension theory of random attractors (in both autonomous and non-autonomous settings), we refer to Crauel and Flandoli (1994), Flandoli and Schmalfuß (1996), Caraballo et al. (2001, 2003), Wang (2012), Wang, Chen et al. (2025), Wang, Guo et al. (2024), Wang, Kinra et al. (2023). Related topics on long-time behavior include mean attractors and invariant measures (Kinra & Mohan, 2022, 2024; Wang, Caraballo et al., 2024), evolution systems of probability measures (Wang, Caraballo et al., 2023), and trajectory attractors as well as trajectory statistical solutions (Zhao & Caraballo, 2019; Zhao et al., 2020).

When $S = 0$ , (1.1) reduces to the deterministic BBM equation, introduced in Benjamin et al. (1972) as a nonlinear dispersive model to describe the physical phenomenon of long waves in shallow water. Well-posedness and long-time dynamics (including global attractors) for BBM-type equations have been studied in many works; see, for example, Astaburuaga et al. (1998), Avrin and Goldstein (1985), Celebi et al. (1999), Chueshov et al. (2004), Stanislavova et al. (2005). For stochastic BBM equations, Wang (2009) obtained a random attractor for the BBM equation with additive noise (in particular, the case $S ψ = h$ ). BBM equations driven by nonlinear colored noise on unbounded domains have also been studied via multivalued random dynamics and robustness analysis; see Chen et al. (2023, 2024).

In this paper, we consider the operator-type (Laplace-multiplier) noise $S = I - Δ$ , rather than the usual multiplicative noise $S = I$ (cf. Caraballo et al., 2001). This choice is naturally adapted to the BBM structure $(I - Δ) ψ_{t}$ . From a modeling viewpoint, the operator-type perturbation $S = I - Δ$ may be interpreted as acting on the same spatial operator as the dispersive component involving $Δ$ . From an analytical viewpoint, it allows an exponential change of variables (see (3.1)–(3.3)) that rewrites (1.1) as a pathwise random PDE without an explicit $d W$ -differential. This reformulation is convenient for establishing pathwise energy estimates and for constructing the associated non-autonomous random dynamical system (NRDS) $Φ$ in the sense of Wang (2012). Random attractors for three-dimensional stochastic BBM equations driven by Laplace-multiplier noise were obtained in Li and Wang (2018), and asymptotically autonomous behavior of the corresponding random attractors was further studied in Wang and Li (2020).

Compared with existing works on BBM-type equations on unbounded domains, the present paper treats a BBM equation on high-dimensional unbounded channel domains and allows for non-autonomous forcing $g (t, x)$ in (1.1). In particular, unlike the three-dimensional setting in Wang and Li (2020), we consider a general spatial dimension $d$ and work in an unbounded channel geometry. Our objective is to show that the NRDS generated by (1.1) admits a unique measurable compact pullback random attractor in $H_{0}^{1} (O)$ . A central step is to verify pullback asymptotic compactness in $H_{0}^{1} (O)$ , which is technically challenging on unbounded domains due to the non-compactness of Sobolev embeddings and the weak dissipativity inherent in the BBM structure.

For deterministic equations on unbounded domains, compactness has often been recovered by either the energy equation approach or the tail-estimates approach. The energy equation method was developed by Ball (1997, 2004) and has been applied in many contexts; see, for example, Chen et al. (2023, 2024), Wang, Li et al. (2025). The tail-estimates method was introduced in Wang (1999) for deterministic PDEs and has been used, for instance, in Wang, Chen et al. (2025), Wang, Guo et al. (2024), Wang, Li et al. (2025), Wang and Li (2020). In the present work, we employ a tail-estimates argument and combine it with a spectral decomposition on bounded truncations of $O$ to obtain the required compactness properties for the weakly dissipative stochastic BBM equation (1.1), and hence to establish the existence of a compact pullback random attractor on the unbounded channel domain.

The paper is organized as follows. In Section 2 we recall basic notions and known results on pullback random attractors for NRDSs. In Section 3 we prove existence and uniqueness of solutions to (1.1) and the cocycle property. In Section 4 we derive uniform a priori estimates. In the last section we establish pullback asymptotic compactness and then prove the existence and uniqueness of the pullback random attractor for (1.1).

2. Preliminaries

In this section, we recall some basic concepts related to random attractors for NRDSs. The reader is referred to Wang (2012) for more details. Let $(X, d)$ be a complete separable metric space with metric $d (\cdot, \cdot)$ . Let $B (X)$ denote the Borel $σ$ -algebras of $X$ . Assume $(Ω, F, P, {θ_{t}}_{t \in R})$ is a metric dynamical system where $(Ω, F, P)$ is a probability space and $θ : R \times Ω \to Ω$ is a measure-preserving group of translations on $Ω$ . Let $D$ be a collection of some families of nonempty subsets of $X$ parametrized by $τ \in R$ and $ω \in Ω$ .

Definition 2.1
Let $θ : R \times Ω \to Ω$ be a $(B (R) \times F, F)$ -measurable mapping. We say $(Ω, F, P, {θ_{t}}_{t \in R})$ is a parametric dynamical system if (i)
$θ (0, \cdot) = I$ ;
(ii)
$θ (s + t, \cdot) = θ (t, \cdot) \circ θ (s, \cdot)$ ;
(iii)
$P \circ θ (t, \cdot) = P$ (preserves measure), that is, $\forall B \in F$ , $θ (t, B) = {θ (t, b) : b \in B}$ ,
$\begin{aligned} P (θ (t, B)) = P (B) \Leftrightarrow P \circ (θ (t, \cdot) (B)) = P (B) . \end{aligned}$

Definition 2.2
A mapping $Φ : R^{+} \times R \times Ω \times X \to X$ is called a continuous cocycle on $X$ if for all $τ \in R$ , $ω \in Ω$ and t, $s \in R^{+}$ , (i)
$Φ (\cdot, τ, \cdot, \cdot) : R^{+} \times Ω \times X \to X$ is $(B (R^{+}) \times F \times B (X), B (X))$ -measurable;
(ii)
$Φ (0, τ, ω, \cdot)$ is the identity on $X$ ;
(iii)
$Φ (t + s, τ, ω, \cdot) = Φ (t, τ + s, θ_{s} ω, \cdot) \circ Φ (s, τ, ω)$ (cocycle property);
(iv)
$Φ (t, τ, ω, \cdot) : X \to X$ is continuous.
If there exists $T > 0$ such that for every $t \in R^{+}$ , $τ \in R$ and $ω \in Ω$ ,
$\begin{aligned} Φ (t, τ + T, ω, \cdot) = Φ (t, τ, ω, \cdot), \end{aligned}$
then we say $Φ$ is a periodic cocycle with period T, that is, for fixed $ω \in Ω$ , $Φ$ is periodic with initial time.
Definition 2.3
We call $K \in D$ is a $D$ -pullback absorbing set for $Φ$ if for every fixed $τ \in R$ , $ω \in Ω$ and $B \in D$ , there exists $T = T (τ, ω, B) > 0$ such that
$\begin{aligned} Φ (t, τ - t, θ_{- t} ω, B (τ - t, θ_{- t} ω)) \subseteq K (τ, ω), \forall t ⩾ T . \end{aligned}$

Definition 2.4
We say $Φ$ is $D$ -pullback asymptotically compact in $X$ if for every $τ \in R$ and $ω \in Ω$ , the sequence ${Φ (t_{n}, τ - t_{n}, θ_{- t_{n}} ω, x_{n})}_{n = 1}^{\infty}$ has a convergent subsequence in $X$ if $t_{n} \to \infty$ , $x_{n} \in B (τ - t_{n}, θ_{- t_{n}} ω)$ with $B \in D$ .
Definition 2.5
Let $A = {A (τ, ω) : τ \in R, ω \in Ω} \in D$ . Then $A$ is called a $D$ -pullback attractor for $Φ$ if (i)
$A$ is measurable and $A (τ, ω)$ is compact in $X$ ;
(ii)
$A$ is invariant: $\forall τ \in R$ and $ω \in Ω$ ,
$\begin{aligned} Φ (t, τ, ω, A (τ, ω)) = A (τ + t, θ_{t} ω); \end{aligned}$

(iii)
$\forall B \in D$ , $τ \in R$ and $ω \in Ω$ ,
$\begin{aligned} lim_{t \to \infty} d i s t_{X} (Φ (t, τ - t, θ_{- t} ω, B (τ - t, θ_{- t} ω)), A (τ, ω)) = 0. \end{aligned}$

If there exists $T > 0$ such that
$\begin{aligned} A (τ + T, ω) = A (τ, ω), \forall τ \in R, ω \in Ω, \end{aligned}$
then we say $A$ is T-periodic.
Proposition 2.6
Let $D$ be an inclusion-closed collection of some families of nonempty subsets of X, and $Φ$ is a continuous cocycle on X. If $Φ$ is $D$ -pullback asymptotically compact and has a closed measurable $D$ -pullback absorbing set $K \in D$ , then $Φ$ has a unique $D$ -pullback attractor $A \in D$ . If $Φ$ and K are T-periodic, then so is $A$ .

In the sequel, we adopt the following notations. Let $‖ \cdot ‖$ and $⟨ \cdot, \cdot ⟩$ denote the norm and inner product in $L^{2} (O)$ , respectively. For a given Banach space $X$ , the norm is denoted by $‖ \cdot ‖_{X}$ . Additionally, we use $‖ \cdot ‖_{p}$ to denote the norm in $L^{p} (O)$ . The constant $c > 0$ represents a generic positive constant, which may change from line to line or even within the same line. Given $ψ \in H_{0}^{1} (O)$ , we define the norm $‖ ψ ‖_{H_{0}^{1} (O)} = ‖ \nabla ψ ‖$ and the norm in $H^{1} (O)$ as $‖ ψ ‖_{H^{1} (O)}^{2} = ‖ ψ ‖^{2} + ‖ \nabla ψ ‖^{2}$ .

Throughout this paper, we will frequently use the Poincaré inequality:
$\begin{aligned} ‖ \nabla ψ ‖^{2} ⩾ λ_{0} ‖ ψ ‖^{2}, \forall ψ \in H_{0}^{1} (O), \end{aligned}$
(2.1)
where $λ_{0} > 0$ is a constant. We will also use the well-known Gagliardo–Nirenberg inequality (see Friedman, 1969):
$\begin{aligned} ‖ u ‖_{W^{j, q} (R^{N})} ⩽ C ‖ u ‖_{W^{m, r} (R^{N})}^{α} \cdot ‖ u ‖_{p}^{1 - α}, \forall u \in W^{m, r} (R^{N}) \cap L^{p} (R^{N}), \end{aligned}$
(2.2)
provided that
$\begin{aligned} \frac{1}{q} = \frac{j}{N} + α (\frac{1}{r} - \frac{m}{N}) + \frac{1 - α}{p}, \end{aligned}$
where $N, j, m \in N$ , $1 ⩽ p, q, r ⩽ \infty$ , $0 ⩽ j ⩽ m$ and $\frac{j}{m} ⩽ α ⩽ 1$ . Note that if $m - j - (N / r)$ is a nonnegative integer, then the inequality holds for $(j / m) ⩽ α < 1$ .
3. Existence and Uniqueness of the Solution

In this section, we study the existence and uniqueness of the solution to problem (1.1). To handle the Laplace-multiplier noise, we introduce an exponential change of variables:

\begin{aligned} v (t, τ, ω, v_{τ}) := e^{- z (θ_{t} ω)} ψ (t, τ, ω, ψ_{τ}), \end{aligned}

(3.1)

where

z (θ_{t} ω) = - \int_{- \infty}^{0} e^{s} (θ_{t} ω) (s) d s

is the pathwise-continuous solution of the stochastic equation

d z + z d t = d W (t)

and such that

\begin{aligned} lim_{t \to \pm \infty} \frac{z (θ_{t} ω)}{t} = lim_{t \to \pm \infty} \frac{1}{t} \int_{0}^{t} z (θ_{s} ω) d s = 0, for each ω \in Ω . \end{aligned}

(3.2)

In this case, the noise term is expressed as:

\begin{aligned} S ψ \circ d W & = e^{z (θ_{t} ω)} v (t, ω) \circ d z (θ_{t} ω) + z (θ_{t} ω) e^{z (θ_{t} ω)} v (t, ω) d t \\ - e^{z (θ_{t} ω)} Δ v (t, ω) \circ d z (θ_{t} ω) - z (θ_{t} ω) e^{z (θ_{t} ω)} Δ v (t, ω) d t . \end{aligned}

Substituting this expression into Equation (1.1), we obtain the following equation for

v

\begin{aligned} v_{t} - Δ v_{t} - ν Δ v = - e^{- z (θ_{t} ω)} \nabla \cdot \vec{F} (e^{z (θ_{t} ω)} v) + z (θ_{t} ω) (v - Δ v) + e^{- z (θ_{t} ω)} g (t, x), \end{aligned}

(3.3)

with initial conditions given by:

\begin{aligned} v (τ, τ, ω, v_{τ}) = v_{τ} = e^{- z (θ_{τ} ω)} ψ_{τ} . \end{aligned}

A solution to the problem (3.3) will be considered in the following sense.

Definition 3.1

Given $τ \in R$ , $ω \in Ω$ and $v_{τ} \in H_{0}^{1} (O)$ , a function $v (\cdot, τ, ω, v_{τ}) : [τ, \infty) \to H_{0}^{1} (O)$ is called a solution of problem (3.3) if $v (\cdot, τ, ω, v_{τ}) \in C ([τ, \infty), H_{0}^{1} (O))$ , $v_{t} \in L^{2} (τ, τ + T; H_{0}^{1} (O))$ for every $T > 0$ , $v (τ, τ, ω, v_{τ}) = v_{τ}$ , and for every $ξ \in C_{0}^{\infty} (τ, τ + T)$ and $ϕ \in H_{0}^{1} (O)$ ,

\begin{aligned} \int_{τ}^{τ + T} (⟨ v_{t} (t), ϕ ⟩ + ⟨ \nabla v_{t} (t), \nabla ϕ ⟩) ξ (t) d t + ν \int_{τ}^{τ + T} ⟨ \nabla v (t), \nabla ϕ ⟩ ξ (t) d t \\ = - \int_{τ}^{τ + T} \int_{O} e^{- z (θ_{t} ω)} \nabla \cdot \vec{F} (e^{z (θ_{t} ω)} v (t)) ϕ ξ (t) d x d t + \int_{τ}^{τ + T} \int_{O} z (θ_{t} ω) (v (t) - Δ v (t)) ϕ ξ (t) d x d t \\ + \int_{τ}^{τ + T} \int_{O} e^{- z (θ_{t} ω)} g (t, x) ϕ (t) ξ (t) d x d t . \end{aligned}

(3.4)

Remark 3.2

If $v$ is a solution of Equation (3.3) in the sense of Definition 3.1, then $v$ satisfies the energy equation:

\begin{aligned} \frac{d}{d t} ‖ v ‖_{H^{1} (O)}^{2} + 2 ν ‖ \nabla v ‖^{2} = - 2 e^{- z (θ_{t} ω)} \int_{O} v \nabla \cdot \vec{F} (e^{z (θ_{t} ω)} v) d x + 2 z (θ_{t} ω) ‖ v ‖_{H^{1}}^{2} + 2 e^{- z (θ_{t} ω)} \int_{O} g (t, x) v d x, \end{aligned}

(3.5)

for almost all

t > τ

. Indeed, by (3.4) we have, for almost all

t > τ

\begin{aligned} ⟨ v_{t} (t), ϕ ⟩ + ⟨ \nabla v_{t} (t), \nabla ϕ ⟩ + ν ⟨ \nabla v, \nabla ϕ ⟩ = - ⟨ e^{- z (θ_{t} ω)} \nabla \cdot \vec{F} (e^{z (θ_{t} ω)} v (t)), ϕ ⟩ + ⟨ z (θ_{t} ω) (v (t) - Δ v (t)), ϕ ⟩ + ⟨ e^{- z (θ_{t} ω)} g (t, x), ϕ ⟩, \end{aligned}

(3.6)

for all

ϕ \in H_{0}^{1} (O)

. Since

v \in C ([τ, \infty), H_{0}^{1} (O))

, by (3.6) with

ϕ = v

we get, for almost all

t > τ

\begin{aligned} ⟨ v_{t} (t), v (t) ⟩ + ⟨ \nabla v_{t} (t), \nabla v (t) ⟩ + ν ⟨ \nabla v, \nabla v ⟩ \\ = - ⟨ e^{- z (θ_{t} ω)} \nabla \cdot \vec{F} (e^{z (θ_{t} ω)} v (t)), v (t) ⟩ + ⟨ z (θ_{t} ω) (v (t) - Δ v (t)), v (t) ⟩ + ⟨ e^{- z (θ_{t} ω)} g (t, x), v (t) ⟩ . \end{aligned}

(3.7)

Since

v \in C ([τ, \infty), H_{0}^{1} (O))

v_{t} \in L^{2} (τ, τ + T; H_{0}^{1} (O))

for every

T > 0

, we see that both

v

and

v_{t}

belong to

L^{2} (τ, τ + T; H_{0}^{1} (O))

. Consequently, by Temam (2024, Lemma 1.2, p.260) we find that for almost all

T > τ

\begin{aligned} ⟨ v_{t} (t), v (t) ⟩_{H^{1} (O)} = \frac{1}{2} \frac{d}{d t} ‖ v (t) ‖_{H^{1} (O)}^{2} . \end{aligned}

(3.8)

Then (3.5) follows from (3.7)–(3.8) immediately.

We establish some energy inequalities, which will be useful frequently.

Lemma 3.3

The solution of Equation (3.3) satisfies: given $δ := m i n (ν / 2, ν λ_{0} / 4)$ ,

\begin{aligned} \frac{d}{d t} ‖ v ‖_{H^{1} (O)}^{2} + (δ - 2 z (θ_{t} ω) ‖ v ‖_{H^{1} (O)}^{2}) ⩽ \frac{2}{ν λ_{0}} e^{2 | z (θ_{t} ω) |} ‖ g (t) ‖^{2} . \end{aligned}

(3.9)

Proof.

Taking the inner product of Equation (3.3) with $v$ in $L^{2} (O)$ , we have

\begin{aligned} \frac{d}{d t} ‖ v ‖_{H^{1} (O)}^{2} + 2 ν ‖ \nabla v ‖^{2} = & - 2 e^{- z (θ_{t} ω)} \int_{O} v \nabla \cdot \vec{F} (e^{z (θ_{t} ω)} v) d x \\ + 2 z (θ_{t} ω) ‖ v ‖_{H^{1} (O)}^{2} + 2 e^{- z (θ_{t} ω)} \int_{O} g (t, x) v d x . \end{aligned}

By the boundary condition in (1.1) and

f_{i} (0) = 0

for

i = 1, 2, \cdot \cdot \cdot, d

\begin{aligned} \int_{O} ψ \nabla \cdot \vec{F} (ψ) d x = - \int_{O} \vec{F} (ψ) \cdot \nabla ψ = - \int_{O} \nabla \cdot \vec{f} (ψ) = - \int_{\partial O} \vec{f} (ψ) \cdot \vec{n} = 0, \end{aligned}

where

\vec{n}

is the outer unit normal vector. Then, by the Poincaré inequality:

‖ \nabla v ‖^{2} ⩾ λ_{0} ‖ v ‖^{2}

, the energy inequality (3.9) follows immediately.

Based on the above estimate, the similar argument as given in Stanislavova et al. (2005) shows the well-posedness of solutions to Equation (3.3).

Theorem 3.4

For each $(τ, ω, v_{τ}) \in R \times Ω \times H_{0}^{1} (O)$ , the problem (3.3) has a unique solution $v (\cdot, τ, ω, v_{τ}) \in C ([τ, \infty), H_{0}^{1} (O))$ with $v (τ, τ, ω, v_{τ}) = v_{τ}$ such that $v$ is continuous in $H_{0}^{1} (O)$ with respect to $v_{τ}$ .

By Theorem 3.4, we define a mapping $Φ : R^{+} \times R \times Ω \times X \to X$ by

\begin{aligned} Φ (t, τ, ω) v_{τ} = v (t + τ, τ, θ_{- τ} ω, v_{τ}), \forall (t, τ, ω) \in R^{+} \times R \times Ω, v_{τ} \in H_{0}^{1} (O), \end{aligned}

(3.10)

where

v

is the solution of Equation (3.3). One can verify that

Φ (t, τ)

is a continuous non-autonomous dynamical system on

H^{1} (O)

From now on, we denote by $D = {D = {D (τ, ω) \in H_{0}^{1} (O) : τ \in R, ω \in Ω}}$ , the collection of all tempered families of bounded nonempty subsets of $H_{0}^{1} (O)$ ; that is, for every $τ \in R$ , $ω \in Ω$ and $μ > 0$ ,

\begin{aligned} lim_{t \to + \infty} e^{- μ t} ‖ D (τ - t, θ_{- t} ω) ‖_{H_{0}^{1}} = 0, \forall (τ, ω) \in R \times Ω, \forall μ > 0, \end{aligned}

(3.11)

where

‖ D (τ - t, θ_{- t} ω) ‖_{H_{0}^{1} (O)} = {sup}_{v \in D (τ - t, θ_{- t} ω)} ‖ v ‖_{H_{0}^{1} (O)}

In this article, we assume that the function $g$ satisfies the following conditions:

\begin{aligned} \int_{- \infty}^{τ} e^{\frac{δ r}{2 (q + 1)}} ‖ g (r) ‖^{2} d r < \infty, \forall τ \in R, \end{aligned}

(3.12)

\begin{aligned} lim_{t \to \infty} e^{- μ t} \int_{- \infty}^{0} e^{\frac{δ r}{2 (q + 1)}} ‖ g (r - t) ‖^{2} d r = 0, \forall μ > 0. \end{aligned}

(3.13)

4. Uniform Estimates of the Solution

In this section, we derive uniform estimates for the solutions of the BBM equation, which are crucial for establishing the existence of pullback attractors.

We derive $D$ -uniform estimates of solutions to Equation (3.3) in $H^{1} (O)$ .

Lemma 4.1
For each $(τ, ω, D) \in R \times Ω \times D$ and $v_{τ - t} \in D (τ - t, θ_{- t} ω)$ , there exists a $T := T (τ, ω, D) > 0$ such that for all $t ⩾ T$ the solution $v$ of problem (3.3) satisfies
$\begin{aligned} ‖ v (τ, τ - t, θ_{- τ} ω, v_{τ - t}) ‖_{H^{1} (O)}^{2} ⩽ \frac{3}{ν λ_{0}} R (τ, ω), \end{aligned}$
(4.1)
where $R (τ, ω)$ is given by
$\begin{aligned} R (τ, ω) := \int_{- \infty}^{0} e^{δ r + 2 | z (θ_{r} ω) | + 2 \int_{r}^{0} z (θ_{σ} ω) d σ} ‖ g (r + τ) ‖^{2} d r . \end{aligned}$
(4.2)
Moreover, for all $s ⩾ τ - t$ , $t ⩾ 0$ and $v_{τ - t} \in H_{0}^{1} (O)$ ,
$\begin{aligned} ‖ v (s, τ - t, θ_{- τ} ω, v_{τ - t}) ‖_{H^{1}}^{2} & ⩽ e^{- δ (s - τ + t) + 2 \int_{- t}^{s - τ} z (θ_{σ} ω) d σ} ‖ v_{τ - t} ‖_{H^{1} (O)}^{2} \\ + \frac{2}{ν λ_{0}} \int_{- t}^{s - τ} e^{δ (r + τ - s) + 2 | z (θ_{r} ω) | + 2 \int_{r}^{s - τ} z (θ_{σ} ω) d σ} ‖ g (r + τ) ‖^{2} d r . \end{aligned}$
(4.3)
In addition, we find that there exists $T := T (s, τ, ω, D) > 0$ such that for all $t ⩾ T$ , by (4.3) we have
$\begin{aligned} ‖ v (s, τ - t, θ_{- τ} ω, v_{τ - t}) ‖_{H^{1} (O)}^{2} ⩽ \frac{3}{ν λ_{0}} \int_{- t}^{s - τ} e^{δ (r + τ - s) + 2 | z (θ_{r} ω) | + 2 \int_{r}^{s - τ} z (θ_{σ} ω) d σ} ‖ g (r + τ) ‖^{2} d r . \end{aligned}$
(4.4)
Proof.
We have rewritten the energy inequality Equation (3.9) for $v (r) = v (r, τ - t, θ_{- τ} ω, v_{τ - t})$ . The result is
$\begin{aligned} \frac{d}{d r} ‖ v ‖_{H^{1} (O)}^{2} + (δ - 2 z (θ_{r - τ} ω)) ‖ v ‖_{H^{1} (O)}^{2} ⩽ \frac{2}{ν λ_{0}} e^{2 | z (θ_{r - τ} ω) |} ‖ g (r) ‖^{2} . \end{aligned}$
(4.5)
Applying the Gronwall inequality to (4.5) with respect to $r \in (τ - t, s)$ , we obtain (4.3) immediately. Letting $s = τ$ in (4.3) yields
$\begin{aligned} ‖ v (τ, τ - t, θ_{- τ} ω, v_{τ - t}) ‖_{H^{1} (O)}^{2} ⩽ e^{- δ t + 2 \int_{- t}^{0} z (θ_{σ} ω) d σ} ‖ v_{τ - t} ‖_{H^{1} (O)}^{2} + \frac{2}{ν λ_{0}} R (τ, ω) . \end{aligned}$
(4.6)
Since $v_{τ - t} \in D (τ - t, θ_{- t} ω)$ with $D \in D$ , it follows from (3.2) and (3.11) that there exists a $T = T (τ, ω, D)$ such that for all $t ⩾ T$ ,
$\begin{aligned} e^{- δ t + 2 \int_{- t}^{0} z (θ_{σ} ω) d σ} ‖ v_{τ - t} ‖_{H^{1} (O)}^{2} & ⩽ e^{- \frac{δ t}{3}} ‖ D (τ - t, θ_{- t} ω) ‖_{H^{1} (O)}^{2} \\ ⩽ \frac{1}{ν λ_{0}} \int_{- t}^{s - τ} e^{δ (r + τ - s) + 2 | z (θ_{r} ω) | + 2 \int_{r}^{s - τ} z (θ_{σ} ω) d σ} ‖ g (r + τ) ‖^{2} d r . \end{aligned}$
This completes the proof.
Lemma 4.2
For every $τ \in R$ and $ω \in Ω$ , the NRDS $Φ$ has a closed $D$ -pullback absorbing set $K = {K (τ, ω) : τ \in R, ω \in Ω} \in D$ , defined by
$\begin{aligned} K (τ, ω) = {v \in H_{0}^{1} (O) : ‖ v ‖_{H^{1} (O)}^{2} ⩽ \frac{3}{ν λ_{0}} R (τ, ω)}, ω \in Ω, \end{aligned}$
(4.7)
where $R (τ, ω) = \int_{- \infty}^{0} e^{δ r + 2 | z (θ_{r} ω) | + 2 \int_{r}^{0} z (θ_{σ} ω) d σ} ‖ g (r + τ) ‖^{2} d r$ .
Proof.
By (3.2), we can infer that
$\begin{aligned} δ r + 2 | z (θ_{r} ω) | + 2 \int_{r}^{0} z (θ_{σ} ω) d σ & ⩽ δ r - \frac{(2 q + 1) δ r}{2 (q + 1)} \\ ⩽ \frac{δ r}{2 (q + 1)} . \end{aligned}$
(4.8)
Then by (3.12) and (4.8) we know that
$\begin{aligned} R (τ, ω) ⩽ c \int_{- \infty}^{0} e^{\frac{δ r}{2 (q + 1)}} ‖ g (r + τ) ‖^{2} d r < + \infty . \end{aligned}$
(4.9)
By (3.10) and (4.1), for every $τ \in R$ and $ω \in Ω$ , there exists $T = T (τ, ω, D) > 0$ such that for all $t ⩾ T$ ,
$\begin{aligned} ‖ Φ (t, τ - t, θ_{- t} ω) v_{τ - t} ‖_{H_{0}^{1} (O)}^{2} = ‖ v (τ, τ - t, θ_{- τ} ω, v_{τ - t}) ‖_{H_{0}^{1} (O)}^{2} ⩽ \frac{3}{ν λ_{0}} R (τ, ω) . \end{aligned}$
(4.10)
This shows that for $t ⩾ T$ ,
$\begin{aligned} Φ (t, τ - t, θ_{- t} ω) D (τ - t, θ_{- t} ω) \subseteq K (τ, ω) . \end{aligned}$
(4.11)
It remains to show $K \in D$ . Given $μ > 0$ . Then by (4.7) and (3.13) we have
$\begin{aligned} lim_{t \to \infty} e^{- μ t} ‖ K (τ - t, θ_{t} ω) ‖_{H^{1}}^{2} & = lim_{t \to \infty} e^{- μ t} \int_{- \infty}^{0} e^{δ r + 2 | z (θ_{r} ω) | + 2 \int_{r}^{0} z (θ_{σ} ω) d σ} ‖ g (r + τ - t) ‖^{2} d r \\ ⩽ lim_{t \to \infty} e^{- μ t} \int_{- \infty}^{0} e^{\frac{δ r}{2 (q + 1)}} ‖ g (r + τ - t) ‖^{2} d r \\ = e^{- μ τ} lim_{t \to \infty} e^{- μ t} \int_{- \infty}^{0} e^{\frac{δ r}{2 (q + 1)}} ‖ g (r - t) ‖^{2} d r \\ = 0. \end{aligned}$
This completes the proof.

4.1. Uniform Estimates Outside Bounded Domains

We derive the following estimates on the derivative of the solutions in $H_{0}^{1} (O)$ .

Lemma 4.3
For each $(t, τ, ω) \in R^{+} \times R \times Ω$ and $v_{τ} \in H_{0}^{1} (O)$ , the solutions of (3.3) satisfy, for almost all $t ⩾ τ$ ,
$\begin{aligned} {‖ \partial_{t} v (t, τ, ω, v_{τ}) ‖}_{H^{1} (O)}^{2} ⩽ c e^{c | z (θ_{t} ω) |} (1 + ‖ v ‖_{H^{1} (O)}^{2 q} + ‖ g (t) ‖^{2}), \end{aligned}$
(4.12)
where $c > 0$ is a constant independent of $t$ , $τ$ , $ω$ and $v_{τ}$ . Proof.
Since $v$ is a solution to (3.3) in the sense of Definition (3.1), by (3.6) we have
$\begin{aligned} ‖ v_{t} ‖_{H^{1} (O)}^{2} + ν ⟨ \nabla v, \nabla v_{t} ⟩ = - e^{- z (θ_{t} ω)} \int_{O} v_{t} \nabla \cdot \vec{F} (e^{z (θ_{t} ω)} v) d x + z (θ_{t} ω) \int_{O} (v - Δ v) v_{t} d x + e^{- z (θ_{t} ω)} \int_{O} g (t, x) v_{t} d x . \end{aligned}$
(4.13)
By Gagliardo–Nirenberg inequality we obtain
$\begin{aligned} ‖ v ‖_{\frac{2 d}{d (2 - q) + 2 q - 2}} ⩽ c ‖ \nabla v ‖^{\frac{(d - 2) (q - 1)}{2}} ‖ v ‖^{\frac{d (1 - q) + 2 q}{2}} . \end{aligned}$
(4.14)
By the Sobolev embedding $H^{1} (O) ↪ L^{2 q} (O)$ , (4.14) and (1.2), we have
$\begin{aligned} - e^{- z (θ_{t} ω)} \int_{O} v_{t} \nabla \cdot \vec{F} (e^{z (θ_{t} ω)} v) d x \\ ⩽ c e^{c | z (θ_{t} ω) |} \int_{O} | {\vec{F}}^{'} (e^{z (θ_{t} ω)} v) | | \nabla v | | v_{t} | d x \\ ⩽ c e^{c | z (θ_{t} ω) |} \int_{O_{2 k}} (1 + | v |^{q - 1}) | | \nabla v | | v_{t} | d x \\ ⩽ c e^{c | z (θ_{t} ω) |} ‖ \nabla v ‖ ‖ v_{t} ‖ + c e^{c | z (θ_{t} ω) |} ‖ v ‖_{\frac{2 d}{d - 2}}^{q - 1} ‖ \nabla v ‖ ‖ v_{t} ‖_{\frac{2 d}{d (2 - q) + 2 q - 2}} \\ ⩽ c e^{c | z (θ_{t} ω) |} ‖ \nabla v ‖ ‖ v_{t} ‖ + c e^{c | z (θ_{t} ω) |} ‖ v ‖_{H^{1}}^{q} ‖ \nabla v_{t} ‖^{\frac{(d - 2) (q - 1)}{2}} ‖ v_{t} ‖^{\frac{d (1 - q) + 2 q}{2}} \\ ⩽ c e^{c | z (θ_{t} ω) |} ‖ \nabla v ‖ ‖ \nabla v_{t} ‖ + c e^{c | z (θ_{t} ω) |} ‖ v ‖_{H^{1}}^{q} ‖ \nabla v_{t} ‖ \\ ⩽ \frac{1}{8} ‖ \nabla v_{t} ‖^{2} + c e^{c | z (θ_{t} ω) |} (1 + ‖ v ‖_{H^{1}}^{2 q}) . \end{aligned}$
(4.15)
Next, we have the following estimate for the second term on the right-hand side of (4.13)
$\begin{aligned} z (θ_{t} ω) \int_{O} (v - Δ v) v_{t} d x & ⩽ c z (θ_{t} ω) ‖ v ‖^{2} + \frac{1}{8} z (θ_{t} ω) ‖ v_{t} ‖^{2} + c z (θ_{t} ω) ‖ \nabla v ‖^{2} + \frac{1}{8} ‖ \nabla v_{t} ‖^{2} \\ ⩽ c z (θ_{t} ω) ‖ v ‖_{H^{1}}^{2} + \frac{1}{8} ‖ v_{t} ‖^{2} + \frac{1}{8} ‖ \nabla v_{t} ‖^{2} . \end{aligned}$
(4.16)
For the last term on the right-hand side of (4.13), we have
$\begin{aligned} e^{- z (θ_{t} ω)} \int_{O} g (t, x) v_{t} d x ⩽ c e^{c z (θ_{t} ω)} ‖ g (t, x) ‖^{2} + \frac{1}{8} ‖ v_{t} ‖^{2} . \end{aligned}$
(4.17)
Therefore, we get from (4.13)–(4.17) that
$\begin{aligned} ‖ v_{t} ‖_{H^{1}}^{2} ⩽ c e^{c | z (θ_{t} ω) |} (1 + ‖ v ‖_{H^{1} (O)}^{2 q} + ‖ g (t) ‖^{2}) . \end{aligned}$
(4.18)
This completes the proof of Lemma 4.3.

Next, we derive the uniform estimates on the tails of the solution, which are essential for proving the $D$ -pullback asymptotic compactness of $Φ$ in $H^{1} (O)$ .
Lemma 4.4
For every $ϵ > 0$ , $τ \in R$ , $ω \in Ω$ , $D = {D (τ, ω) : τ \in R, ω \in Ω} \in D$ and $v_{τ - t} \in D (τ - t, θ_{- t} ω)$ , there exist $T := T (τ, ω, D, ϵ) > 0$ and $\bar{K} := \bar{K} (τ, ω, ϵ) ⩾ 1$ such that for all $t ⩾ T$ and $k ⩾ \bar{K}$ ,
$\begin{aligned} \int_{| x_{d} | ⩾ k} (| v (τ, τ - t, θ_{- τ} ω, v_{τ - t}) |^{2} + | \nabla v (τ, τ - t, θ_{- τ} ω, v_{τ - t} |^{2}) d x ⩽ ϵ . \end{aligned}$
(4.19)
Proof.
Take a smooth function $ρ$ such that $0 ⩽ ρ (s) ⩽ 1$ for all $s \in R^{+}$ and
$\begin{aligned} ρ (s) = {\begin{cases} 0, & 0 ⩽ s ⩽ 1, \\ 1, & s ⩾ 2. \end{cases} \end{aligned}$
(4.20)
Then $| ρ^{'} (s) | + | ρ^{″} (s) | ⩽ c$ for all $s \in R^{+}$ , where $c$ is a positive constant independent of $s$ .

Let $ρ_{k} (x) = ρ (x_{d}^{2} / k^{2})$ for $x = (x_{1}, x_{2}, \cdot \cdot \cdot, x_{d}) \in O$ . Since $ρ$ is a smooth function and $v$ is a solution to (3.3) in the sense of Definition (3.1), we know that $ρ_{k}^{2} v \in C ([τ, \infty), H_{0}^{1} (O))$ . Then by (3.6) with $ϕ = ρ_{k}^{2} v (t)$ we get for almost all $t > τ$ ,
$\begin{aligned} ⟨ v_{t} (t), ρ_{k}^{2} v (t) ⟩ + ⟨ \nabla v_{t} (t), \nabla (ρ_{k}^{2} v (t)) ⟩ + ν ⟨ \nabla v, \nabla (ρ_{k}^{2} v (t)) ⟩ \\ = - ⟨ e^{- z (θ_{t} ω)} \nabla \cdot \vec{F} (e^{z (θ_{t} ω)} v (t)), ρ_{k}^{2} v (t) ⟩ + ⟨ z (θ_{t} ω) (v (t) - Δ v (t)), ρ_{k}^{2} v (t) ⟩ \\ + ⟨ e^{- z (θ_{t} ω)} g (t, x), ρ_{k}^{2} v (t) ⟩ . \end{aligned}$
(4.21)
Since $v \in C ([τ, \infty), H_{0}^{1} (O))$ and $v_{t} \in L^{2} (τ, τ + T; H_{0}^{1} (O))$ for every $T > 0$ ,we infer that $ρ_{k} v$ and $(ρ_{k} v)_{t}$ belong to $L^{2} (τ, τ + T; H_{0}^{1} (O))$ . By Temam (2024, Lemma 1.2, p.260) we find that for almost all $t > τ$ ,
$\begin{aligned} ⟨ {(ρ_{k} v)}_{t}, ρ_{k} v ⟩ + ⟨ \nabla {(ρ_{k} v)}_{t}, \nabla (ρ_{k} v) ⟩ = \frac{1}{2} \frac{d}{d t} ({‖ ρ_{k} v (t) ‖}^{2} + {‖ \nabla (ρ_{k} v (t)) ‖}^{2}) . \end{aligned}$
(4.22)
Note that
$\begin{aligned} ⟨ {(ρ_{k} v)}_{t}, ρ_{k} v ⟩ + ⟨ \nabla {(ρ_{k} v)}_{t}, \nabla (ρ_{k} v) ⟩ \\ = \int_{O} ρ_{k}^{2} (x) v_{t} (t) v (t) d x + \int_{O} \nabla v_{t} (t) \cdot \nabla (ρ_{k}^{2} (x) v (t)) d x \\ + \int_{O} {| \nabla ρ_{k} |}^{2} v v_{t} d x + \int_{O} ρ_{k} v_{t} \nabla ρ_{k} \cdot \nabla v (t) d x - \int_{O} ρ_{k} (x) v (t) \nabla ρ_{k} \cdot \nabla v_{t} d x . \end{aligned}$
(4.23)
For the right-hand side of (4.22) we have
$\begin{aligned} \frac{1}{2} \frac{d}{d t} ({‖ ρ_{k} v (t) ‖}^{2} + {‖ \nabla (ρ_{k} v (t)) ‖}^{2}) \\ = \frac{1}{2} \frac{d}{d t} \int_{O} ρ_{k}^{2} (x) (v^{2} + | \nabla v |^{2}) d x + \frac{1}{2} \frac{d}{d t} \int_{O} {| \nabla ρ_{k} |}^{2} v^{2} d x + \frac{d}{d t} \int_{O} ρ_{k} v \nabla ρ_{k} \cdot \nabla v d x . \end{aligned}$
(4.24)
Note that for each $i = 1, 2, 3, \cdot \cdot \cdot, d$ , $\frac{\partial ρ_{k}}{\partial x_{i}} v$ , $\frac{\partial ρ_{k}}{\partial x_{i}} v_{t}$ , $ρ_{k} \frac{\partial v}{\partial x_{i}}$ and $ρ_{k} \frac{\partial v_{t}}{\partial x_{i}}$ belong to $L^{2} (τ, τ +$ $T; H_{0}^{1} (O))$ . By Temam (2024, Lemma 1.2, p.260) again we find that for almost all $t > τ$ ,
$\begin{aligned} \frac{1}{2} \frac{d}{d t} \int_{O} {| \nabla ρ_{k} |}^{2} v^{2} d x = \int_{O} {| \nabla ρ_{k} |}^{2} v v_{t} d x, \end{aligned}$
(4.25)
and
$\begin{aligned} \frac{d}{d t} \int_{O} ρ_{k} v \nabla ρ_{k} \cdot \nabla v d x = \int_{O} ρ_{k} v_{t} \nabla ρ_{k} \cdot \nabla v d x + \int_{O} ρ_{k} v \nabla ρ_{k} \cdot \nabla v_{t} d x . \end{aligned}$
(4.26)
It follows from (4.24)–(4.26) that
$\begin{aligned} \frac{1}{2} \frac{d}{d t} ({‖ ρ_{k} v (t) ‖}^{2} + {‖ \nabla (ρ_{k} v (t)) ‖}^{2}) & = \frac{1}{2} \frac{d}{d t} \int_{O} ρ_{k}^{2} (x) (v^{2} + | \nabla v |^{2}) d x + \int_{O} {| \nabla ρ_{k} |}^{2} v v_{t} d x \\ + \int_{O} ρ_{k} v_{t} \nabla ρ_{k} \cdot \nabla v d x + \int_{O} ρ_{k} v \nabla ρ_{k} \cdot \nabla v_{t} d x . \end{aligned}$
(4.27)
By (4.22), (4.23) and (4.27) we obtain
$\begin{aligned} \int_{O} ρ_{k}^{2} (x) v_{t} (t) v (t) d x + \int_{O} \nabla v_{t} (t) \cdot \nabla (ρ_{k}^{2} (x) v (t)) d x \\ = \frac{1}{2} \frac{d}{d t} \int_{O} ρ_{k}^{2} (x) (v^{2} + | \nabla v |^{2}) d x + 2 \int_{O} ρ_{k} (x) v (t) \nabla ρ_{k} \cdot \nabla v_{t} d x . \end{aligned}$
(4.28)
We also have
$\begin{aligned} ν \int_{O} \nabla v (t) \cdot \nabla (ρ_{k}^{2} (x) v (t)) d x \\ = 2 ν \int_{O} ρ_{k} (x) v (t) \nabla v (t) \cdot \nabla ρ_{k} d x + ν \int_{O} ρ_{k}^{2} (x) | \nabla v (t) |^{2} d x . \end{aligned}$
(4.29)
By (4.21), (4.28) and (4.29), we get for almost all $t > τ$ ,
$\begin{aligned} \frac{d}{d s} \int_{O} ρ_{k}^{2} (| v |^{2} + | \nabla v |^{2}) d x + 2 ν \int_{O} ρ_{k}^{2} | \nabla v |^{2} d x \\ = - 4 \int_{O} ρ_{k} v \nabla ρ_{k} \cdot \nabla v_{t} d x + 2 z (θ_{t} ω) \int_{O} v (\nabla v \cdot \nabla ρ_{k}^{2}) d x \\ - 4 ν \int_{O} ρ_{k} v \nabla v \cdot \nabla ρ_{k} d x + 2 z (θ_{t} ω) \int_{O} ρ_{k}^{2} (| v |^{2} + | \nabla v |^{2}) d x \\ + 2 z (θ_{t} ω) \int_{O} v (\nabla v \cdot \nabla ρ_{k}^{2}) d x - 2 e^{- z (θ_{t} ω)} \int_{Q} ρ_{k}^{2} v \nabla \cdot \vec{F} (v) d x \\ + 2 e^{- z (θ_{t} ω)} \int_{O} ρ_{k}^{2} v g (t, x) d x . \end{aligned}$
(4.30)
Next, we estimate all the terms on the right-hand side of (4.30). For the first term, we get from Lemma (4.3) that
$\begin{aligned} - 4 \int_{O} ρ_{k} (x) v (t) \nabla ρ_{k} \cdot \nabla u_{t} d x & ⩽ 8 \int_{O} | ρ (\frac{x_{d}^{2}}{k^{2}}) | | ρ^{'} (\frac{x_{d}^{2}}{k^{2}}) | \frac{| x_{d} |}{k^{2}} | \nabla v_{t} | | v | d x \\ ⩽ \frac{c}{k} \int_{k ⩽ | x_{d} | ⩽ \sqrt{2} k} | \nabla v_{t} | | v | d x ⩽ \frac{c}{k} ({‖ \nabla v_{t} ‖}^{2} + ‖ v ‖^{2}) \\ ⩽ \frac{c}{k} (1 + ‖ g (t) ‖^{2} + ‖ v ‖_{H^{1}}^{2 q}) . \end{aligned}$
(4.31)
Similarly with (4.31), we have
$\begin{aligned} 2 z (θ_{t} ω) \int_{O} v (\nabla v \cdot \nabla ρ_{k}^{2}) d x - 4 ν \int_{O} ρ_{k} (x) v (t) \nabla v (t) \cdot \nabla ρ_{k} d x \\ ⩽ \frac{c}{k} (1 + | z (θ_{t} ω) |) (‖ \nabla v ‖^{2} + ‖ v ‖^{2}) . \end{aligned}$
(4.32)

$\begin{aligned} 2 z (θ_{t} ω) \int_{O} ρ_{k}^{2} (| v |^{2} + | \nabla v |^{2}) d x ⩽ \frac{c}{k} e^{| z (θ_{t} ω) |} (‖ \nabla v ‖^{2} + ‖ v ‖^{2}) . \end{aligned}$
(4.33)

$\begin{aligned} 2 z (θ_{t} ω) \int_{O} v (\nabla v \cdot \nabla ρ_{k}^{2}) d x ⩽ \frac{c}{k} e^{| z (θ_{t} ω) |} (‖ \nabla v ‖^{2} + ‖ v ‖^{2}) . \end{aligned}$
(4.34)
By (1.3) and $H^{1} (O) ↪ L^{q + 1} (O)$ , we get
$\begin{aligned} - 2 e^{- z (θ_{t} ω)} \int_{Q} ρ_{k}^{2} v \nabla \cdot \vec{F} (v) d x \\ = 2 e^{- 2 z (θ_{t} ω)} \int_{O} ρ_{k}^{2} (\nabla v \cdot \vec{F} (v)) d x + 2 e^{- 2 z (θ_{t} ω)} \int_{O} v (\nabla ρ_{k}^{2} \cdot \vec{F} (v)) d x \\ = 2 e^{- 2 z (θ_{t} ω)} \int_{O} ρ_{k}^{2} (\nabla \cdot \vec{f} (v)) d x + 2 e^{- 2 z (θ_{t} ω)} \int_{O} v (\nabla ρ_{k}^{2} \cdot \vec{F} (v)) d x \\ = - 2 e^{- 2 z (θ_{t} ω)} \int_{O} \nabla ρ_{k}^{2} \cdot \vec{f} (v) d x + 2 e^{- 2 z (θ_{t} ω)} \int_{O} v (\nabla ρ_{k}^{2} \cdot \vec{F} (v)) d x \\ ⩽ \frac{c}{k} e^{c | z (θ_{t} ω) |} \int_{O} (| v |^{2} + | v |^{q + 1}) d x ⩽ \frac{c}{k} e^{c | z (θ_{t} ω) |} (‖ v ‖^{2} + ‖ v ‖_{q + 1}^{q + 1}) \\ ⩽ \frac{c}{k} e^{c | z (θ_{t} ω) |} (‖ v ‖_{H^{1}}^{2} + ‖ v ‖_{H^{1}}^{q + 1}) ⩽ \frac{c}{k} e^{c | z (θ_{t} ω) |} (1 + ‖ v ‖_{H^{1}}^{q + 1}) . \end{aligned}$
(4.35)
For the last term, we obtain
$\begin{aligned} 2 e^{- z (θ_{t} ω)} \int_{O} ρ_{k}^{2} v g (t, x) d x ⩽ \frac{ν λ_{0}}{4} \int_{O} ρ_{k}^{2} | v |^{2} d x + c e^{c | z (θ_{t} ω) |} \int_{O} ρ_{k}^{2} | g (t, x) |^{2} d x . \end{aligned}$
(4.36)
Applying the Poincaré inequality on $ρ_{k} v$ , we get
$\begin{aligned} 2 ν \int_{O} ρ_{k}^{2} | \nabla v |^{2} d x ⩾ ν \int_{O} ρ_{k}^{2} | \nabla v |^{2} d x + \frac{ν λ_{0}}{2} \int_{O} ρ_{k}^{2} | v |^{2} d x - \frac{c}{k} ‖ v ‖^{2} . \end{aligned}$
(4.37)
Substituting all above estimates into (4.30) and recalling $δ := m i n (ν / 2, ν λ_{0} / 4)$ , we have
$\begin{aligned} \frac{d}{d t} \int_{O} ρ_{k}^{2} (v^{2} + | \nabla v |^{2}) d x + (δ - 2 z (θ_{t} ω)) \int_{O} ρ_{k}^{2} (v^{2} + | \nabla v |^{2}) d x \\ ⩽ \frac{c}{k} e^{c | z (θ_{t} ω) |} (1 + ‖ g (t) ‖^{2} + ‖ v ‖_{H^{1} (O)}^{q + 1}) + c e^{c | z (θ_{t} ω) |} \int_{O_{k}^{c}} | g (s, x) |^{2} d x . \end{aligned}$
(4.38)
Applying the Gronwall inequality to (4.38) over $(τ - t, τ)$ and replacing $ω$ by $θ_{- τ} ω$ , we get
$\begin{aligned} \int_{O} ρ_{k}^{2} ({| v (τ, τ - t, θ_{- τ} ω, v_{τ - t}) |}^{2} + | \nabla v (τ, τ - t, θ_{- τ} ω, v_{τ - t}) |^{2}) d x \\ ⩽ e^{- δ t + 2 \int_{- t}^{0} z (θ_{σ} ω) d σ} \int_{O} ρ_{k}^{2} ({| v_{τ - t} |}^{2} + {| \nabla v_{τ - t} |}^{2}) d x \\ + \int_{τ - t}^{τ} e^{δ (s - τ) + c | z (θ_{s - τ} ω) | + 2 \int_{s}^{τ} z (θ_{σ - τ} ω) d σ} \int_{O_{k}^{c}} | g (s, x) |^{2} d x d s \\ + \frac{c}{k} \int_{τ - t}^{τ} e^{δ (s - τ) + c | z (θ_{s - τ} ω) | + 2 \int_{s}^{τ} z (θ_{σ - τ} ω) d σ} (1 + ‖ g (s) ‖^{2}) d s \\ + \frac{c}{k} \int_{τ - t}^{τ} e^{δ (s - τ) + c | z (θ_{s - τ} ω) | + 2 \int_{s}^{τ} z (θ_{σ - τ} ω) d σ} {‖ v (s, τ - t, θ_{- τ} ω, v_{τ - t}) ‖}_{H^{1} (O)}^{q + 1} d s . \end{aligned}$
(4.39)
Next, we estimate all the terms on the right-hand side of (4.39). For the first term, we get from the fact $v_{τ - t} \in D (τ - t, θ_{- t} ω)$ with $D \in D$ that as $t \to \infty$ ,
$\begin{aligned} e^{- δ t + 2 \int_{- t}^{0} z (θ_{σ} ω) d σ} \int_{O} ρ_{k}^{2} ({| v_{τ - t} |}^{2} + {| \nabla v_{τ - t} |}^{2}) d x & ⩽ c e^{- \frac{δ}{3} t} {‖ v_{τ - t} ‖}_{H^{1} (O)}^{2} \\ ⩽ c e^{- \frac{δ}{3} t} {‖ D (τ - t, θ_{- t} ω) ‖}_{H^{1} (O)}^{2} \to 0. \end{aligned}$
(4.40)
By (3.2), we have the following estimate for the second term on the right-hand side of (4.39)
$\begin{aligned} \int_{τ - t}^{τ} e^{δ (s - τ) + c | z (θ_{s - τ} ω) | + 2 \int_{s}^{τ} z (θ_{σ - τ} ω) d σ} & ⩽ c \int_{- \infty}^{0} e^{δ s + c | z (θ_{s} ω) | + 2 \int_{s}^{0} z (θ_{σ} ω) d σ} \int_{O_{k}^{c}} | g (s + τ, x) |^{2} d x d s \\ ⩽ c \int_{- \infty}^{τ} e^{\frac{δ}{4} (r - τ)} \int_{O_{k}^{c}} | g (r, x) |^{2} d x d r \to 0, \end{aligned}$
(4.41)
as $k \to + \infty$ . For the third term on the right-hand side of (4.39), we get
$\begin{aligned} \frac{c}{k} \int_{τ - t}^{τ} e^{δ (s - τ) + c | z (θ_{s - τ} ω) | + 2 \int_{s}^{τ} z (θ_{σ - τ} ω) d σ} (1 + ‖ g (s) ‖^{2}) d s \\ ⩽ \frac{c}{k} \int_{- \infty}^{0} e^{δ (s) + c | z (θ_{s} ω) | + 2 \int_{s}^{τ} z (θ_{σ - τ} ω) d σ} (1 + ‖ g (s + τ) ‖^{2}) d s \to 0, \end{aligned}$
(4.42)
as $k \to \infty$ . For the last term on the right-hand side of (4.39), we get from (4.4) in Lemma 4.1 and (3.12) that there exists $T (s, τ, ω, D) > 0$ such for $t ⩾ T$ as $k \to \infty$ ,
$\begin{aligned} \frac{c}{k} \int_{τ - t}^{τ} e^{δ (s - τ) + c | z (θ_{s - τ} ω) | + 2 \int_{s}^{τ} z (θ_{σ - τ} ω) d σ} ‖ v (s, τ - t, θ_{- τ} ω, v_{τ - t}) ‖_{H^{1}}^{q + 1} d s \\ ⩽ \frac{c}{k} \int_{τ - t}^{τ} e^{δ (s - τ) + c | z (θ_{s - τ} ω) | + 2 \int_{s}^{τ} z (θ_{σ - τ} ω) d σ} {[\int_{- \infty}^{s - τ} e^{δ (r + τ - s) + 2 | z (θ_{r} ω) | + 2 \int_{r}^{s - τ} z (θ_{σ} ω) d σ} ‖ g (r + τ) ‖^{2} d r]}^{\frac{q + 1}{2}} d s \\ ⩽ \frac{c}{k} \int_{τ - t}^{τ} e^{δ (s - τ) + c | z (θ_{s - τ} ω) | + 2 \int_{s}^{τ} z (θ_{σ - τ} ω) d σ} {[\int_{- \infty}^{s - τ} e^{\frac{δ}{q + 1} (r + τ - s) + 2 | z (θ_{r} ω) | + 2 \int_{r}^{s - τ} z (θ_{σ} ω) d σ} ‖ g (r + τ) ‖^{2} d r]}^{\frac{q + 1}{2}} d s \\ ⩽ \frac{c}{k} \int_{τ - t}^{τ} e^{\frac{δ (s - τ)}{2} (s - τ) + c | z (θ_{s - τ} ω) | + 2 \int_{s}^{τ} z (θ_{σ - τ} ω) d σ} {[\int_{- \infty}^{s - τ} e^{\frac{δ}{q + 1} r + 2 | z (θ_{r} ω) | + 2 \int_{r}^{s - τ} z (θ_{σ} ω) d σ} ‖ g (r + τ) ‖^{2} d r]}^{\frac{q + 1}{2}} d s \\ ⩽ \frac{c}{k} \int_{τ - t}^{τ} e^{\frac{δ}{4} (s - τ)} d s {[\int_{- \infty}^{s - τ} e^{\frac{δ}{q + 1} r + 2 | z (θ_{r} ω) | + 2 \int_{r}^{s - τ} z (θ_{σ} ω) d σ} ‖ g (r + τ) ‖^{2} d r]}^{\frac{q + 1}{2}} \\ ⩽ \frac{c}{k} \int_{τ - t}^{τ} e^{\frac{δ}{4} (s - τ)} d s {[\int_{- \infty}^{0} e^{\frac{δ r}{2 (q + 1)}} ‖ g (r + τ) ‖^{2} d r]}^{\frac{q + 1}{2}} \to 0. \end{aligned}$
(4.43)
By (4.39)–(4.43) we know that for given $ϵ > 0$ , there exist $T = T (τ, ω, D, δ, ϵ) > 0$ and $\bar{K} = \bar{K} (τ, ω, δ, ϵ) ⩾ 1$ such that for all $t ⩾ T$ and $k ⩾ \bar{K}$ ,
$\begin{aligned} \int_{O} ρ^{2} (\frac{x_{d}^{2}}{k^{2}}) (| v (τ, τ - t, θ_{- τ} ω, v_{τ - t}) |^{2} + | \nabla v (τ, τ - t, θ_{- τ} ω, v_{τ - t}) |^{2}) d x ⩽ ϵ, \end{aligned}$
which combined with the property of the function $ρ$ implies that
$\begin{aligned} \begin{aligned} \int_{| x_{d} | ⩾ \sqrt{2} k} (| v (τ, τ - t, θ_{- τ} ω, v_{τ - t}) |^{2} + | \nabla v (τ, τ - t, θ_{- τ} ω, v_{τ - t}) |^{2}) d x \\ ⩽ \int_{O} ρ^{2} (\frac{x_{d}^{2}}{k^{2}}) (| v (τ, τ - t, θ_{- τ} ω, v_{τ - t}) |^{2} + | \nabla v (τ, τ - t, θ_{- τ} ω, v_{τ - t}) |^{2}) d x ⩽ ϵ . \end{aligned} \end{aligned}$
This completes the proof of Lemma 4.4.

4.2. Uniform Estimates on Spectral Projections of Solutions in Bounded Domains

In order to prove the asymptotic compactness of the solution operators, we also need to establish the uniform estimates of the solutions $v$ of (3.3) in bounded domains. Let $φ_{k} = 1 - ρ (x_{d}^{2} / k^{2})$ , where $ρ$ is the function given in (4.20). Fix $k ⩾ 1$ and let $\bar{v} (t, τ, ω) (x) = φ_{k} v (t, τ, ω) (x)$ for $x = (x_{1}, x_{2}, \cdot \cdot \cdot, x_{d}) \in O$ . Then $\bar{v} (t, τ, ω) \in H_{0}^{1} (O_{2 k})$ and

\begin{aligned} ‖ \bar{v} (t, τ, ω) ‖_{H_{0}^{1} (O_{2 k})} ⩽ c ‖ v (t, τ, ω) ‖_{H^{1} (O_{2 k})}, \forall t ⩾ τ, ω \in Ω . \end{aligned}

(4.44)

Note that

\begin{aligned} φ_{k} Δ v = & Δ \bar{v} - v Δ φ_{k} - 2 \nabla φ_{k} \cdot \nabla v and φ_{k} Δ v_{t} = Δ {\bar{v}}_{t} - v_{t} Δ φ_{k} - 2 \nabla φ_{k} \cdot \nabla v_{t} . \end{aligned}

(4.45)

Multiplying equation (3.3) by

φ_{k}

and applying the relations of (4.45), we get

\begin{aligned} {\bar{v}}_{t} - Δ {\bar{v}}_{t} - ν Δ \bar{v} = z (θ_{t} ω) (\bar{v} - Δ \bar{v}) - e^{- z (θ_{t} ω)} φ_{k} \nabla \cdot \vec{F} (e^{z (θ_{t} ω)} v) + e^{- z (θ_{t} ω)} φ_{k} g \\ - v_{t} Δ φ_{k} - 2 \nabla φ_{k} \cdot \nabla v_{t} + (z (θ_{t} ω) - ν) v Δ φ_{k} + 2 (z (θ_{t} ω) - ν) \nabla φ_{k} \cdot \nabla v . \end{aligned}

(4.46)

Consider the eigenvalue problem:

\begin{aligned} - Δ \bar{v} = & λ \bar{v} in O_{2 k}, \bar{v} = 0 in \partial O_{2 k} . \end{aligned}

(4.47)

Then, the eigenvalue problem (4.47) has a family of eigenfunctions

{e_{i}}_{i = 1}^{\infty}

with corresponding eigenvalues

{λ_{i}}_{i = 1}^{\infty}

such that

{e_{i}}_{i = 1}^{\infty}

is an orthonormal basis of

L^{2} (O_{2 k})

and

\begin{aligned} 0 < λ_{1} ⩽ λ_{2} ⩽ \dots ⩽ λ_{i} \to \infty as i \to \infty . \end{aligned}

Given

n \in N

, let

X_{n} = span {e_{1}, e_{2}, \dots, e_{n}}

and

P_{n} : L^{2} (O_{2 k}) \to X_{n}

be the projection operator.

We have the following uniform estimates in $H_{0}^{1} (O_{2 k})$ .

Lemma 4.5

For every $k \in N$ , $τ \in R$ , $ω \in Ω$ , $D = {D (τ, ω) : τ \in R, ω \in Ω} \in D$ , and $ϵ > 0$ , there exist $n_{0} > 0$ and $t_{0} > 0$ , such that $n > n_{0}$ and $t > t_{0}$ , we have

\begin{aligned} ‖ (I - P_{n}) \bar{v} (τ, τ - t, θ_{- τ} ω, {\bar{v}}_{τ - t}) ‖_{H^{1} (O_{2 k})} < ϵ, \end{aligned}

(4.48)

where

{\bar{v}}_{τ - t} = φ (x_{d}^{2} / k^{2}) v_{τ - t}

and

v_{τ - t} \in D (τ - t, θ_{- t} ω)

. Proof.

Denote by $\bar{v} = P_{n} \bar{v} + (I - P_{n}) \bar{v} := {\bar{v}}_{n, 1} + {\bar{v}}_{n, 2}$ . Applying $I - P_{n}$ to Equation (4.46) and then taking the inner product of the resulting equation with ${\bar{v}}_{n, 2}$ in $L^{2} (O_{2 k})$ , we obtain that

\begin{aligned} \frac{d}{d t} {‖ {\bar{v}}_{n, 2} ‖}_{H^{1}}^{2} + 2 ν {‖ \nabla {\bar{v}}_{n, 2} ‖}^{2} - 2 z (θ_{t} ω) {‖ {\bar{v}}_{n, 2} ‖}_{H^{1}}^{2} \\ = - 2 e^{- z (θ_{t} ω)} ⟨ φ_{k} \nabla \cdot \vec{F} (e^{z (θ_{t} ω)} v), {\bar{v}}_{n, 2} ⟩ + 2 e^{- z (θ_{t} ω)} ⟨ φ_{k} g, {\bar{v}}_{n, 2} ⟩ \\ + 2 ⟨ (z - ν) v Δ φ_{k} - v_{t} Δ φ_{k} - 2 \nabla φ_{k} \cdot \nabla v_{t} + 2 (z - ν) \nabla φ_{k} \cdot \nabla v, {\bar{v}}_{n, 2} ⟩ . \end{aligned}

(4.49)

Now, we estimate all the terms on the right-hand side of (4.49). For the first term, by (4.14) and

‖ \nabla {\bar{v}}_{n, 2} ‖^{2} ⩾ λ_{n + 1} ‖ {\bar{v}}_{n, 2} ‖^{2}

we get

\begin{aligned} - 2 e^{- z (θ_{t} ω)} ⟨ φ_{k} \nabla \cdot \vec{F} (e^{z (θ_{t} ω)} v), {\bar{v}}_{n, 2} ⟩ \\ ⩽ c e^{c | z (θ_{t} ω) |} \int_{O_{2 k}} φ_{k} | {\vec{F}}^{'} (e^{z (θ_{t} ω)} v) | | \nabla v | | {\bar{v}}_{n, 2} | d x \\ ⩽ c e^{c | z (θ_{t} ω) |} \int_{O_{2 k}} (1 + | v |^{q - 1}) | | \nabla v | | {\bar{v}}_{n, 2} | d x \\ ⩽ c e^{c | z (θ_{t} ω) |} ‖ \nabla v ‖ ‖ {\bar{v}}_{n, 2} ‖ + c e^{c | z (θ_{t} ω) |} ‖ v ‖_{\frac{2 d}{d - 2}}^{q - 1} ‖ \nabla v ‖ ‖ {\bar{v}}_{n, 2} ‖_{\frac{2 d}{d (2 - q) + 2 q - 2}} \\ ⩽ c e^{c | z (θ_{t} ω) |} ‖ \nabla v ‖ ‖ {\bar{v}}_{n, 2} ‖ + c e^{c | z (θ_{t} ω) |} ‖ v ‖_{H^{1}}^{q} ‖ \nabla {\bar{v}}_{n, 2} ‖^{\frac{(d - 2) (q - 1)}{2}} ‖ {\bar{v}}_{n, 2} ‖^{\frac{d (1 - q) + 2 q}{2}} \\ ⩽ c λ_{n + 1}^{- \frac{1}{2}} e^{c | z (θ_{t} ω) |} ‖ \nabla v ‖ ‖ \nabla {\bar{v}}_{n, 2} ‖ + c λ_{n + 1}^{- \frac{d (1 - q) + 2 q}{4}} e^{c | z (θ_{t} ω) |} ‖ v ‖_{H^{1}}^{q} ‖ \nabla {\bar{v}}_{n, 2} ‖ \\ ⩽ \frac{ν}{8} ‖ \nabla {\bar{v}}_{n, 2} ‖^{2} + c (λ_{n + 1}^{- 1} + λ_{n + 1}^{- \frac{d (1 - q) + 2 q}{2}}) e^{c | z (θ_{t} ω) |} (1 + ‖ v ‖_{H^{1}}^{2 q}) . \end{aligned}

(4.50)

For the second term, we get from the Young’s inequality that

\begin{aligned} 2 e^{- z (θ_{t} ω)} ⟨ φ_{k} g, {\bar{v}}_{n, 2} ⟩ ⩽ \frac{ν}{4} {‖ \nabla {\bar{v}}_{n, 2} ‖}^{2} + c λ_{n + 1}^{- 1} e^{c | z (θ_{t} ω) |} ‖ g (t) ‖^{2} . \end{aligned}

(4.51)

Similarly, by Lemma 4.3 and

‖ Δ φ_{k} ‖_{\infty} ⩽ c

, we find that the last term on the right-hand side of (4.49) satisfies

\begin{aligned} 2 ⟨ (z - ν) v Δ φ_{k} - v_{t} Δ φ_{k} - 2 \nabla φ_{k} \cdot \nabla v_{t} + 2 (z - ν) \nabla φ_{k} \cdot \nabla v, {\bar{v}}_{n, 2} ⟩ \\ ⩽ c λ_{n + 1}^{- \frac{1}{2}} (1 + | z (θ_{t} ω) |) ‖ \nabla {\bar{v}}_{n, 2} ‖ ({‖ v_{t} ‖}_{H^{1}} + ‖ v ‖_{H^{1}}) \\ ⩽ \frac{ν}{8} {‖ \nabla {\bar{v}}_{n, 2} ‖}^{2} + c λ_{n + 1}^{- 1} (1 + | z (θ_{t} ω) | + {| z (θ_{t} ω) |}^{2}) ({‖ v_{t} ‖}_{H^{1}}^{2} + ‖ v ‖_{H^{1}}^{2}) \\ ⩽ \frac{ν}{8} {‖ \nabla {\bar{v}}_{n, 2} ‖}^{2} + c λ_{n + 1}^{- 1} e^{c | z (θ_{t} ω) |} (1 + ‖ g (t) ‖^{2} + ‖ v ‖_{H^{1}}^{2 q}) . \end{aligned}

(4.52)

Therefore, it follows from (4.49)–(4.52) that

\begin{aligned} \frac{d}{d t} {‖ {\bar{v}}_{n, 2} ‖}_{H^{1}}^{2} + (δ - 2 z (θ_{t} ω)) {‖ {\bar{v}}_{n, 2} ‖}_{H^{1} (O_{2 k})}^{2} \\ ⩽ c (λ_{n + 1}^{- 1} + λ_{n + 1}^{- \frac{d (1 - q) + 2 q}{2}}) e^{c | z (θ_{t} ω) |} (1 + ‖ g (t) ‖^{2} + ‖ v ‖_{H^{1}}^{2 q}), \end{aligned}

(4.53)

where

δ := m i n (ν / 2, ν λ_{0} / 4)

. Hence, the Gronwall lemma over (4.53) and integrating over

(τ - t, τ)

with

t ⩾ 0

, we get that

\begin{aligned} {‖ {\bar{v}}_{n, 2} (τ, τ - t, θ_{- τ} ω, (I - P_{n}) (φ_{k} v_{τ - t})) ‖}_{H^{1}}^{2} \\ ⩽ e^{- δ t + 2 \int_{- t}^{0} z (θ_{σ} ω) d σ} {‖ (I - P_{n}) (φ_{k} v_{τ - t}) ‖}_{H^{1}}^{2} \\ + c (λ_{n + 1}^{- 1} + λ_{n + 1}^{- \frac{d (1 - q) + 2 q}{2}}) (\int_{τ - t}^{τ} e^{δ (s - τ) + c | z (θ_{s - τ} ω) | + 2 \int_{s}^{τ} z (θ_{σ - τ} ω) d σ} (1 + ‖ g (s) ‖^{2}) d s) \\ + c (λ_{n + 1}^{- 1} + λ_{n + 1}^{- \frac{d (1 - q) + 2 q}{2}}) \int_{τ - t}^{τ} e^{δ (s - τ) + c | z (θ_{s - τ} ω) | + 2 \int_{s}^{τ} z (θ_{σ - τ} ω) d σ} {‖ v (s, τ - t, θ_{- τ} ω, v_{τ - t}) ‖}^{2 q} d s \\ ⩽ e^{- δ t + 2 \int_{- t}^{0} z (θ_{σ} ω) d σ} {‖ (I - P_{n}) (φ_{k} v_{τ - t}) ‖}_{H^{1}}^{2} \\ + c (λ_{n + 1}^{- 1} + λ_{n + 1}^{- \frac{d (1 - q) + 2 q}{2}}) (\int_{- \infty}^{0} e^{\frac{δ s}{2 (q + 1)}} (1 + ‖ g (τ + τ) ‖^{2}) d s) \\ + c (λ_{n + 1}^{- 1} + λ_{n + 1}^{- \frac{d (1 - q) + 2 q}{2}}) \int_{- \infty}^{0} e^{\frac{δ s}{2 (q + 1)}} {‖ v (s + τ, τ - t, θ_{- τ} ω, v_{τ - t}) ‖}^{2 q} d s, \end{aligned}

(4.54)

where the last term is obtained by the argument of (4.43).

By $v_{τ - t} \in D (τ - t, θ_{- t} ω)$ with $D \in D$ we find that as $t \to \infty$ ,

\begin{aligned} e^{- δ t + 2 \int_{- t}^{0} z (θ_{σ} ω) d σ} {‖ (I - P_{n}) (φ_{k} v_{τ - t}) ‖}_{H^{1}}^{2} ⩽ c e^{- \frac{1}{3} δ t} {‖ D (τ - t, θ_{- t} ω) ‖}_{H^{1}}^{2} \to 0. \end{aligned}

(4.55)

Note that

(λ_{n + 1}^{- 1} + λ_{n + 1}^{- \frac{d (1 - q) + 2 q}{2}}) \to 0

n \to + \infty

. Then exist

n_{0} > 0

and

t_{0} > 0

, such that

n > n_{0}

and

t > t_{0}

, we have

\begin{aligned} ‖ (I - P_{n}) \bar{v} (τ, τ - t, θ_{- τ} ω, {\bar{v}}_{τ - t}) ‖_{H^{1} (O_{2 k})} < ϵ . \end{aligned}

(4.56)

5. Existence and Uniqueness of Pullpack Random Attractors

Based on the results established in the previous subsections, we now prove the $D$ -pullback asymptotic compactness of solutions to (3.3) in $H_{0}^{1} (O)$ .

Lemma 5.1

For $τ \in R$ , $ω \in Ω$ , $D = {D (τ, ω) : τ \in R, ω \in Ω} \in D,$ $t_{n} \to \infty$ and $v_{τ - t_{n}} \in D (τ - t_{n}, θ_{- t_{n}} ω)$ the sequence $v (τ, τ - t_{n}, θ_{- τ} ω, v_{τ - t_{n}})$ of solutions of Equation (3.3) has a convergent subsequence in $H_{0}^{1} (O)$ .

Proof.

Let $ϵ > 0$ be an arbitrary number. By Lemma 4.4, there are $N_{1} := N_{1} (τ, ω, D, ϵ) ⩾ 1$ and $k_{0} := k_{0} (τ, ω, ϵ) ⩾ 1$ such that for all $n ⩾ N_{1}$ ,

\begin{aligned} \int_{O ∖ O_{k_{0}}} ({| v (τ, τ - t_{n}, θ_{- τ} ω, v_{τ - t_{n}}) |}^{2} + {| \nabla v (τ, τ - t_{n}, θ_{- τ} ω, v_{τ - t_{n}}) |}^{2}) d x ⩽ ϵ . \end{aligned}

(5.1)

By Lemma 4.5 with

k = 2 k_{0}

, there exists

N_{2} := N_{2} (τ, ω, D, ϵ) ⩾ N_{1}

such that for all

n ⩾ N_{2}

\begin{aligned} {‖ (I - P_{N}) \bar{v} (τ, τ - t_{n}, θ_{- τ} ω, {\bar{v}}_{τ - t_{n}}) ‖}_{H_{0}^{1} (O_{2 k_{0}})} ⩽ ϵ, \end{aligned}

(5.2)

where

\bar{v} (τ, τ - t_{n}, θ_{- τ} ω, {\bar{v}}_{τ - t_{n}}) = φ_{k} (x_{d}^{2} / k^{2}) v (τ, τ - t_{n}, θ_{- τ} ω, v_{τ - t_{n}})

with

{\bar{v}}_{τ - t_{n}} = φ (x_{d}^{2} / k^{2}) v_{τ - t_{n}}

. In addition, by Lemma 4.1 we know that there exists

N_{3} := N_{3} (τ, ω, D, ϵ) ⩾ N_{2}

such that for all

n ⩾ N_{3}

\begin{aligned} {‖ v (τ, τ - t_{n}, θ_{- τ} ω, v_{τ - t_{n}}) ‖}_{H_{0}^{1} (O)} ⩽ c \sqrt{R (τ, ω)}, \end{aligned}

(5.3)

where

R (τ, ω) = \int_{- \infty}^{0} e^{δ r + 2 | z (θ_{r} ω) | + 2 \int_{r}^{0} z (θ_{σ} ω) d σ} ‖ g (r + τ) ‖^{2} d r < \infty

. By (4.44) and (5.3) we find that for all

n ⩾ N_{3}

\begin{aligned} {‖ \bar{v} (τ, τ - t_{n}, θ_{- τ} ω, {\bar{v}}_{τ - t_{n}}) ‖}_{H_{0}^{1} (O_{2 k_{0}})} ⩽ c \sqrt{R (τ, ω)} . \end{aligned}

(5.4)

Note that (5.4) shows that the sequence

{P_{N} \bar{v} (τ, τ - t_{n}, θ_{- τ} ω, {\bar{v}}_{τ - t_{n}})}_{n = 1}^{\infty}

is bounded in the finite dimensional space

P_{N} H_{0}^{1} (O_{2 k_{0}})

, and thus it is precompact in

P_{N} H_{0}^{1} (O_{2 k_{0}})

. This fact combined with (5.2) means that

{\bar{v} (τ, τ - t_{n}, θ_{- τ} ω, {\bar{v}}_{τ - t_{n}})}_{n = 1}^{\infty}

is precompact in

H_{0}^{1} (O_{2 k_{0}})

. By the definition of

φ_{k}

we know that

\bar{v} (τ, τ - t_{n}, θ_{- τ} ω, {\bar{v}}_{τ - t_{n}}) = v (τ, τ - t_{n}, θ_{- τ} ω, v_{τ - t_{n}})

O_{k_{0}}

, and hence

{v (τ, τ - t_{n}, θ_{- τ} ω, v_{τ - t_{n}})}_{n = 1}^{\infty}

is precompact in

H^{1} (O_{k_{0}})

, which together with (5.1) shows that

{v (τ, τ - t_{n}, θ_{- τ} ω, v_{τ - t_{n}})}_{n = 1}^{\infty}

is precompact in

H_{0}^{1} (O)

, so it’s easy to know that

{ψ (τ, τ - t_{n}, θ_{- τ} ω, ψ_{τ - t_{n}})}_{n = 1}^{\infty}

is also precompact.

Theorem 5.2

The NRDS $Φ$ associated with the Equation (3.3) has a unique $D$ -pullback random attractor $A = {A (τ, ω) : τ \in R, ω \in Ω} \in D$ in $H_{0}^{1} (O)$ . So we know for the NRDS $Φ$ associated with the problem (1.1) has a unique $D$ -pullback random attractor $A = {A (τ, ω) : τ \in R, ω \in Ω} \in D$ in $H_{0}^{1} (O)$ . If, in addition, there exists $T > 0$ such that $g (t + T) = g (t)$ in $L^{2} (O)$ for $t \in R$ , then the attractor $A$ is also $T$ -periodic, that is, $A (τ + T, ω) = A (τ, ω)$ for all $τ \in R$ and $ω \in Ω$ .

Proof.

Note that $Φ$ is $D$ -pullback asymptotically compact in $H_{0}^{1} (O)$ by Lemma 5.1, $Φ$ has a closed $D$ -pullback absorbing set $K \in D$ by Lemma 4.2. As a result, the existence, uniqueness, and periodicity of the $D$ -pullback random attractor $A$ of $Φ$ in $H_{0}^{1} (O)$ follow from Proposition 2.6 immediately.

Footnotes

ORCID iDs

Jibing Leng

Author Contributions

Liguang Zhou: Writing-review & editing; writing-original draft.

Dexin Li: Writing-review & editing.

Guofu Li: Writing-review & editing.

Jibing Leng: Writing-review & editing.

All authors reviewed the final version of the manuscript.

Funding

The authors received the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (12301299, 12426606, 12161019), the Guiyang City Science and Technology Plan Project ([2024]2-17), the Natural Science Research Project of Guizhou Provincial Department of Education (QJJ[2024]322,QJJ[2023]011), the Young Elite Scientist Sponsorship Program (GASTYESS202403), the Guizhou Provincial Basic Research Program (Natural Science) (QKHJC-MS[2025]279, QKHJC-ZK [2022]YB318).

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data Availability

No data was used for the research described in the article.

References

Astaburuaga

M. A.

Bisognin

Fernandez

(1998). Global attractors and finite dimensionality for a class of dissipative equations of BBM’s type. Electron Journal of Differential Equations, 25, 1–14.

Avrin

Goldstein

J. A.

(1985). Global existence for the Benjamin-Bona-Mahony equation in arbitrary dimensions. Nonlinear Analysis, 9, 861–865.

Babin

A. V.

Vishik

M. I.

(1992). Attractors of evolution equations. North-Holland, Amsterdam.

Ball

J. M.

(1997). Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations. Journal of Nonlinear Science, 7, 475–502.

Ball

J. M.

(2004). Global attractors for damped semilinear wave equations. Discrete and Continuous Dynamical Systems, 10, 31–52.

Benjamin

T. B.

Bona

J. L.

Mahony

J. J.

(1972). Model equations for long waves in nonlinear dispersive systems. Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences, 272, 47–78.

Caraballo

Langa

J. A.

Melnik

V. S.

Valero

(2003). Pullback attractors for nonautonomous and stochastic multivalued dynamical systems. Set-Valued and Variational Analysis, 11, 153–201.

Caraballo

Langa

J. A.

Valero

(2001). Global attractors for multivalued random dynamical systems generated by random differential inclusions with multiplicative noise. Journal of Mathematical Analysis and Applications, 260, 602–622.

Celebi

A. O.

Kalantarov

V. K.

Polat

(1999). Attractors for the generalized Benjamin-Bona-Mahony equation. Journal of Differential Equations, 157, 439–451.

10.

Chen

Wang

Zhang

(2023). Multivalued random dynamics of Benjamin-Bona-Mahony equations driven by nonlinear colored noise on unbounded domains. Mathematische Annalen, 386, 343–373.

11.

Chen

Wang

Zhang

(2024). Asymptotically autonomous robustness of random attractors for 3D BBM equations driven by nonlinear colored noise. SIAM Journal on Mathematical Analysis, 56, 254–274.

12.

Chueshov

Polat

Siegmund

(2004). Gevrey regularity of global attractor for generalized Benjamin-Bona-Mahony equation. Mathematical Physics Analysis And Geometry, 11, 226–242.

13.

Crauel

Flandoli

(1994). Attractors for random dynamical systems. Probability Theory and Related Fields, 100, 365–393.

14.

Flandoli

Schmalfuß

(1996). Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise. Stochastics and Stochastics Reports, 59, 21–45.

15.

Friedman

(1969). Partial differential equations. Holt, Reinhart and Winston.

16.

Hale

J. K.

(1988). Asymptotic behavior of dissipative systems. American Mathematical Society, Providence, Rhode Island.

17.

Hao

Caraballo

(2025). Solvability and convergence results for the temperature and concentration field in incompressible Navier-Stokes equations with boundary conditions. Journal of Differential Equations, 450, 113735.

18.

Kinra

Mohan

M. T.

(2022). Weak pullback mean random attractors for the stochastic convective Brinkman-Forchheimer equations and locally monotone stochastic partial differential equations. Infinite Dimensional Analysis. Quantum Probability and Related Topics, 25, 2250005.

19.

Kinra

Mohan

M. T.

(2024). Random attractors and invariant measures for stochastic convective Brinkman-Forchheimer equations on 2D and 3D unbounded domains. Discrete and Continuous Dynamical Systems-B, 29, 377–425.

20.

Wang

(2018). Random attractors for 3D Benjamin-Bona-Mahony equations derived by a Laplace-multiplier noise. Stochastics and Dynamics, 18, 1850–004.

21.

Robinson

J. C.

(2001). Infinite-dimensional dynamical systems. Cambridge University Press.

22.

Stanislavova

Stefanov

Wang

(2005). Asymptotic smoothing and attractors for the generalized Benjamin-Bona-Mahony equation on

R^{3}

. Journal of Differential Equations, 219, 451–483.

23.

Temam

(2024). Navier-Stokes Equations: Theory and numerical analysis. American Mathematical Society.

24.

Wang

(1999). Attractors for reaction-diffusion equations in unbounded domains. Physica D: Nonlinear Phenomena, 128, 41–52.

25.

Wang

(2009). Random attractors for for the stochastic Benjamin-Bona-Mahony equation on unbounded domains. Journal of Differential Equations, 246, 2506–2537.

26.

Wang

(2012). Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems. Journal of Differential Equations, 253, 1544–1583.

27.

Wang

Caraballo

Nguyen

H. T.

(2024). Mean attractors and invariant measures of locally monotone and generally coercive SPDEs driven by superlinear noise. Journal of Differential Equations, 381, 209–259.

28.

Wang

Caraballo

Tuan

N. H.

(2023). Asymptotic stability of evolution systems of probability measures for nonautonomous stochastic systems: Theoretical results and applications. Proceedings of the American Mathematical Society, 151, 2449–2458.

29.

Wang

Chen

Wang

(2025). Asymptotic behavior of weakly dissipative fractional wave equations driven by superlinear colored noise on

R^{N}

. The Journal of Geometric Analysis, 35, 1–51.

30.

Wang

Guo

Liu

Nguyen

D. T.

(2024). Fractal dimension of random invariant sets and regular random attractors for stochastic hydrodynamical equations. Mathematische Annalen, 389, 671–718.

31.

Wang

Kinra

Mohan

M. T.

(2023). Asymptotically autonomous robustness in probability of random attractors for stochastic Navier-Stokes equations on unbounded poincaré domains. SIAM Journal on Mathematical Analysis, 55, 2644–2676.

32.

Wang

(2020). Asymptotic autonomy of random attractors for BBM equations with Laplace-multiplier noise. Journal of Applied Analysis and Computation, 10, 1199–1222.

33.

Wang

Guo

(2025). Pullback asymptotic compactness and asymptotically autonomous robustness of non-autonomous Klein-Gordon-Schrödinger system on unbounded domains. International Mathematics Research Notices, 12, rnaf171.

34.

Zhao

Caraballo

(2019). Asymptotic regularity of trajectory attractor and trajectory statistical solution for the 3D globally modified Navier-Stokes equations. Journal of Differential Equations, 266, 7205–7229.

35.

Zhao

Caraballo

(2020). Trajectory statistical solutions and Liouville type equations for evolution equations: Abstract results and applications. Journal of Differential Equations, 269, 467–494.