Multivalued Random Dynamics of Fractional Reaction–Diffusion–AdvectionEquations Driven by Superlinear Colored Noise on Unbounded Domains

Abstract

This article investigates the multivalued random dynamics generated by solution operators of fractional random reaction–diffusion–advection equations driven by superlinear colored noise on unbounded domains. The equations under consideration exhibit non-Lipschitz continuous nonlinear drift and diffusion terms, which lead to the non-uniqueness of solutions and hence generate multivalued random dynamical systems. Our goals are to demonstrate: (i) the existence of solutions for fractional reaction–diffusion–advection equations driven by superlinear colored noise on unbounded domains; and (ii) the existence and uniqueness of pullback random attractors for these systems. The measurability of the random attractors is demonstrated using a method that relies on the weak upper semicontinuity of the solutions. The asymptotic compactness of solution operators is obtained by employing Ball’s approach to energy equations in order to address the non-compactness of Sobolev embeddings on the entire space.

Keywords

Reaction–diffusion–advection equations multivalued random dynamical system fractional Laplacian superlinear colored noise pullback random attractor unbounded domain

1. Introduction

In this article, we investigate the random attractors for multivalued random dynamics of fractional random reaction–diffusion–advection equations driven by superlinear colored noise on the entire space $R^{d}$ :

\begin{aligned} {\begin{cases} ϕ_{t} + (- Δ)^{β} ϕ + λ ϕ + \vec{α} \cdot \nabla ϕ^{γ} = f (t, y, ϕ) + g (t, y) + ℏ (t, y, ϕ) Z_{δ} (θ_{t} ω), t > ι, \\ ϕ (ι, y) = ϕ_{ι} (y), \end{cases} \end{aligned}

(1.1)

where

ι \in R

y \in R^{d}

λ > 0

, and

γ > 1

are constants,

(- Δ)^{β}

is the fractional Laplacian operator with

β \in (0, 1)

\vec{α} = (α_{1}, α_{2}, \dots, α_{d})

is a given advective direction,

g

is a non-autonomous deterministic external term in

L_{loc}^{2} (R, L^{2} (R^{d}))

f

and

ℏ

are nonlinear functions satisfying certain conditions, and the term

Z_{δ} (θ_{t} ω)

is the colored noise with a correlation time

δ > 0

Reaction–diffusion–advection models are extensively employed to elucidate the diverse aspects of population dynamics. In this framework, the entire space $R^{d}$ represents the habitat of a species where the population density at any given time $t ⩾ ι$ and position $y \in R^{d}$ is denoted by $ϕ (y, t)$ . These models encapsulate the spatial movement of organisms, which may occur randomly (diffusion) or directionally along a preferred vector (advection). Advection may arise from individual behaviors or physical transport mechanisms, such as wind, river currents, or blood flow (see Cosner, 2014; Cantrell & Cosner, 2004; Okubo, 1980). The advective term $\vec{α} \cdot \nabla ϕ^{γ}$ represents movement in the direction of the vector $\vec{α}$ , at a velocity proportional to $| \vec{α} |$ and the population density $ϕ$ . Studies addressing this nonlinear advection term can be found by Cantrell et al. (2008), Cintra et al. (2022), and He & Liu (2022).

Colored noise is utilized to approximate the Wiener process, effectively addressing the challenges posed by the non-differentiability of the sample paths. From a practical perspective, accounting for random influences in systems using colored noise is more rational, given that random fluctuations in complex systems often exhibit some degree of correlation (as noted by Anishchenko et al. (2007), Chen et al. (2021), Caraballo et al. (2023), Gu et al. (2020), Gu & Wang (2019, 2020, 2018), and Ridolfi et al. (2011)). Recent investigations, such as those by Chen et al. (2023), Freitas (2024), and Wang et al. (2023a), have delved into the long-term dynamics of the multivalued partial differential equations subjected to colored noise.

Random attractors are essential tools for understanding the dynamic behavior of dynamical systems (see Carvalho et al., 2012; Crauel et al., 1997; Caraballo et al., 1998, 2004, 2012; Freitas et al., 2018; Kloeden, 2011; Li et al., 2015, 2024; She et al., 2022; Wang, 1999; Wang & Li, 2019; Wang & Wang, 2020a, 2020b; Wang, 2021). Multivalued random dynamical systems more accurately capture the phenomenon of non-uniqueness by describing multiple possible evolutions of the system’s state, helping to better understand the long-term behavior and multiple steady states of uncertain systems. Extensive studies have been conducted on the random attractors of multivalued random dynamical systems, as evidenced by Caraballo et al. (2003), Caraballo et al. (2008), Caraballo et al. (2010), Cintra et al. (2024), Li et al. (2019), Wang & Wang (2015), and Xu & Caraballo (2023). Notably, random attractors for SPDEs with the standard Laplacian operator ( $β = 1$ ) have been widely examined in the literature (see Costa et al., 2022; Caraballo & Sonner, 2017; Caraballo et al., 2021; Freitas et al., 2018; Kinra et al., 2024; Li et al., 2018, 2019; Li & Wang, 2018; Lu & Wang, 2019; Wang, 2012; Wang et al., 2023b, 2023c, 2018; Xu & Caraballo, 2022). Additionally, SPDEs featuring the fractional Laplacian operator for $β \in (0, 1)$ have also received substantial attention (see Caffarelli et al., 2010; Li & Freitas, 2023; Lu et al., 2019; Ros-Oton & Serra, 2014; Wang, 2017, 2019; Wang et al., 2019a, 2019b, 2021). For $β = 1$ , the existence and uniqueness of attractors for the multivalued random dynamics of reaction–diffusion–advection equations driven by superlinear colored noise on $b o u n d e d$ domains has been explored by Freitas (2024). As far as we know, there are currently no reported results in the literature regarding the multivalued random dynamics of reaction–diffusion–advection equations driven by superlinear colored noise on $u n b o u n d e d$ domains for $β = 1$ . This article aims to investigate the existence and uniqueness of attractors for the multivalued random dynamics of reaction–diffusion–advection equations driven by superlinear colored noise on $u n b o u n d e d$ domains for $β \in (0, 1)$ .

Throughout this article, we assume that the nonlinear drift term $f$ and the nonlinear diffusion term $ℏ$ are continuous but not necessarily Lipschitz continuous, this non-Lipschitz continuity leads to the non-uniqueness of solutions for (1.1). To demonstrate the existence and uniqueness of random attractors for the multivalued dynamical system in (1.1), it is imperative to demonstrate the measurability of the random attractors and the asymptotic compactness of solutions. The challenges are (i) the proof of measurability of multivalued dynamical systems and random attractors is much more complex than that of single-valued systems; (ii) the non-compactness of Sobolev embeddings on unbounded domains; and (iii) the difficulty of obtaining uniform estimates due to the time nowhere differentiability in time of colored noise and the Wiener process. To address the above issues, we use the weak upper semicontinuity method of multivalued functions as shown by Wang (2017) to prove the measurability. Then, we employ Ball’s energy equations (see Ball (1997, 2004)) to overcome the non-compactness of Sobolev embeddings on the entire space $R^{d}$ . We decompose the nonlinear terms into parts defined within a bounded ball $O_{n} \subseteq R^{d}$ with radius $n$ and parts defined in its complement. By utilizing the dissipative properties of the nonlinearity term, we demonstrate the convergence of nonlinear terms related to $f$ and $h$ in $O_{n}$ and the uniform smallness of their tails outside $O_{n}$ when $n$ is large enough, which leads to the desired pullback asymptotic compactness.

The article is structured as follows. In Section 2, we review some known results on random attractors, colored noise, and the fractional Laplacian operator for multivalued non-autonomous random dynamical systems. Section 3 is dedicated to demonstrating the weak upper semicontinuity and measurability of the solutions. Additionally, we define a multivalued non-autonomous random dynamical system for (1.1). In Section 4, we derive uniform estimates of solutions and prove the existence and uniqueness of pullback random attractors.

2. Preliminaries

For the convenience of readers, it is necessary to review some fundamental results on the pullback random attractors, colored noise, and the fractional Laplacian operator for multivalued non-autonomous random dynamical systems, as described by Di Nezza et al. (2012) and Gu & Wang (2020).

2.1. Pullback Attractors for Multivalued Non-Autonomous Random Dynamical Systems

Consider two metric spaces $Y$ and $X$ with their respective Borel $σ$ -alegebras $B (Y)$ and $B (X)$ , The collection of all subsets of $X$ is denoted by $2^{X}$ . Assume $(Ω, F, P, {θ_{t}}_{t \in R})$ is a metric dynamical system where $(Ω, F, P)$ is a probability space and $θ : R \times Ω \to Ω$ represents a measure-preserving translation group on $Ω$ , refer to Arnold (1998) for more details. Let $D$ be a collection of some families of nonempty subsets of $Y$ parameterized by $ι \in R$ and $ω \in Ω$ .

Definition 2.1
Let $y_{0} \in Y$ . If for any $y_{m} \to y_{0}$ in $Y$ and $z_{m} \in Z (y_{m})$ , there esists $z_{0} \in Z (y_{0})$ and a subsequence ${z_{m_{k}}}$ of ${z_{m}}$ such that ${z_{m_{k}}} \to z_{0}$ in $X$ , then $Z$ is upper semicontinuous at $y_{0}$ .
Definition 2.2
Let $Y$ be a metric space and $X$ be a Banach space. A multivalued function $Z : Y \to 2^{X}$ is said to be weakly upper semicontinuous at $y_{0} \in Y$ if for any $y_{m} \to y_{0}$ in $Y$ and $z_{m} \in Z (y_{m})$ , there exist $z_{0} \in Z (y_{0})$ and a subsequence ${z_{m_{k}}}$ of ${z_{m}}$ such that $z_{m_{k}} \to z_{0}$ weakly in $X$ . If in addition, $Z$ is weakly upper semicontinuous at every $y \in Y$ , then we say $Z$ : $Y \to 2^{X}$ is weakly upper semicontinuous.
Definition 2.3
A multivalued mapping $Ψ : R^{+} \times R \times Ω \times Y \to 2^{Y}$ with nonempty closed images is called a multivalued non-autonomous random dynamical system on $Y$ over $(Ω, F, P, {θ_{t}}_{t \in R})$ if for all $ι \in R, ω \in Ω$ and $t, s \in R^{+}$ , (i)
$Ψ (t, ι, \cdot, \cdot) : Ω \times Y \to 2^{Y}$ is $F \times B (Y)$ -measurable;
(ii)
$Ψ (0, ι, ω, \cdot)$ is the identity on $Y$ ;
(iii)
$Ψ (t + s, ι, ω, \cdot)$ = $Ψ (t, ι + s, θ_{s} ω, \cdot) \circ Ψ (s, ι, ω, \cdot)$ .

Lemma 2.4
Let $Ψ : R^{+} \times R \times Ω \times Y \to 2^{Y}$ be a multivalued function. Suppose $Ω$ is a metric space and $Y$ a separable Banach space. If for every $ι \in R$ and t $⩾ 0$ , the mapping $Ψ (t, ι, \cdot, \cdot) : Ω \times Y \to 2^{Y}$ is weakly upper semicontinuous in the sense of Definition 2.2, then $Ψ (t, ι, \cdot, \cdot)$ : $Ω \times Y \to 2^{Y}$ is $B (Ω) \times B (Y)$ -measurable.
Definition 2.5
A family $A = {A (ι, ω) : ι \in R, ω \in Ω} \in D$ is called a $D$ -pullback attractor for $Ψ$ if for every $ι \in R$ and $ω \in Ω$ , the following holds: (i)
$A (ι, \cdot) : Ω \to 2^{Y}$ is measurable with respect to $F$ and $A (ι, ω)$ is compact in $Y$ ;
(ii)
$A$ is invariant $: Ψ (t, ι, ω, A (ι, ω))$ = $A (t + ι, θ_{t} ω)$ for all $t ⩾ 0$ ;
(iii)
$A$ attracts every member of $D :$ for every D $\in D$ ,

$lim_{t \to \infty} d_{X} (Ψ (t, ι - t, θ_{- t} ω, D (ι - t, θ_{- t} ω)), A (ι, ω)) = 0,$
where $d_{X} (\cdot, \cdot)$ is the Hausdorff semi-distance in $Y$ .

The result of existence of pullback random attractors is stated as follows (see Gu & Wang (2020); Wang (2017)).
Theorem 2.6
Let $Y$ be a separable Banach space and $Ω$ be a metric space with $Ω = ⋃_{n = 1}^{\infty} Ω_{n}$ for $Ω_{n} \in B (Y)$ . Suppose further: (i)
$Ψ$ is $D$ -pullback asymptotically compact in $Y$ ;
(ii)
$Ψ (t, ι, ω, \cdot) : Y \to 2^{Y}$ is upper semicontinuous for each $t \in R^{+}, ι \in R$ and $ω \in Ω$ ;
(iii)
$Ψ$ has a closed $D$ -pullback absorbing set $K = {K (t, ω) : ι \in R, ω \in Ω} \in D$ ;
(iv)
$Ψ (t, ι, \cdot, K (ι, \cdot)) : Ω_{n} \to 2^{Y}$ is weakly upper semicontinuous in the sence of Definition 2.2 for every $n \in N$ , $t \in R^{+}$ , and $ι \in R$ .

Then $Ψ$ has a unique $D$ -pullback attractor $A$ in $D$ .

2.2. Colored Noise

To characterize the colored noise used in this article, we define a probability space $(Ω, F, P)$ , where $Ω = {ω \in C (R, R) : ω (0) = 0}$ furnished with the compact-open topology, $F = B (Ω)$ denotes the Borel sigma-algebra of $Ω$ and $P$ is the Wiener measure. Subsequently, we define a measure-preserving transformation group ${θ_{t}}_{t \in R}$ on $Ω$ by

θ_{t} ω (\cdot) = ω (t + \cdot) - ω (t), ω \in Ω, t \in R .

Consider

W

as a bilateral real-valued Wiener process defined on the probability space

(Ω, F, P)

, which can be represented by

W (t, ω) = ω (t)

for

t \in R

and

ω \in Ω

. Define a random variable

Z_{δ} : Ω \to R

as follows:

Z_{δ} (ω) = \frac{1}{δ} \int_{- \infty}^{0} e^{\frac{s}{δ}} d W (s, ω) for δ > 0.

The process

Z_{δ} (θ_{t} ω)

is referred to as colored noise. It is a stationary Gaussian process with

E (Z_{δ}) = 0

and represents the unique stationary solution of the following stochastic equation:

d Z_{δ} + \frac{1}{δ} Z_{δ} d t = \frac{1}{δ} d W .

According to Gu & Wang (2020), there exists a subset of

Ω

(still denoted by

Ω

), which is invariant under the transformation group

{θ_{t}}_{t \in R}

and has full measure, such that for all

ω \in Ω

lim_{t \to \pm \infty} \frac{| Z_{δ} (θ_{t} ω) |}{t} = 0 for every δ \in (0, 1] .

(2.1)

In order to demonstrate the weak and strong upper semicontinuity of the solutions to problems (3.1) and (3.2), for each

n \in N

, we denote by

Ω_{n} = {ω \in Ω : | ω (t) | ⩽ | t | for all | t | ⩾ n} .

(2.2)

According to Lemma 3.3 by Gu & Wang (2020), it is established that

Ω_{n} \subseteq Ω

possesses the following properties.

Lemma 2.7

Let $δ \in (0, 1]$ and $n \in N$ . (i)

If $ω_{m} \to ω$ with $ω_{m}$ , $ω \in Ω_{n}$ , then $Z_{δ} (θ_{t} ω_{m}) \to Z_{δ} (θ_{t} ω)$ uniformly for $t$ in a compact interval as $m \to \infty$ .

(ii)

Every $Ω_{n}$ is a closed subset of $Ω$ and $Ω = ⋃_{n = 1}^{\infty} Ω_{n}$ .

(iii)

Given $ω \in Ω_{n}$ , we have for all $t ⩽ - n$ , $| Z_{δ} (θ_{t} ω) | ⩽ \frac{2}{δ} | t | + 1$ .

2.3. Fractional Laplacian Operator

Let $S$ denote the Schwartz space of rapidly decaying $C^{\infty}$ functions on $R^{d}$ . The fractional Laplacian operator $(- Δ)^{β}$ with $0 < β < 1$ is defined for all $ϕ \in S$ as follows:

(- Δ)^{β} ϕ (y) = - \frac{1}{2} C (d, β) \int_{R^{d}} \frac{ϕ (y + x) + ϕ (y - x) - 2 ϕ (y)}{| y |^{d + 2 β}} d x, y \in R^{d},

provided the integral exists, where

C (d, β)

is a positive constant that depends on

d

and

β

given by the following equation:

C (d, β) = {(\int_{R^{d}} \frac{1 - \cos (ξ_{1})}{| ξ |^{d + 2 β}} d ξ)}^{- 1}, ξ = (ξ_{1}, ξ_{2}, \dots, ξ_{d}) \in R^{d} .

It follows from Di Nezza et al. (2012) that

(- Δ)^{β} ϕ = F^{- 1} (| ξ |^{2 β} F (ϕ)), ξ \in R^{d},

where

F

is the Fourier transform. As usual, the fractional Sobolev space

H^{β} (R^{d})

defined by the following equation:

H^{β} (R^{d}) = {ϕ \in L^{2} (R^{d}) : \int_{R^{d}} \int_{R^{d}} \frac{| ϕ (y) - ϕ (x) |^{2}}{| y - x |^{d + 2 β}} d y d x < \infty}

with norm

‖ ϕ ‖_{H^{β} (R^{d})} = (\int_{R^{d}} | ϕ (y) |^{2} d x + \int_{R^{d}} \int_{R^{d}} \frac{| ϕ (y) - ϕ (x) |^{2}}{| y - x |^{d + 2 β}} d y d x)^{\frac{1}{2}} .

Throughout this article, we denote the norm and the inner product of

L^{2} (R^{d})

‖ \cdot ‖

and

⟨ \cdot, \cdot ⟩

, respectively. For convenience, the Gagliardo semi-norm of

H^{β} (R^{d})

is denoted

‖ \cdot ‖_{{\dot{H}}^{β} (R^{d})}

, that is,

‖ ϕ ‖_{{\dot{H}}^{β} (R^{d})} = \int_{R^{d}} \int_{R^{d}} \frac{| ϕ (y) - ϕ (x) |^{2}}{| y - x |^{d + 2 β}} d y d x, ϕ \in H^{β} (R^{d}) .

We also use the notation

⟨ ϕ_{1}, ϕ_{2} ⟩_{{\dot{H}}^{β} (R^{d})} = \int_{R^{d}} \int_{R^{d}} \frac{(ϕ_{1} (y) - ϕ_{1} (x)) (ϕ_{2} (y) - ϕ_{2} (x))}{| y - x |^{d + 2 β}} d y d x, ϕ_{1}, ϕ_{2} \in H^{β} (R^{d}) .

Then for all

ϕ \in H^{β} (R^{d})

, we have

‖ ϕ ‖_{H^{β} (R^{d})}^{2} = ‖ ϕ ‖^{2} + ‖ ϕ ‖_{{\dot{H}}^{β} (R^{d})}^{2}

. Note that

H^{β} (R^{d})

is a Hilbert space with inner product given by the following equation:

⟨ ϕ_{1}, ϕ_{2} ⟩_{H^{β} (R^{d})} = ⟨ ϕ_{1}, ϕ_{2} ⟩ + ⟨ ϕ_{1}, ϕ_{2} ⟩_{{\dot{H}}^{β} (R^{d})}, ϕ_{1}, ϕ_{2} \in H^{β} (R^{d}) .

According to Di Nezza et al. (2012), we have

‖ (- Δ)^{\frac{β}{2}} ϕ ‖^{2} = \frac{1}{2} C (d, β) ‖ ϕ ‖_{{\dot{H}}^{β} (R^{d})}^{2}, for all ϕ \in H^{β} (R^{d}),

and hence

‖ ϕ ‖_{H^{β} (R^{d})}^{2} = ‖ ϕ ‖^{2} + \frac{2}{C (d, β)} ‖ (- Δ)^{\frac{β}{2}} ϕ ‖^{2}, for all ϕ \in H^{β} (R^{d}) .

(2.3)

This implies that

(‖ ϕ ‖^{2} + ‖ (- Δ)^{\frac{β}{2}} ϕ ‖^{2})^{\frac{1}{2}}

is an equivalent norm of

H^{β} (R^{d})

3. Multivalued Non-Autonomous Cocycles

In this section, we establish a multivalued non-autonomous cocycle for the fractional reaction–diffusion–advection equation driven by superlinear colored noise:

ϕ_{t} + (- Δ)^{β} ϕ + λ ϕ + \vec{α} \cdot \nabla ϕ^{γ} = f (t, y, ϕ) + g (t, y) + ℏ (t, y, ϕ) Z_{δ} (θ_{t} ω), t > ι, y \in R^{d},

(3.1)

with initial condition

ϕ (ι, y) = ϕ_{ι} (y), y \in R^{d},

(3.2)

for all

ϕ \in R

and

y \in R^{d}

\begin{aligned} f (t, y, ϕ) ϕ ⩽ - β_{1} | ϕ |^{p} + ψ_{1} (t, y), ψ_{1} \in L_{loc}^{1} (R, L^{1} (R^{d})), \end{aligned}

(3.3)

\begin{aligned} | f (t, y, ϕ) | ⩽ β_{2} | ϕ |^{p - 1} + ψ_{2} (t, y), ψ_{2} \in L_{loc}^{p^{'}} (R, L^{p^{'}} (R^{d})), \end{aligned}

(3.4)

\begin{aligned} | ℏ (t, y, ϕ) | ⩽ ψ_{3} (t, y) | ϕ |^{q - 1} + ψ_{4} (t, y), ψ_{3} \in L^{\infty} (R, L^{q^{'}} (R^{d})), ψ_{4} \in L^{\infty} (R, L^{p^{'}} (R^{d})), \end{aligned}

(3.5)

where

β_{1}

β_{2}

p

, and

q

are positive constants with

2 ⩽ q < p = 2 γ

p^{'} = \frac{p}{p - 1}

, and

q^{'} = \frac{p}{p - q}

We will look for the solutions of (3.1) and (3.2) in the following sense.

3.1. Definition and Existence of Solutions

Definition 3.1
Given $ι \in R$ , $ω \in Ω$ and $ϕ_{ι} \in L^{2} (R^{d})$ , a continuous function $ϕ (\cdot, ι, ω, ϕ_{ι}) : [ι, \infty) \to L^{2} (R^{d})$ is called a solution of problems (3.1) and (3.2) if $ϕ (ι, ι, ω, ϕ_{ι})$ = $ϕ_{ι}$ and
$\begin{aligned} ϕ \in L_{loc}^{2} (ι, \infty; H^{β} (R^{d})) ⋂ L_{loc}^{p} (ι, \infty; L^{p} (R^{d})), \\ \frac{d ϕ}{d t} \in L_{loc}^{2} (ι, \infty; H^{- β} (R^{d})) + L_{loc}^{p^{'}} (ι, \infty; L^{p^{'}} (R^{d})), \end{aligned}$
for every $η \in H^{β} (R^{d}) ⋂ L^{p} (R^{d})$ , and $ϕ$ satisties
$\begin{aligned} \frac{d}{d t} ⟨ ϕ, η ⟩ + \frac{1}{2} C (d, β) \int_{R^{d}} \int_{R^{d}} \frac{(ϕ (y) - ϕ (x)) (η (y) - η (x))}{| y - x |^{d + 2 β}} d y d x + λ ⟨ ϕ, η ⟩ + ⟨ \vec{α} \cdot \nabla ϕ^{γ}, η ⟩ \\ = \int_{R^{d}} f (t, y, ϕ) η (y) d y + \int_{R^{d}} g (t, y) η (y) d y + Z_{δ} (θ_{t} ω) \int_{R^{d}} ℏ (t, y, ϕ) η (y) d y \end{aligned}$
(3.6)
in the sense of distribution on $(ι, \infty)$ .

To demonstrate the existence of solution for equations (3.1) and (3.2) as defined in Definition 3.1, we approximate the entire space $R^{d}$ with a bounded domain $O_{n} = {y \in R^{d} : | y | < n}$ and subsequently take the limit as $n \to \infty$ . Let $ρ : [0, \infty) \to R$ be a smooth function such that $0 ⩽ ρ (s) ⩽ 1$ for all $0 ⩽ s < \infty$ and
$ρ (s) = 1 for 0 ⩽ s ⩽ \frac{1}{2} and ρ (s) = 0 for s ⩾ 1.$
(3.7)
We consider the subsequent non-autonomous fractional equation on the bounded domain $O_{n}$ :
$\frac{d ϕ_{n}}{d t} + (- Δ)^{β} ϕ_{n} + λ ϕ_{n} + \vec{α} \cdot \nabla ϕ_{n}^{γ} = f (t, y, ϕ_{n}) + g (t, y) + Z_{δ} (θ_{t} ω) ℏ (t, y, ϕ_{n}), y \in O_{n}, t > ι$
(3.8)
with boundary condition
$ϕ_{n} (t, y) = 0, y \in R^{d} ∖ O_{n}, t > ι,$
(3.9)
and initial condition
$ϕ_{n} (ι, y) = ρ (\frac{| y |}{n}) ϕ_{ι} (y), y \in O_{n},$
(3.10)
where $ϕ_{ι} \in L^{2} (R^{d})$ . Note that for the boundary condition in (3.9), we require $ϕ_{n} = 0$ on the complement of $O_{n}$ (i.e. on $R^{d} ∖ O_{n})$ , not just on its boundary. This boundary condition aligns with the definition of the non-local fractional operator $(- Δ)^{β}$ . To prove the existence of solutions of problems (3.8)–(3.10), for every $n \in N$ , we define $H_{n} = {ϕ \in L^{2} (R^{d}) : ϕ = 0$ a.e. for $| y | ⩾ n}$ and $V_{n} = {ϕ \in H^{β} (R^{d}) : ϕ = 0$ a.e. for $| y | ⩾ n}$ . The dual space of $V_{n}$ is denoted $V_{n}^{}$ .

Let $a : H^{β} (R^{d}) \times H^{β} (R^{d}) \to R$ be a bilinear form given by, for $ϕ_{1}, ϕ_{2} \in H^{β} (R^{d})$ ,
$a ⟨ ϕ_{1}, ϕ_{2} ⟩ = λ ⟨ ϕ_{1}, ϕ_{2} ⟩ + \frac{1}{2} C (d, β) \int_{R^{d}} \int_{R^{d}} \frac{(ϕ_{1} (y) - ϕ_{1} (x)) (ϕ_{2} (y) - ϕ_{2} (x))}{| y - x |^{d + 2 β}} d y d x .$
(3.11)
Using the bilinear form $a$ , we define the operator $A : H^{β} (R^{d}) \to H^{- β} (R^{d})$ by
$_{H^{- β}} ⟨ A (ϕ_{1}), ϕ_{2} ⟩_{H^{β}} = a ⟨ ϕ_{1}, ϕ_{2} ⟩ for all ϕ_{1}, ϕ_{2} \in H^{β} (R^{d}),$
(3.12)
where $_{H^{- β}} ⟨ \cdot, \cdot ⟩_{H^{β}}$ is the duality product of $H^{- β} (R^{d})$ and $H^{β} (R^{d})$ . Since $H_{n}$ and $V_{n}$ are subspaces of $L^{2} (R^{d})$ and $H^{β} (R^{d})$ , respectively, it follows that $a : V_{n} \times V_{n} \to R$ and $A : V_{n} \to V _{n}$ are well defined. Specifically,
$_{V _{n}} ⟨ A (ϕ_{1}), ϕ_{2} ⟩_{V_{n}} = a ⟨ ϕ_{1}, ϕ_{2} ⟩ for all ϕ_{1}, ϕ_{2} \in V_{n},$
where $_{V _{n}} ⟨ \cdot, \cdot ⟩_{V_{n}}$ is the duality product of $V _{n}$ and $V_{n}$ . Under conditions (3.3)–(3.5), it can be demonstrated that for every $ι \in R$ , $ω \in Ω$ , and $ϕ_{ι} \in L^{2} (R^{d})$ , problems (3.8)–(3.10) posses at least one solution $ϕ_{n}$ in the sense that $ϕ_{n} (\cdot, ι, ω, ϕ_{ι}) : [ι, \infty) \to H_{n}$ is continuous, $ϕ_{n} (ι, ι, ω, ϕ_{ι}) (y) = ρ (\frac{| y |}{n}) ϕ_{ι} (y)$ . Moreover,
$\begin{aligned} ϕ_{n} \in L_{loc}^{2} ((ι, \infty), V_{n}) ⋂ L_{loc}^{p} ((ι, \infty), L^{p} (R^{d})), \\ \frac{d ϕ_{n}}{d t} \in L_{loc}^{2} ((ι, \infty), V _{n}) + L_{loc}^{p^{'}} ((ι, \infty), L^{p^{'}} (R^{d})), \end{aligned}$
and $ϕ_{n}$ satisfies, for every $η \in V_{n} ⋂ L^{p} (R^{d})$ ,
$\begin{aligned} \frac{d}{d t} ⟨ ϕ_{n}, η ⟩ + \frac{1}{2} C (d, β) \int_{R^{d}} \int_{R^{d}} \frac{(ϕ_{n} (y) - ϕ_{n} (x)) (η (y) - η (x))}{| y - x |^{d + 2 β}} d y d x + λ ⟨ ϕ_{n}, η ⟩ + ⟨ \vec{α} \cdot \nabla ϕ_{n}^{γ}, η ⟩ \\ = \int_{O_{n}} f (t, y, ϕ_{n}) η (y) d y + \int_{O_{n}} g (t, y) η (y) d y + Z_{δ} (θ_{t} ω) \int_{O_{n}} ℏ (t, y, ϕ_{n}) η (y) d y \end{aligned}$
(3.13)
in the distributional sense on $(ι, \infty)$ . We then derive uniform estimates for the solution $ϕ_{n}$ with respect to $n \in N$ and demonstrate the existence of solutions to (3.1) and (3.2) by taking the limit of $ϕ_{n}$ when $n \to \infty$ .
Remark 3.2
Since a solution to (3.1) and (3.2) satisfies $ϕ \in L^{p} (ι, ι + T; L^{p} (R^{d})) = L^{2 γ} (ι, ι + T; L^{2 γ} (R^{d}))$ for any $T > 0$ , then $\vec{a} \cdot \nabla ϕ^{γ} \in L^{2} (ι, ι + T; H^{- β} (R^{d}))$ . Specifically, for any $φ \in H^{β} (R^{d})$ , we have
$| \int_{R^{d}} \vec{α} \cdot \nabla ϕ^{γ} φ d y | = | \int_{R^{d}} ϕ^{γ} \vec{α} \cdot \nabla φ d y | ⩽ | \vec{α} | ‖ ϕ ‖_{L^{2 γ} (R^{d})}^{γ} ‖ \nabla φ ‖ ⩽ | ‖ ϕ ‖_{L^{p} (R^{d})}^{p / 2} ‖ \nabla φ ‖ .$
From the above and by using (3.4), we conclude that $\frac{d ϕ}{d t} \in L^{2} (ι, ι + T; H^{- β} (R^{d})) + L^{p^{'}} (ι, ι + T; L^{p^{'}} (R^{d}))$ for any $T > 0$ (where $p$ and $p^{'}$ are conjugate exponents). Therefore, $ϕ \in C ([ι, \infty); L^{2} (R^{d}))$ and the initial data $ϕ (ι)$ is well-defined. Moreover, integrating by parts $⟨ z^{γ}, \vec{a} \cdot \nabla z ⟩ = 0$ holds for any $z \in H^{β} (R^{d}) ⋂ L^{p} (R^{d})$ . Indeed,
$⟨ z^{γ}, \vec{α} \cdot \nabla z ⟩ = \frac{1}{γ + 1} \int_{R^{d}} \vec{α} \cdot \nabla z^{γ + 1} d y = - \frac{1}{γ + 1} \int_{R^{d}} z^{γ + 1} \nabla \cdot \vec{α} d y = 0.$

Theorem 3.3
Suppose (3.3)–(3.5) hold. Then for every $ι \in R$ , $ω \in Ω$ and $ϕ_{ι} \in L^{2} (R^{d})$ , problems (3.1) and (3.2) has a solution $ϕ (t, ι, ω, ϕ_{ι})$ in the sense of Definition 3.1. This solution is $(F, B (L^{2} (R^{d})))$ -measurable in $ω$ and continuous in the initial data $ϕ_{ι}$ in $L^{2} (R^{d})$ . Moreover, the solution $ϕ$ satisfies the energy equation:
$\begin{aligned} \frac{d}{d t} ‖ ϕ (t, ι, ω, ϕ_{ι}) ‖^{2} + C (d, β) ‖ ϕ ‖_{{\dot{H}}^{β}}^{2} + 2 λ ‖ ϕ ‖^{2} \\ = 2 \int_{R^{d}} f (t, y, ϕ) ϕ d y + 2 \int_{R^{d}} g (t, y) ϕ d y + 2 Z_{δ} (θ_{t} ω) \int_{R^{d}} ℏ (t, y, ϕ) ϕ d y \end{aligned}$
(3.14)
for almost all $t ⩾ ι$ . Proof.
For problems (3.1) and (3.2) defined on the unbounded domain $R^{d}$ , it is necessary to show that all estimates on the solutions of (3.8)–(3.10) are uniform with respect to all $n \in N$ .

Step (i). Uniform estimates for the solutions of (3.8)–(3.10). By (3.8) and using the Remark 3.2, we obtain
$\begin{aligned} \frac{1}{2} \frac{d}{d t} \int_{O_{n}} | ϕ_{n} (y) |^{2} d y + \int_{O_{n}} ϕ_{n} (y) (- Δ)^{β} ϕ_{n} (y) d y + λ \int_{O_{n}} | ϕ_{n} (y) |^{2} d y \\ = \int_{O_{n}} f (t, y, ϕ_{n} (y)) ϕ_{n} (y) d y + \int_{O_{n}} g (t, y) ϕ_{n} (y) d y + Z_{δ} (θ_{t} ω) \int_{O_{n}} ℏ (t, y, ϕ_{n} (y)) ϕ_{n} (y) d y . \end{aligned}$
Utilizing the boundary condition (3.9), all above integrals over the bounded domain $O_{n}$ can be extended to the entire space $R^{d}$ , yielding
$\frac{1}{2} \frac{d}{d t} ‖ ϕ_{n} ‖^{2} + a ⟨ ϕ_{n}, ϕ_{n} ⟩ = \int_{R^{d}} f (t, y, ϕ_{n}) ϕ_{n} d y + ⟨ g (t), ϕ_{n} ⟩ + Z_{δ} (θ_{t} ω) \int_{R^{d}} ℏ (t, y, ϕ_{n}) ϕ_{n} d y .$
(3.15)
By (3.3), the first nonlinear term in (3.15) satisfies
$\int_{R^{d}} f (t, y, ϕ_{n}) ϕ_{n} d y ⩽ - β_{1} ‖ ϕ_{n} ‖_{L^{p}}^{p} + ‖ ψ_{1} (t) ‖_{L^{1}} .$
(3.16)
Note that $ψ_{3} \in L^{\infty} (R, L^{q^{'}} (R^{d}))$ and $ψ_{4} \in L^{\infty} (R, L^{p^{'}} (R^{d}))$ , by (3.5) and Young’s inequality, we get
$\begin{aligned} Z_{δ} (θ_{t} ω) \int_{R^{d}} ℏ (t, y, ϕ_{n}) ϕ_{n} d y & ⩽ | Z_{δ} (θ_{t} ω) | \int_{R^{d}} | ψ_{3} (t, y) | | ϕ_{n} |^{q} d y + | Z_{δ} (θ_{t} ω) | \int_{R^{d}} | ψ_{4} (t, y) | | ϕ_{n} | d y \\ ⩽ \frac{1}{2} β_{1} ‖ ϕ_{n} ‖_{L^{p}}^{p} + c | Z_{δ} (θ_{t} ω) |^{q^{'}} ‖ ψ_{3} (t) ‖_{L^{q^{'}}}^{q^{'}} + c | Z_{δ} (θ_{t} ω) |^{p^{'}} ‖ ψ_{4} (t) ‖_{L^{p^{'}}}^{p^{'}} \\ ⩽ \frac{1}{2} β_{1} ‖ ϕ_{n} ‖_{L^{p}}^{p} + c (| Z_{δ} (θ_{t} ω) |^{q^{'}} + | Z_{δ} (θ_{t} ω) |^{p^{'}}), \end{aligned}$
(3.17)
where $p^{'} = \frac{p}{p - 1}$ and $q^{'} = \frac{p}{p - q}$ . Applying Young’s inequality, we obtain
$⟨ g (t), ϕ_{n} ⟩ ⩽ \frac{λ}{2} ‖ ϕ_{n} ‖^{2} + \frac{1}{2 λ} ‖ g (t) ‖^{2} .$
(3.18)
By combining (3.11) and (3.15)–(3.18), we obtain that for every $n \in N$ ,
$\begin{aligned} \frac{d}{d t} ‖ ϕ_{n} ‖^{2} + C (d, β) ‖ ϕ_{n} ‖_{{\dot{H}}^{β}}^{2} + λ ‖ ϕ_{n} ‖^{2} + β_{1} ‖ ϕ_{n} ‖_{L^{p}}^{p} \\ ⩽ c (‖ g (t) ‖^{2} + ‖ ψ_{1} (t) ‖_{L^{1}} + | Z_{δ} (θ_{t} ω) |^{q^{'}} + | Z_{δ} (θ_{t} ω) |^{p^{'}}) . \end{aligned}$
(3.19)
By (3.19) and (3.12), we observe that for every fixed $ω \in Ω$ and $T > 0$ , the sequence ${ϕ_{n}}_{n = 1}^{\infty}$ is bounded in
$L^{\infty} (ι, ι + T; L^{2} (R^{d})) ⋂ L^{2} (ι, ι + T; H^{β} (R^{d})) ⋂ L^{p} (ι, ι + T; L^{p} (R^{d}))$
(3.20)
and
${A (ϕ_{n})}_{n = 1}^{\infty} is bounded in L^{2} (ι, ι + T; H^{- β} (R^{d})) .$
(3.21)
By (3.4), (3.5) and (3.20), it can be verified that
${f (t, \cdot, ϕ_{n}) + Z_{δ} (θ_{t} ω) ℏ (t, \cdot, ϕ_{n})}_{n = 1}^{\infty} is bounded in L^{p^{'}} (ι, ι + T; L^{p^{'}} (R^{d})) .$
(3.22)
As a consequence of (3.8) and (3.21) and (3.22), we find that for each fixed $N \in N$ ,
${\frac{d ϕ_{n}}{d t}}_{n = 1}^{\infty} is bounded in L^{p^{'}} (ι, ι + T; (V_{N} ⋂ L^{p} (R^{d}))^{}) .$
(3.23)
Note that $1 < p^{'} < 2$ since $p > 2$ , $p$ and $p^{'}$ are conjugate exponents.

Step (ii). Existence of solutions of problems (3.1) and (3.2). By applying a diagonal process from (3.20) to (3.23), we conclude that there exists $\tilde{ϕ} \in L^{2} (R^{d})$ ,
$ϕ \in L^{\infty} (ι, ι + T; L^{2} (R^{d})) ⋂ L^{2} (ι, ι + T; H^{β} (R^{d})) ⋂ L^{p} (ι, ι + T; L^{p} (R^{d})),$
and $X_{1} \in L^{p^{'}} (ι, ι + T; L^{p^{'}} (R^{d}))$ and $X_{2} \in L^{2} (ι, ι + T; H^{- β} (R^{d}))$ such that, up to a subsequence,
$\begin{aligned} ϕ_{n} \to ϕ weak-star in L^{\infty} (ι, ι + T; L^{2} (R^{d})), \end{aligned}$
(3.24)

$\begin{aligned} ϕ_{n} \to ϕ weakly in L^{2} (ι, ι + T; H^{β} (R^{d})) and in L^{p} (ι, ι + T; L^{p} (R^{d})), \end{aligned}$
(3.25)

$\begin{aligned} f (t, \cdot, ϕ_{n}) + Z_{δ} (θ_{t} ω) ℏ (t, \cdot, ϕ_{n}) \to X_{1} weakly in L^{p^{'}} (ι, ι + T; L^{p^{'}} (R^{d})), \end{aligned}$
(3.26)

$\begin{aligned} \frac{d ϕ_{n}}{d t} \to \frac{d ϕ}{d t} weakly in L^{p^{'}} (ι, ι + T; (V_{N} ⋂ L^{p} (R^{d}))^{}), \forall N \in N, \end{aligned}$
(3.27)

$\begin{aligned} \vec{a} \cdot \nabla ϕ_{n}^{γ} \to X_{2} weakly in L^{2} (ι, ι + T; H^{- β} (R^{d})) . \end{aligned}$
(3.28)
and
$ϕ_{n} (ι + T, ι, ω) \to \tilde{ϕ} weakly in L^{2} (R^{d}) .$
(3.29)
Note that the embedding $H^{β} (O_{N}) ↪ L^{2} (O_{N})$ is compact. Additionally, the embedding $L^{2} (O_{N}) ↪ (V_{N} ⋂ L^{p} (R^{d}))^{*}$ is continuous. And by (3.20), (3.23) and the compactness result by Lions (1969), after applying an appropriate diagonal process, we obtain that, up to a subsequence,
$ϕ_{n} \to ϕ strongly in L^{2} (ι, ι + T; L^{2} (O_{N})), \forall N \in N .$
(3.30)
By (3.30) and a diagonal again, there exists a further subsequence (still denoted by ${ϕ_{n}}_{n = 1}^{\infty}$ ) such that
$ϕ_{n} \to ϕ for almost every (t, y) \in (ι, ι + T) \times R^{d} .$
(3.31)
Since both $f$ and $ℏ$ are continuous, it follows from (3.31) that
$f (t, y, ϕ_{n}) + Z_{δ} (θ_{t} ω) ℏ (t, y, ϕ_{n}) \to f (t, y, ϕ) + Z_{δ} (θ_{t} ω) ℏ (t, y, ϕ)$
(3.32)
for almost every $(t, y) \in (ι, ι + T) \times R^{d}$ . By (3.22) and (3.32), we infer from Mazur’s lemma that
$f (t, \cdot, ϕ_{n}) + Z_{δ} (θ_{t} ω) ℏ (t, \cdot, ϕ_{n}) \to f (t, \cdot, ϕ) + Z_{δ} (θ_{t} ω) ℏ (t, \cdot, ϕ) weakly in L^{p^{'}} (ι, ι + T; L^{p^{'}} (R^{d})) .$
(3.33)
Consequently, from (3.26) and (3.33), we derive
$X_{1} = f (t, \cdot, ϕ) + Z_{δ} (θ_{t} ω) ℏ (t, \cdot, ϕ) .$
(3.34)
Moreover, since $ϕ_{n}^{γ} \to ϕ^{γ}$ a.e. on $(ι, ι + T) \times R^{d}$ , from the uniform estimate of ${ϕ_{n}}$ in $L^{p} (ι, ι + T; L^{p} (R^{d}))$ and $L^{p} (ι, ι + T; L^{p} (R^{d})) = L^{2 γ} (ι, ι + T; L^{2 γ} (R^{d}))$ , we obtain
$ϕ_{n}^{γ} \to ϕ^{γ} weakly in L^{2} (ι, ι + T; L^{2} (R^{d})) .$
Therefore,
$\begin{aligned} \int_{ι}^{ι + T} ⟨ \vec{a} \cdot \nabla ϕ_{n}^{γ} (s), ϕ (s) ⟩ d s = - \int_{ι}^{ι + T} (ϕ_{n}^{γ} (s), \vec{a} \cdot \nabla ϕ (s)) d s \\ \to - \int_{ι}^{ι + T} (ϕ^{γ} (s), \vec{a} \cdot \nabla ϕ (s)) d s = \int_{ι}^{ι + T} ⟨ \vec{a} \cdot \nabla ϕ^{γ} (s), ϕ (s) ⟩ d s, \end{aligned}$
for all $ϕ \in L^{2} (ι, ι + T; H^{β} (R^{d}))$ . Hence, we conclude that $X_{2} = \vec{a} \cdot \nabla ϕ^{γ}$ .

Now, given $η \in H^{β} (R^{d}) ⋂ L^{p} (R^{d})$ , let
$η_{N} (y) = ρ (\frac{| y |}{N}) η (y) for all y \in R^{d} .$
Through simple calculations, we can verify that for each $N \in N$ , $η_{N} \in H^{β} (R^{d}) ⋂ L^{p} (R^{d})$ and
$η_{N} \to η in H^{β} (R^{d}) ⋂ L^{p} (R^{d}) .$
(3.35)
For every $n > N$ and $ξ \in C_{0}^{\infty} (ι, ι + T)$ , using (3.8)–(3.10), we get
$\begin{aligned} - \int_{ι}^{ι + T} ⟨ ϕ_{n}, η_{N} ⟩ ξ^{'} d t + \int_{ι}^{ι + T} a ⟨ ϕ_{n}, η_{N} ⟩ ξ d t + \int_{ι}^{ι + T} ⟨ \vec{α} \cdot \nabla ϕ_{n}^{γ}, η_{N} ⟩ ξ d t \\ = \int_{ι}^{ι + T}_{L^{p^{'}}} ⟨ f (t, \cdot, ϕ_{n}), η_{N} ⟩_{L^{p}} ξ d t + \int_{ι}^{ι + T} ⟨ g, η_{N} ⟩ ξ d t \\ + \int_{ι}^{ι + T} Z_{δ} (θ_{t} ω)_{L^{p^{'}}} ⟨ ℏ (t, \cdot, ϕ_{n}), η_{N} ⟩_{L^{p}} ξ d t . \end{aligned}$
(3.36)
Taking the limit of (3.36) as $n \to \infty$ , by (3.24)–(3.26) and (3.34), we get
$\begin{aligned} - \int_{ι}^{ι + T} ⟨ ϕ, η_{N} ⟩ ξ^{'} d t + \int_{ι}^{ι + T} a ⟨ ϕ, η_{N} ⟩ ξ d t + \int_{ι}^{ι + T} ⟨ \vec{α} \cdot \nabla ϕ^{γ}, η_{N} ⟩ ξ d t \\ = \int_{ι}^{ι + T}_{L^{p^{'}}} ⟨ f (t, \cdot, ϕ), η_{N} ⟩_{L^{p}} ξ d t + \int_{ι}^{ι + T} ⟨ g, η_{N} ⟩ ξ d t + \int_{ι}^{ι + T} Z_{δ} (θ_{t} ω)_{L^{p^{'}}} ⟨ ℏ (t, \cdot, ϕ), η_{N} ⟩_{L^{p}} ξ d t . \end{aligned}$
(3.37)
Taking the limit of (3.37) as $N \to \infty$ , by (3.35) we find that,
$\begin{aligned} - \int_{ι}^{ι + T} ⟨ ϕ, η ⟩ ξ^{'} d t + \int_{ι}^{ι + T} a ⟨ ϕ, η ⟩ ξ d t + \int_{ι}^{ι + T} ⟨ \vec{α} \cdot \nabla ϕ^{γ}, η ⟩ ξ d t \\ = \int_{ι}^{ι + T}_{L^{p^{'}}} ⟨ f (t, \cdot, ϕ), η ⟩_{L^{p}} ξ d t + \int_{ι}^{ι + T} ⟨ g, η ⟩ ξ d t + \int_{ι}^{ι + T} Z_{δ} (θ_{t} ω)_{L^{p^{'}}} ⟨ ℏ (t, \cdot, ϕ), η ⟩_{L^{p}} ξ d t . \end{aligned}$
(3.38)
Hence, for any given $η \in H^{β} (R^{d}) ⋂ L^{p} (R^{d})$ , we have that
$\frac{d}{d t} ⟨ ϕ, η ⟩ + a ⟨ ϕ, η ⟩ + ⟨ \vec{α} \cdot \nabla ϕ^{γ}, η ⟩ =_{L^{p^{'}}} ⟨ f (t, \cdot, ϕ), η ⟩_{L^{p}} + ⟨ g, η ⟩ + Z_{δ} (θ_{t} ω)_{L^{p^{'}}} ⟨ ℏ (t, \cdot, ϕ), η ⟩_{L^{p}}$
(3.39)
in the sense of distribution on $(ι, ι + T)$ .

To prove the continuity of $ϕ : [ι, \infty) \to L^{2} (R^{d})$ , we observe that $ϕ \in L^{2} (ι, ι + T; H^{β} (R^{d})) ⋂ L^{p} (ι, ι + T; L^{p} (R^{d}))$ and $\frac{d ϕ}{d t} \in L^{2} (ι, ι + T; H^{- β} (R^{d}))$ + $L^{p^{'}} (ι, ι + T; L^{p^{'}} (R^{d}))$ by (3.25) and (3.27), respectively. Then by the argument of Lions (1969) we infer that $ϕ \in C ([ι, ι + T], L^{2} (R^{d}))$ and
$\frac{1}{2} \frac{d}{d t} ‖ ϕ ‖^{2} =_{H^{- β} + L^{p^{'}}} ⟨ \frac{d ϕ}{d t}, ϕ ⟩_{H^{β} ⋂ L^{p}} for almost every t \in (ι, ι + T) .$
(3.40)
From (3.39), (3.40), and Remark 3.2, we get that
$\frac{1}{2} \frac{d}{d t} ‖ ϕ ‖^{2} + a ⟨ ϕ, ϕ ⟩ =_{L^{p^{'}}} ⟨ f (t, \cdot, ϕ), ϕ ⟩_{L^{p}} + ⟨ g, ϕ ⟩ + Z_{δ} (θ_{t} ω)_{L^{p^{'}}} ⟨ ℏ (t, \cdot, ϕ), ϕ ⟩_{L^{p}},$
(3.41)
which leads to the desired energy equation (3.14).

Next, we demonstrate that $ϕ (ι) = ϕ_{ι}$ and $ϕ (ι + T) = \tilde{ϕ}$ . For this purpose, we let $ξ \in C^{1} ([ι, ι + T])$ and $η \in H^{β} (R^{d}) ⋂ L^{p} (R^{d})$ . Similar to (3.36), by (3.8)–(3.10) we find for every $n > N$ ,
$\begin{aligned} ⟨ ϕ_{n} (ι + T), η_{N} ⟩ ξ (ι + T) - ⟨ ϕ_{n} (ι), η_{N} ⟩ ξ (ι) = \int_{ι}^{ι + T} ⟨ ϕ_{n}, η_{N} ⟩ ξ^{'} d t - \int_{ι}^{ι + T} a ⟨ ϕ_{n}, η_{N} ⟩ ξ d t \\ - \int_{ι}^{ι + T} ⟨ \vec{α} \cdot \nabla ϕ_{n}^{γ}, η_{N} ⟩ ξ d t + \int_{ι}^{ι + T}_{L^{p^{'}}} ⟨ f (t, \cdot, ϕ_{n}), η_{N} ⟩_{L^{p}} ξ d t + \int_{ι}^{ι + T} ⟨ g, η_{N} ⟩ ξ d t \\ + \int_{ι}^{ι + T} Z_{δ} (θ_{t} ω)_{L^{p^{'}}} ⟨ ℏ (t, \cdot, ϕ_{n}), η_{N} ⟩_{L^{p}} ξ d t . \end{aligned}$
(3.42)
Following similar steps as in (3.10), (3.24)–(3.26), (3.29), and (3.34) we get from (3.36), as $n \to \infty$ ,
$\begin{aligned} ⟨ \tilde{ϕ}, η_{N} ⟩ ξ (ι + T) - ⟨ ϕ_{ι}, η_{N} ⟩ ξ (ι) = \int_{ι}^{ι + T} ⟨ ϕ, η_{N} ⟩ ξ^{'} d t - \int_{ι}^{ι + T} a ⟨ ϕ, η_{N} ⟩ ξ d t - \int_{ι}^{ι + T} ⟨ \vec{α} \cdot \nabla ϕ^{γ}, η_{N} ⟩ ξ d t \\ + \int_{ι}^{ι + T}_{L^{p^{'}}} ⟨ f (t, \cdot, ϕ), η_{N} ⟩_{L^{p}} ξ d t + \int_{ι}^{ι + T} ⟨ g, η_{N} ⟩ ξ d t + \int_{ι}^{ι + T} Z_{δ} (θ_{t} ω)_{L^{p^{'}}} ⟨ ℏ (t, \cdot, ϕ), η_{N} ⟩_{L^{p}} ξ d t . \end{aligned}$
(3.43)
As $N \to \infty$ in the above equality, by (3.35) we obtain for all $η \in H^{β} (R^{d}) ⋂ L^{p} (R^{d})$ ,
$\begin{aligned} ⟨ \tilde{ϕ}, η ⟩ ξ (ι + T) - ⟨ ϕ_{ι}, η ⟩ ξ (ι) = \int_{ι}^{ι + T} ⟨ ϕ, η ⟩ ξ^{'} d t - \int_{ι}^{ι + T} a ⟨ ϕ, η ⟩ ξ d t - \int_{ι}^{ι + T} ⟨ \vec{α} \cdot \nabla ϕ^{γ}, η ⟩ ξ d t \\ + \int_{ι}^{ι + T}_{L^{p^{'}}} ⟨ f (t, \cdot, ϕ), η ⟩_{L^{p}} ξ d t + \int_{ι}^{ι + T} ⟨ g, η ⟩ ξ d t + \int_{ι}^{ι + T} Z_{δ} (θ_{t} ω)_{L^{p^{'}}} ⟨ ℏ (t, \cdot, ϕ), η ⟩_{L^{p}} ξ d t . \end{aligned}$
(3.44)
On the other hand, by (3.39), we find that the right hand side of the equality (3.44) can be written as follows:
$⟨ ϕ (ι + T), η ⟩ ξ (ι + T) - ⟨ ϕ (ι), η ⟩ ξ (ι),$
furthermore, we have
$\begin{aligned} ⟨ ϕ (ι + T), η ⟩ ξ (ι + T) - ⟨ ϕ (ι), η ⟩ ξ (ι) = ⟨ \tilde{ϕ}, η ⟩ ξ (ι + T) - ⟨ ϕ_{ι}, η ⟩ ξ (ι) . \end{aligned}$
(3.45)
By selecting $ξ \in C^{1} ([ι, ι + T])$ with $ξ (ι) = 1$ and $ξ (ι + T) = 0$ , from (3.45), we derive for all $η \in H^{β} (R^{d}) ⋂ L^{p} (R^{d})$ ,
$⟨ ϕ (ι), η ⟩ = ⟨ ϕ_{ι}, η ⟩ .$
(3.46)
Similarly, by selecting $ξ \in C^{1} ([ι, ι + T])$ with $ξ (ι) = 0$ and $ξ (ι + T) = 1$ , we obtain from (3.45) that for all $η \in H^{β} (R^{d}) ⋂ L^{p} (R^{d})$ ,
$⟨ ϕ (ι + T), η ⟩ = ⟨ \tilde{ϕ}, η ⟩ .$
(3.47)
From (3.46) and (3.47), we conclude that
$ϕ (ι) = ϕ_{ι} and ϕ (ι + T) = \tilde{ϕ} in L^{2} (R^{d}),$
(3.48)
combining together with (3.29) infers that
$ϕ_{n} (ι + T, ι, ω) \to ϕ (ι + T) weakly in L^{2} (R^{d}) .$
(3.49)
Analogue to (3.49), it can be verified that for every $t ⩾ ι$ , as $n \to \infty$ ,
$ϕ_{n} (t, ι, ω) \to ϕ (t) weakly in L^{2} (R^{d}) .$
(3.50)
Note that (3.39) and (3.48) indicate that $ϕ$ is a solution of problems (3.1) and (3.2) in terms of Definition 3.1, and (3.41) shows that $ϕ$ satisfies the energy equation (3.14). The proof is complete.

3.2. Uniform Estimates on Compact Time Intervals

The subsequent uniform estimate of solutions to (3.1) and (3.2) within a finite time interval is crucial for proving the weak and strong upper semicontinuity of solutions.

Lemma 3.4
Suppose that (3.3)–(3.5) hold. Then for every $δ >$ 0, $ι \in R$ , $M > 0$ , $T > 0$ , and $ω_{m} \to ω$ with $ω_{m}$ , $ω \in Ω_{n}$ , there exists $C := C (δ, ι, T, M, n, ω$ ) > 0 such that the solutions of problems (3.1) and (3.2) satisfy
$‖ ϕ (t, ι, ω_{m}, ϕ_{ι}) ‖^{2} + C (d, β) \int_{ι}^{t} e^{λ (s - t)} ‖ ϕ ‖_{{\dot{H}}^{β}}^{2} d s + β_{1} \int_{ι}^{t} e^{λ (s - t)} ‖ ϕ ‖_{L^{p}}^{p} d s ⩽ C,$
(3.51)
for all $m \in N, t \in [ι, ι + T]$ and $ϕ_{ι} \in L^{2} (R^{d})$ with $‖ ϕ_{ι} ‖$ $⩽ M$ .
Proof.
From the energy equalities (3.14) and (3.3), we have that
$\begin{aligned} \frac{d}{d t} ‖ ϕ ‖^{2} + 2 ‖ (- Δ)^{\frac{β}{2}} ϕ ‖^{2} + 2 λ ‖ ϕ ‖^{2} + 2 β_{1} ‖ ϕ ‖_{L^{p}}^{p} \\ ⩽ 2 ‖ ψ_{1} (t) ‖_{L^{1}} + 2 \int_{R^{d}} g (t, y) ϕ d y + 2 Z_{δ} (θ_{t} ω) \int_{R^{d}} ℏ (t, y, ϕ) ϕ d y . \end{aligned}$
(3.52)
Note that $ψ_{3} \in L^{\infty} (R, L^{q^{'}} (R^{d}))$ and $ψ_{4} \in L^{\infty} (R, L^{p^{'}} (R^{d}))$ , by (3.5) and Young’s inequality, we see
$\begin{aligned} Z_{δ} (θ_{t} ω) \int_{R^{d}} ℏ (t, y, ϕ) ϕ d y & ⩽ | Z_{δ} (θ_{t} ω) | \int_{R^{d}} | ψ_{3} (t, y) | | ϕ |^{q} d y + | Z_{δ} (θ_{t} ω) | \int_{R^{d}} | ψ_{4} (t, y) | | ϕ | d y \\ ⩽ \frac{1}{2} β_{1} ‖ ϕ ‖_{L^{p}}^{p} + c | Z_{δ} (θ_{t} ω) |^{q^{'}} ‖ ψ_{3} (t) ‖_{L^{q^{'}}}^{q^{'}} + c | Z_{δ} (θ_{t} ω) |^{p^{'}} ‖ ψ_{4} (t) ‖_{L^{p^{'}}}^{p^{'}} \\ ⩽ \frac{1}{2} β_{1} ‖ ϕ ‖_{L^{p}}^{p} + c (| Z_{δ} (θ_{t} ω) |^{q^{'}} + | Z_{δ} (θ_{t} ω) |^{p^{'}}), \end{aligned}$
(3.53)
where $p^{'} = \frac{p}{p - 1}$ and $q^{'} = \frac{p}{p - q}$ . It follows from Young’s inequality that
$⟨ g (t), ϕ ⟩ ⩽ \frac{λ}{2} ‖ ϕ ‖^{2} + \frac{1}{2 λ} ‖ g (t) ‖^{2} .$
(3.54)
Then it follows from (3.52) to (3.54) that
$\begin{aligned} \frac{d}{d t} ‖ ϕ ‖^{2} & + C (d, β) ‖ ϕ ‖_{{\dot{H}}^{β}}^{2} + λ ‖ ϕ ‖^{2} + β_{1} ‖ ϕ ‖_{L^{p}}^{p} \\ ⩽ c ‖ g (t) ‖^{2} + c ‖ ψ_{1} (t) ‖_{L^{1}} + c | Z_{δ} (θ_{t} ω) |^{q^{'}} + c | Z_{δ} (θ_{t} ω) |^{p^{'}} . \end{aligned}$
(3.55)
Multiplying the above expression by $e^{λ t}$ , we have
$\begin{aligned} \frac{d}{d t} (e^{λ t} ‖ ϕ ‖^{2}) & + C (d, β) e^{λ t} ‖ ϕ ‖_{{\dot{H}}^{β}}^{2} + β_{1} e^{λ t} ‖ ϕ ‖_{L^{p}}^{p} \\ ⩽ c e^{λ t} (‖ g (t) ‖^{2} + ‖ ψ_{1} (t) ‖_{L^{1}} + | Z_{δ} (θ_{t} ω) |^{q^{'}} + | Z_{δ} (θ_{t} ω) |^{p^{'}}) . \end{aligned}$
(3.56)
Integrating (3.56) from $ι$ to t and then replacing $ω$ by $ω_{m}$ , we get
$\begin{aligned} ‖ ϕ (t, ι, ω_{m}, ϕ_{ι}) ‖^{2} + C (d, β) \int_{ι}^{t} e^{λ (s - t)} ‖ ϕ ‖_{{\dot{H}}^{β}}^{2} d s + β_{1} \int_{ι}^{t} e^{λ (s - t)} ‖ ϕ ‖_{L^{p}}^{p} d s \\ ⩽ e^{λ (ι - t)} ‖ ϕ_{ι} ‖^{2} + c \int_{ι}^{t} e^{λ (s - t)} (‖ g (t) ‖^{2} + ‖ ψ_{1} (t) ‖_{L^{1}} + | Z_{δ} (θ_{t} ω_{m}) |^{q^{'}} + | Z_{δ} (θ_{t} ω_{m}) |^{p^{'}}) d s . \end{aligned}$
(3.57)
From Lemma 2.7(i) and $ω_{m} \to ω$ , there exists $K_{1} = K_{1} (T, ι, ω) ⩾ 1$ such that for all $m ⩾ K_{1}$ and s $\in [ι, ι + T]$ ,
$| Z_{δ} (θ_{s} ω_{m}) | ⩽ 1 + | Z_{δ} (θ_{s} ω) |,$
which together with the pathwise continuity of $Z_{δ} (θ_{s} ω)$ in $s$ , exist a constant $c = c (δ, T, ι, ω) > 0$ such that for all $m ⩾ K_{1}$ and $s \in$ [ $ι, ι + T]$ ,
$| Z_{δ} (θ_{s} ω) | ⩽ c and | Z_{δ} (θ_{s} ω_{m}) | ⩽ 1 + c .$
(3.58)
Utilizing the inequalities (3.57) and (3.58), $g \in L_{loc}^{2} (R; L^{2} (R^{d}))$ and the fact $‖ ϕ_{ι} ‖ ⩽ M$ , we obtain the result (3.51) immediately. This completes the proof.
3.3. Weak and Strong Continuity of Solutions

In this subsection, we discuss the weak and strong continuity of solutions to (3.1) and (3.2).

Lemma 3.5
Suppose that (3.3)–(3.5) hold. Given $ι \in R, t ⩾ ι$ and $ω_{m}$ , $ω \in Ω_{n}$ , let $ϕ (\cdot, ι, ω_{m}, ϕ_{ι, m})$ be a solution of problems (3.1) and (3.2) with the initial data $ϕ_{ι, m}$ .

(i) If $ϕ_{ι, m} \to ϕ_{ι} weakly in L^{2} (R^{d})$ , then problems (3.1) and (3.2) has a solution $ϕ$ with the initial condition $ϕ_{ι}$ such that, up to subsequence
$ϕ (t, ι, ω_{m}, ϕ_{ι, m}) \to ϕ (t, ι, ω, ϕ_{ι}) weakly in L^{2} (R^{d}), \forall t ⩾ ι,$
(3.59)
and
$ϕ (\cdot, ι, ω_{m}, ϕ_{ι, m}) \to ϕ (\cdot, ι, ω, ϕ_{ι}) weakly in L^{2} (ι, ι + T; H^{β} (R^{d})), \forall T > 0.$
(3.60)
(ii) If $ϕ_{ι, m} \to ϕ_{ι} s t r o n g l y i n L^{2} (R^{d})$ , then problems (3.1) and (3.2) has a solution $ϕ$ with the initial condition $ϕ_{ι}$ such that, up to a subsequence
$ϕ (t, ι, ω_{m}, ϕ_{ι, m}) \to ϕ (t, ι, ω, ϕ_{ι}) i n L^{2} (R^{d}), \forall t ⩾ ι .$
Proof.
(i) Due to $ϕ_{ι, m} \to ϕ_{ι}$ weakly in $L^{2} (R^{d})$ , there exists $c > 0$ independent of $d$ such that
$‖ ϕ_{ι, m} ‖ ⩽ c, \forall m \in N .$
(3.61)
Let us denote $ϕ_{m} (r)$ = $ϕ (r, ι, ω_{m}, ϕ_{ι, m})$ for $r \in$ [ $ι, ι + T$ ]. Since $ω_{m} \to ω$ with $ω_{m}$ , $ω \in Ω_{n}$ , we get from (3.4) and Lemma 3.4 that
${ϕ_{m}} is bounded in L^{\infty} (ι, ι + T; L^{2} (R^{d})) ⋂ L^{2} (ι, ι + T; H^{β} (R^{d})) ⋂ L^{p} (ι, ι + T; L^{p} (R^{d})) .$
In addition, ${f (t, y, ϕ_{m})}_{m = 1}^{\infty}$ and ${Z_{δ} (θ_{t} ω_{m}) ℏ (t, y, ϕ_{m})}_{m = 1}^{\infty}$ are bounded in $L^{p^{'}} (ι, ι + T; L^{p^{'}} (R^{d}))$ , ${\vec{α} \cdot \nabla ϕ_{m}^{γ}}_{m = 1}^{\infty}$ is bounded in $L^{2} (ι, ι + T; H^{- β} (R^{d}))$ and ${\frac{d ϕ_{m}}{d t}}_{m = 1}^{\infty}$ is bounded in $L^{2} (ι, ι + T; H^{- β} (R^{d})) + L^{p^{'}} (ι, ι + T; L^{p^{'}} (R^{d}))$ . Then there exist $ϕ \in L^{\infty} (ι, ι + T; L^{2} (R^{d})) ⋂ L^{2} (ι, ι + T; H^{β} (R^{d})) ⋂ L^{p} (ι, ι + T; L^{p} (R^{d})$ , $Y_{1} \in L^{p^{'}} (ι, ι + T; L^{p^{'}} (R^{d}))$ and $Y_{2} \in L^{2} (ι, ι + T; H^{- β} (R^{d}))$ , then $\bar{ϕ} (t_{s}) \in L^{2} (R^{d})$ for any fixed $t_{s} \in [ι, ι + T]$ , such that, up to a subsequence,
$\begin{aligned} ϕ_{m} \to ϕ weak-star in L^{\infty} (ι, ι + T; L^{2} (R^{d})), \end{aligned}$
(3.62)

$\begin{aligned} ϕ_{m} \to ϕ weakly in L^{2} (ι, ι + T; H^{β} (R^{d})) and in L^{p} (ι, ι + T; L^{p} (R^{d})), \end{aligned}$
(3.63)

$\begin{aligned} f (t, y, ϕ_{m}) + Z_{δ} (θ_{t} ω) ℏ (t, y, ϕ_{m}) \to Y_{1} weakly in L^{p^{'}} (ι, ι + T; L^{p^{'}} (R^{d})), \end{aligned}$
(3.64)

$\begin{aligned} \frac{d ϕ_{m}}{d t} \to \frac{d ϕ}{d t} weakly in L^{2} (ι, ι + T; H^{- β} (R^{d})) + L^{p^{'}} (ι, ι + T; L^{p^{'}} (R^{d})), \end{aligned}$
(3.65)

$\begin{aligned} \vec{a} \cdot \nabla ϕ_{m}^{γ} \to Y_{2} weakly in L^{2} (ι, ι + T; H^{- β} (R^{d})), \end{aligned}$
(3.66)

$\begin{aligned} ϕ_{m} (t_{s}, ι, ω, ϕ_{ι}) \to \bar{ϕ} (t_{s}) weakly in L^{2} (R^{d}) . \end{aligned}$
(3.67)
By the standard method one can show that $Y_{1} = f (t, y, ϕ) + ζ_{δ} (θ_{t} ω) ℏ (t, y, ϕ)$ and $Y_{2} = \vec{a} \cdot \nabla ϕ^{γ}$ , so $ϕ$ is a solution of (3.1) and (3.2) with the initial data $ϕ (ι) = ϕ_{ι}$ . Moreover,
$ϕ_{m} (r) \to ϕ (r) weakly in L^{2} (R^{d}), for every r \in [ι, ι + T] .$
(3.68)
(ii) $Let ϕ (t) = ϕ (t, ι, ω, ϕ_{ι})$ . By the weak convergence (i), we know that
$‖ ϕ (t) ‖ ⩽ \underset{m \to \infty}{lim inf} ‖ ϕ_{m} (t) ‖ .$
Therefore, we only need to prove that
$\underset{m \to \infty}{lim sup} ‖ ϕ_{m} (t) ‖ ⩽ ‖ ϕ (t) ‖ .$
Given $t ⩾ ι$ , choose a number $T > 0$ such that $t \in [ι, ι + T]$ . From the energy equation (3.14), we have
$\frac{d}{d t} ‖ ϕ ‖^{2} + φ (ϕ) = 2 \int_{R^{d}} f (t, y, ϕ) ϕ d y + 2 \int_{R^{d}} g (t, y) ϕ d y + 2 Z_{δ} (θ_{t} ω) \int_{R^{d}} ℏ (t, y, ϕ) ϕ d y,$
(3.69)
where $φ (\cdot)$ is a functional on $L^{2} (R^{d})$ given by
$φ (ϕ) = λ ‖ ϕ ‖^{2} + 2 ‖ (- Δ)^{\frac{β}{2}} ϕ ‖^{2} for all ϕ \in L^{2} (R^{d}) .$
Multiplying (3.69) by $e^{λ t}$ and integrating from $ι$ to t, we obtain
$\begin{aligned} ‖ ϕ (t, ι, ω, ϕ_{ι}) ‖^{2} = & e^{λ (ι - t)} ‖ ϕ_{ι} ‖^{2} - \int_{ι}^{t} e^{λ (s - t)} φ (ϕ (s)) d s \\ + 2 \int_{ι}^{t} e^{λ (s - t)} ⟨ f (s, y, ϕ (s)), ϕ (s) ⟩ d s \\ + 2 \int_{ι}^{t} e^{λ (s - t)} ⟨ g (s, y), ϕ (s) ⟩ d s \\ + 2 \int_{ι}^{t} e^{λ (s - t)} Z_{δ} (θ_{s} ω) ⟨ ℏ (s, y, ϕ (s)), ϕ (s) ⟩ d s . \end{aligned}$
(3.70)
Replacing $ω$ by $ω_{m}$ and $ϕ_{ι}$ by $ϕ_{ι, m}$ in (3.70), yields
$\begin{aligned} ‖ ϕ_{m} (t) ‖^{2} = & e^{λ (ι - t)} ‖ ϕ_{ι, m} ‖^{2} - \int_{ι}^{t} e^{λ (s - t)} φ (ϕ_{m} (s)) d s \\ + 2 \int_{ι}^{t} e^{λ (s - t)} ⟨ g (s, y), ϕ_{m} (s) ⟩ d s \\ + 2 \int_{ι}^{t} e^{λ (s - t)} (⟨ f (s, y, ϕ_{m} (s)), ϕ_{m} (s) ⟩ + Z_{δ} (θ_{s} ω_{m}) ⟨ ℏ (s, y, ϕ_{m} (s)), ϕ_{m} (s) ⟩) d s \\ ≜ & I_{1} (m) + I_{2} (m) + I_{3} (m) + I_{4} (m) . \end{aligned}$
(3.71)
Next, we need to handle each term in (3.71). Since $ϕ_{ι, m} \to ϕ_{ι}$ in $L^{2} (R^{d})$ , we have
$lim_{m \to \infty} I_{1} (m) = e^{λ (ι - t)} ‖ ϕ_{ι} ‖^{2} .$
(3.72)
By (3.60), we deduce
$\underset{m \to \infty}{lim sup} I_{2} (m) ⩽ - \int_{ι}^{t} e^{λ (s - t)} φ (ϕ) d s .$
(3.73)
By (3.60), we deduce
$lim_{n \to \infty} I_{3} (m) = 2 \int_{ι}^{t} e^{λ (s - t)} ⟨ g (s, y), ϕ (s) ⟩ d s .$
Finally, we address the nonlinear term, for $n \in N$ , we have
$\begin{aligned} I_{4} (m) = & 2 \int_{ι}^{t} e^{λ (s - t)} \int_{| y | ⩾ n} ((f (s, y, ϕ_{m} (s)) ϕ_{m} (s)) + Z_{δ} (θ_{t} ω_{m}) ℏ (s, y, ϕ_{m} (s)) ϕ_{m} (s)) d y d s \\ + 2 \int_{ι}^{t} e^{λ (s - t)} \int_{O_{n}} ((f (s, y, ϕ_{m} (s)) ϕ_{m} (s)) + Z_{δ} (θ_{t} ω_{m}) ℏ (s, y, ϕ_{m} (s)) ϕ_{m} (s)) d y d s \\ ≜ I_{41} (m) + I_{42} (m) . \end{aligned}$
(3.74)
By (3.3) and (3.5) and Young’s inequality, we find that there exists $K_{2} (ε) > 0$ such that for $n ⩾ K_{2}$ ,
$I_{41} (m) ⩽ c \int_{ι}^{t} e^{λ (s - t)} \int_{| y | ⩾ n} (| ψ_{1} (s, y) | + c (| Z_{δ} (θ_{s} ω) |^{q^{'}} | ψ_{3} (s, y) |^{q^{'}} + | Z_{δ} (θ_{s} ω) |^{p^{'}} | ψ_{4} (s, y) |^{p^{'}})) d y d s ⩽ ε .$
(3.75)
On the other hand, by (3.62), (3.63), and (3.65), a compactness argument and a diagonal process, we find that, up to a subsequence, for every $n \in N$ ,
$ϕ_{m} \to ϕ in L^{2} (ι, ι + T; L^{2} (O_{n})),$
which implies that $ϕ_{m} (t, y) \to ϕ (t, y)$ for a.e. $(t, y) \in (ι, ι + T) \times O_{n}$ . Since $f (y, ϕ) + Z_{δ} (θ_{t} ω) ℏ (t, y, ϕ)$ is continuous in $ϕ$ , by (i) of Lemma 2.7, we get
$f (t, y, ϕ_{m}) ϕ_{m} + Z_{δ} (θ_{t} ω_{m}) ℏ (t, y, ϕ_{m}) \to f (t, y, ϕ) ϕ + Z_{δ} (θ_{t} ω) ℏ (t, y, ϕ)$
for a.e. $(t, y) \in (ι, ι + T) \times O_{n}$ . Now, using (3.3) and (3.5) again, we find that
$| ψ_{1} (t, y) | + c (| Z_{δ} (θ_{t} ω) |^{q^{'}} | ψ_{3} (t, y) |^{q^{'}} + | Z_{δ} (θ_{t} ω) |^{p^{'}} | ψ_{4} (t, y) |^{p^{'}}) - f (t, y, ϕ_{m}) ϕ_{m} - Z_{δ} (θ_{t} ω_{m}) ℏ (t, y, ϕ_{m}) ⩾ 0.$
Then, from Fatou’s lemma implies and Lemma 2.7(i) it follows that:
$\underset{m \to \infty}{lim sup} I_{42} (m) ⩽ 2 \int_{ι}^{t} e^{λ (s - t)} \int_{O_{n}} ((f (s, y, ϕ (s)) ϕ (s)) + Z_{δ} (θ_{t} ω_{m}) (ℏ (s, y, ϕ (s)) ϕ (s))) d y d s,$
(3.76)
which together with (3.75) implies that
$\underset{m \to \infty}{lim sup} I_{4} (m) ⩽ 2 \int_{ι}^{t} e^{λ (s - t)} \int_{O_{n}} ((f (s, y, ϕ (s)) ϕ (s)) + Z_{δ} (θ_{t} ω_{m}) (ℏ (s, y, ϕ (s)) ϕ (s))) d y d s + ε .$
(3.77)
Now taking the limit of (3.77) as $m \to \infty$ and using (3.70), (3.72), and (3.73), we conclude that
$\underset{m \to \infty}{lim sup} ‖ ϕ_{m} (t) ‖^{2} ⩽ ‖ ϕ (t) ‖^{2} + ε .$
Letting $ε \to 0$ , we can get what we want, which completes the proof.

Lemma 3.6
Suppose that (3.3)–(3.5) hold. Given $ι \in R$ and $ω \in Ω$ , let $ϕ (\cdot, ι, ω, ϕ_{ι, m})$ be a solution of problems (3.1) and (3.2) with initial data $ϕ_{ι, m}$ . If $ϕ_{ι, m} \to ϕ_{ι}$ weakly in $L^{2} (R^{d})$ , then problems (3.1) and (3.2) has a solution $ϕ (\cdot, ι, ω, ϕ_{ι})$ with initial condition $ϕ_{ι}$ at the initial time $ι$ such that, up to a subsequence
$ϕ (t, ι, ω, ϕ_{ι, m}) \to ϕ (t, ι, ω, ϕ_{ι}) weakly in L^{2} (R^{d}), \forall t ⩾ ι,$
and
$ϕ (\cdot, ι, ω, ϕ_{ι, m}) \to ϕ (\cdot, ι, ω, ϕ_{ι}) weakly in L^{2} (ι, ι + T; H^{β} (R^{d})), \forall T > 0,$

Proof.
Since $Ω = ⋃_{n = 1}^{\infty} Ω_{n}$ by (ii) of Lemma 2.7, then given $ω \in Ω$ there exists $n \in N$ such that $ω \in Ω_{n}$ . Therefore, the results follows by replacing $ω_{m}$ by $ω$ in Lemma 3.5. The proof is complete.
3.4. Generation of a Multivalued Random Dynamical System

We now define a multivalued non-autonomous random dynamical system for the given problems (3.1) and (3.2). In what follows, for $ι \in R$ , $ω \in Ω$ and $t ⩾ 0$ , we use $U (t + ι, ι, ω, ϕ_{ι})$ to represent the set of solutions to (3.1) and (3.2) over the interval [ $ι, \infty$ ) with the initial datum $ϕ_{ι} \in L^{2} (R^{d})$ :

U (t + ι, ι, θ_{- ι} ω, ϕ_{ι}) = {ϕ (t + ι, ι, θ_{- ι} ω, ϕ_{ι}) : ϕ is a solution of (3.1) and (3.2)} .

Subsequently, the set

U (t + ι, ι, θ_{- ι} ω, ϕ_{ι})

has the following properties.

Lemma 3.7
Suppose that assumptions (3.3)–(3.5) hold. Then $:$ (a)
For each $ι \in R$ and t $⩾ 0$ , the multivalued mapping $U (t + ι, ι, \cdot, \cdot) : Ω \times L^{2} (R^{d}) \to 2^{L^{2} (R^{d})}$ is measurable with respect to $B (Ω) \times B (L^{2} (R^{d})) .$
(b)
For every t $\in R^{+}, ι \in R$ and $ω \in Ω$ , the multivalued mapping $U (t + ι, ι, ω, \cdot) : L^{2} (R^{d}) \to 2^{L^{2} (R^{d})}$ is upper semicontinuous.
(c)
For every t $\in R^{+}$ , $ι \in R$ , $ω \in Ω$ , and $ϕ_{ι} \in L^{2} (R^{d})$ , the set $U (t + ι, ι, ω, ϕ_{ι})$ is closed in $L^{2} (R^{d})$ .

Proof.
(a) For each $n \in N$ fixed, Lemma 3.5(i) implies that $U (t + ι, ι, \cdot, \cdot$ ): $Ω_{n} \times L^{2} (R^{d}) \to 2^{L^{2} (R^{d})}$ is weakly upper semicontinuous. Then, by Lemma 2.4, $U (t + ι, ι, \cdot, \cdot)$ : $Ω_{n} \times L^{2} (R^{d}) \to 2^{L^{2} (R^{d})}$ is measurable with respect to $B (Ω_{n}) \times B (L^{2} (R^{d}))$ . According to Lemma 2.7(ii), we have that $Ω = ⋃_{n = 1}^{\infty} Ω_{n}$ and $Ω_{n}$ is measurable. Hence, we conclude that $U (t + ι, ι, \cdot, \cdot$ ): $Ω_{n} \times L^{2} (R^{d}) \to 2^{L^{2} (R^{d})}$ is measurable with respect to $B (Ω) \times B (L^{2} (R^{d}))$ . The properties (b) and (c) follow from Lemma 3.5 immediately. The proof is complete.
Now, we define a multivalued map $Ψ$ : $R^{+} \times R \times Ω \times L^{2} (R^{d}) \to 2^{L^{2} (R^{d})}$ as
$Ψ (t, ι, ω, ϕ_{ι}) = U (t + ι, ι, θ_{- ι}, ϕ_{ι}) = {ϕ (t + ι, ι, θ_{- ι} ω, ϕ_{ι}) : ϕ is a solution of (3.1) and (3.2)} .$
(3.78)
Since $θ_{- ι}$ : $Ω \to Ω$ is measurable, by (3.78) and Lemma 3.7, we know that for every $ι \in R$ and $t ⩾ 0$ , $Ψ (t, ι, \cdot, \cdot) : Ω \times L^{2} (R^{d}) \to 2^{L^{2} (R^{d})}$ is measurable. Clearly, $Ψ (0, ι, ω, \cdot)$ is the identity on $L^{2} (R^{d})$ and the non-autonomous cocycle property $Ψ (t + s, ι, ω, \cdot) = Ψ (t, ι + s, θ_{s} ω, \cdot) \circ Ψ (s, ι, ω, \cdot)$ also holds. Therefore, $Ψ$ is a multivalued non-autonomous cocycle on ( $Ω, F, P, {θ_{t}}_{t \in R}$ ).
4. Existence of Pullback Random Attractors

From now on, we denote by $D = {D = {D (ι, ω) : ι \in R, ω \in Ω}}$ , the collection of all tempered families of bounded nonempty subsets of $L^{2} (R^{d})$ , that is,

lim_{t \to - \infty} e^{μ t} ‖ D (ι + t, θ_{t} ω) ‖^{2} = 0, \forall ι \in R, ω \in Ω, μ > 0.

(4.1)

In the next subsections, we prove the existence and uniqueness of

D

-pullback random attractors for

Ψ

. For this purpose, we assume that the function

g

satisfies the following conditions:

\int_{- \infty}^{0} e^{λ t} (‖ g (r + ι, \cdot) ‖^{2} + ‖ ψ_{1} (r + ι, \cdot) ‖_{L^{1}}) d r < \infty, \forall ι \in R,

(4.2)

and

lim_{t \to - \infty} e^{ς} t \int_{- \infty}^{0} e^{λ t} (‖ g (r + t, \cdot) ‖^{2} + ‖ ψ_{1} (r + t, \cdot) ‖_{L^{1}}) d r = 0, \forall ς > 0.

4.1. Uniform Estimates of Solutions

In this subsection, we derive uniform estimates of the solutions to (3.1) and (3.2) in $L^{2} (R^{d})$ .

Lemma 4.1
Then for every $δ \in (0, 1]$ , $ι \in R, t ⩾ 0$ , $ω \in Ω$ , $D = {D = {D (ι, ω) : ι \in R, ω \in Ω}},$ there exists $T := T (ι, ω, D, δ) > 0$ such that for all $t ⩾ T$ and s $\in [- t, 0]$ , the solution of problems (3.1) and (3.2) satisfies
$\begin{aligned} ‖ ϕ (ι + s, ι - t, θ_{- ι} ω, ϕ_{ι - t}) ‖^{2} + C (d, β) \int_{ι - t}^{ι + s} & e^{λ (r - ι - s)} ‖ ϕ (r) ‖_{{\dot{H}}^{β}}^{2} d r \\ + & β_{1} \int_{ι - t}^{ι + s} e^{λ (r - ι - s)} ‖ ϕ (r) ‖_{L^{q}}^{q} d r ⩽ e^{- λ s} R (ι, ω), \end{aligned}$
where
$R (ι, ω) = M \int_{- \infty}^{0} e^{λ r} (‖ g (r + ι) ‖^{2} + ‖ ψ_{1} (r + ι) ‖_{L^{1}} + | Z_{δ} (θ_{r} ω) |^{q^{'}} + | Z_{δ} (θ_{r} ω) |^{p^{'}}) d r,$
(4.3)
and M $> 0$ is a constant independent of $ι, ω, D$ , and $δ$ .
Proof.
Integrating (3.56) over ( $ι - t, ι + s$ ) with t $⩾ 0$ and $s \in [- t, 0]$ , and then replacing $ω$ by $θ_{- ι} ω$ , we obtain
$\begin{aligned} ‖ ϕ (ι + s, ι - t, θ_{- ι} ω, ϕ_{ι - t}) ‖^{2} + C (d, β) \int_{ι - t}^{ι + s} e^{λ (r - ι - s)} ‖ ϕ (r) ‖_{{\dot{H}}^{β}}^{2} d r + β_{1} \int_{ι - t}^{ι + s} e^{λ (r - ι - s)} ‖ ϕ (r) ‖_{L^{q}}^{q} d r \\ ⩽ e^{- λ (t + s)} ‖ ϕ_{ι - t} ‖^{2} + c_{1} \int_{ι - t}^{ι + s} e^{λ (r - ι - s)} (‖ g (r) ‖^{2} + ‖ ψ_{1} (r) ‖_{L^{1}} + | Z_{δ} (θ_{r - ι} ω) |^{q^{'}} + | Z_{δ} (θ_{r - ι} ω) |^{p^{'}}) d r \\ ⩽ e^{- λ (t + s)} ‖ ϕ_{ι - t} ‖^{2} + c_{1} \int_{- t}^{s} e^{λ (r - s)} (‖ g (r + ι) ‖^{2} + ‖ ψ_{1} (r + ι) ‖_{L^{1}} + | Z_{δ} (θ_{r} ω) |^{q^{'}} + | Z_{δ} (θ_{r} ω) |^{p^{'}}) d r \\ ⩽ e^{- λ (t + s)} ‖ ϕ_{ι - t} ‖^{2} + c_{1} e^{- λ s} \int_{- \infty}^{0} e^{λ r} (‖ g (r + ι) ‖^{2} + ‖ ψ_{1} (r + ι) ‖_{L^{1}} + | Z_{δ} (θ_{r} ω) |^{q^{'}} + | Z_{δ} (θ_{r} ω) |^{p^{'}}) d r . \end{aligned}$
(4.4)
It follows from (2.1) and (4.2) that for every $ι, s \in R$ and $ω \in Ω$ ,
$\int_{- \infty}^{0} e^{λ r} (‖ g (r + ι) ‖^{2} + ‖ ψ_{1} (r + ι) ‖_{L^{1}} + | Z_{δ} (θ_{r} ω) |^{q^{'}} + | Z_{δ} (θ_{r} ω) |^{p^{'}}) d r < \infty .$
Since $ϕ_{ι - t} \in D (ι - t, θ_{- t} ω$ ) with D $\in D$ , it follows from (4.1) that
$e^{- λ t} ‖ ϕ_{ι - t} ‖^{2} ⩽ e^{- λ t} ‖ D (ι - t, θ_{- t} ω) ‖^{2} \to 0 as t \to \infty,$
this implies the desired conclusion.
4.2. $D$ -Pullback Asymptotic Compactness of Solutions

In this subsection, we prove the $D$ -pullback asymptotic compactness of solutions to (3.1) and (3.2) in $L^{2} (R^{d})$ .

Lemma 4.2
Suppose that assumptions (3.3)–(3.5) hold. Then, for any $ι \in R$ and $ω \in Ω$ , $D = {D = D (ι$ , $ω) : ι \in R, ω \in Ω}$ , $t_{m} \to \infty$ and $ϕ_{ι - t_{m}} \in D (ι - t_{m}, θ_{- t_{m}} ω)$ , the sequence ${ϕ (ι, ι - t_{m}, θ_{- ι} ω, ϕ_{ι - t_{m}})}$ has a convergent subsequence in $L^{2} (R^{d})$ .
Proof.
From Lemma 4.1 (with s = 0), we infer that ${ϕ (ι, ι - t_{m}, θ_{- ι} ω, ϕ_{ι - t_{m}})}$ is bounded in $L^{2} (R^{d})$ . Then there exists a subsequence satisfing,
$ϕ (ι, ι - t_{m}, θ_{- ι} ω, ϕ_{ι - t_{m}}) \to \bar{ϕ} weakly in L^{2} (R^{d}),$
(4.5)
so we have,
$‖ \bar{ϕ} ‖ ⩽ \underset{m \to \infty}{lim inf} ‖ ϕ (ι, ι - t_{m}, θ_{- ι} ω, ϕ_{ι - t_{m}}) ‖ .$
Thus, it is sufficient to show that
$\underset{n \to \infty}{lim sup} ‖ ϕ (ι, ι - t_{m}, θ_{- ι} ω, ϕ_{ι - t_{m}}) ‖ ⩽ ‖ \bar{ϕ} ‖ .$
By Lemma 4.1, there exists $m_{1} = m_{1} (ι, ω, D) ⩾ 1$ such that for all $m ⩾ m_{1}$ and $s \in [- t_{m}, 0]$ ,
$‖ ϕ (ι + s, ι - t_{m}, θ_{- ι} ω, ϕ_{ι - t_{m}}) ‖^{2} ⩽ e^{- λ s} R (ι, ω) .$
(4.6)
Let $k$ be a positive ingeter and $m_{2} = m_{2} (ι, ω, D, k) ⩾ m_{1}$ such that $t_{m} ⩾ k$ for all $m ⩾ m_{2}$ . It follows from (4.6) that for $m ⩾ m_{2}$ ,
$‖ ϕ (ι - k, ι - t_{m}, θ_{- ι} ω, ϕ_{ι - t_{m}}) ‖^{2} ⩽ e^{λ k} R (ι, ω),$
which implies that, for $k$ fixed, the sequence ${ϕ (ι - k, ι - t_{m}, θ_{- ι} ω, ϕ_{ι - t_{m}})}$ is bounded in $L^{2} (R^{d})$ , and hence, by a diagonal process we deduce that there exists a subsequence (not relabeled) and construct ${\bar{ϕ}}_{k} \in L^{2} (R^{d})$ for every $k \in N$ such that, as $m \to \infty$ ,
$ϕ (ι - k, ι - t_{m}, θ_{- ι} ω, ϕ_{ι - t_{m}}) \to {\bar{ϕ}}_{k} weakly in L^{2} (R^{d}) .$
We note that for s $\in [- k, 0]$ ,
$ϕ (ι + s, ι - t_{m}, θ_{- ι} ω, ϕ_{ι - t_{m}}) = ϕ (ι + s, ι - k, θ_{- ι} ω, ϕ (ι - k, ι - t_{m}, θ_{- ι} ω, ϕ_{ι - t_{m}})) .$
(4.7)
Hence, by Lemma 3.6, for every k $\in N$ , there exists a subsequence ( $t_{m^{'}}, ϕ_{ι - t_{m^{'}}})$ and a solution $ϕ_{k} (\cdot, ι - k, θ_{- ι} ω, {\bar{ϕ}}_{k}$ ) of (3.1) and (3.2) with initial data ${\bar{ϕ}}_{k}$ such that for all s $\in [- k, 0]$ , as $m^{'} \to \infty$ , we obtain
$ϕ (ι + s, ι - t_{m^{'}}, θ_{- ι} ω, ϕ_{ι - t_{m^{'}}}) \to ϕ_{k} (ι + s, ι - k, θ_{- ι} ω, {\bar{ϕ}}_{k}) weakly in L^{2} (R^{d}),$
(4.8)
and
$\begin{aligned} ϕ (\cdot, ι - k, θ_{- ι} ω, & ϕ (ι - k, ι - t_{m^{'}}, θ_{- ι} ω, ϕ_{ι - t_{m^{'}}})) \\ \to ϕ_{k} (\cdot, ι - k, θ_{- ι} ω, {\bar{ϕ}}_{k}) weakly in L^{2} ((ι - k, ι), H^{β} (R^{d})) . \end{aligned}$
Note that this subsequence $(t_{m}^{'}, ϕ_{ι - t_{m}^{'}})$ might be dependent on $k$ . Nevertheless, through an appropriate diagonal process, we can choose a subsequence (still denoted as $(t_{m}, ϕ_{ι - t_{m}})$ ) such that for all $k \in N$ and $s \in [- k, 0]$ , as $m \to \infty$ ,
$ϕ (ι + s, ι - t_{m}, θ_{- ι} ω, ϕ_{ι - t_{m}}) \to ϕ_{k} (ι + s, ι - k, θ_{- ι} ω, {\tilde{ϕ}}_{k}) weakly in L^{2} (R^{d}),$
(4.9)
and
$\begin{aligned} ϕ (\cdot, ι - k, θ_{- ι} ω, & ϕ (ι - k, ι - t_{m}, θ_{- ι} ω, ϕ_{ι - t_{m}})) \\ \to ϕ_{k} (\cdot, ι - k, θ_{- ι} ω, {\tilde{ϕ}}_{k}) weakly in L^{2} ((ι - k, ι), H^{β} (R^{d})) . \end{aligned}$
(4.10)
From (4.5), (4.7), and (4.9), when $s = 0$ , yields:
$\bar{ϕ} = ϕ_{k} (ι, ι - k, θ_{- ι} ω, {\bar{ϕ}}_{k}) .$
(4.11)
Let us denote $ϕ_{m, ι - k} = ϕ (ι - k, ι - t_{m}, θ_{- ι} ω, ϕ_{ι - t_{m}})$ and $ϕ_{m} (r) = ϕ (r, ι - k, θ_{- ι} ω, ϕ_{m, ι - k})$ . For convenience, we denote $ϕ_{k} (ι + s) = ϕ_{k} (ι + s, ι - k, θ_{- ι} ω, {\bar{ϕ}}_{k})$ . Using the energy equation (3.69) to $ϕ (ι, ι - k, θ_{- ι} ω, ϕ_{m, ι - k})$ and relation (4.7), we deduce that
$\begin{aligned} ‖ ϕ (ι, ι - t_{m}, θ_{- ι} ω, ϕ_{ι - t_{m}}) ‖^{2} = & e^{- λ k} ‖ ϕ_{m, ι - k} ‖^{2} - \int_{ι - k}^{ι} e^{λ (s - ι)} φ (ϕ_{m} (s)) d s \\ + 2 \int_{ι - k}^{ι} e^{λ (s - ι)} ⟨ g (s, y), ϕ_{m} (s) ⟩ d s \\ + 2 \int_{ι - k}^{ι} e^{λ (s - ι)} ⟨ f (s, y, ϕ_{m} (s)), ϕ_{m} (s) ⟩ d s \\ + 2 \int_{ι - k}^{ι} e^{λ (s - ι)} Z_{δ} (θ_{s} ω) ⟨ ℏ (s, y, ϕ_{m} (s)), ϕ_{m} (s) ⟩ d s \\ = & e^{- λ k} ‖ ϕ_{m, ι - k} ‖^{2} - \int_{- k}^{0} e^{λ s} φ (ϕ_{m} (s + ι)) d s \\ + 2 \int_{- k}^{0} e^{λ s} ⟨ g (s + ι, y), ϕ_{m} (s + ι) ⟩ d s \\ + 2 \int_{- k}^{0} e^{λ s} (⟨ f (s + ι, y, ϕ_{m} (s + ι)), ϕ_{m} (s + ι) ⟩ \\ + Z_{δ} (θ_{s} ω) ⟨ ℏ (s, y, ϕ_{m} (s)), ϕ_{m} (s + ι) ⟩) \\ ≜ & I_{1} (k, m) + I_{2} (k, m) + I_{3} (k, m) + I_{4} (k, m) . \end{aligned}$
(4.12)
We shall estimate all terms on the right hand side of (4.12). Using (4.4), we obtain for $m > m_{2}$ ,
$\begin{aligned} I_{1} (k, m) ⩽ e^{- λ t_{m}} ‖ & D (ι - t_{m}, θ_{- t_{m}} ω) ‖^{2} \\ + c_{1} \int_{- \infty}^{- k} e^{λ s} (‖ g (s + ι) ‖^{2} + ‖ ψ_{1} (s + ι) ‖_{L^{1}} + | Z_{δ} (θ_{s} ω) |^{q^{'}} + | Z_{δ} (θ_{s} ω) |^{p^{'}}) d s . \end{aligned}$
Since $D \in D,$ we deduce
$\underset{m \to \infty}{lim sup} I_{1} (k, m) ⩽ c_{1} \int_{- \infty}^{- k} e^{λ s} (‖ g (s + ι) ‖^{2} + ‖ ψ_{1} (s + ι) ‖_{L^{1}} + | Z_{δ} (θ_{s} ω) |^{q^{'}} + | Z_{δ} (θ_{s} ω) |^{p^{'}}) d s .$
(4.13)
Since $φ^{\frac{1}{2}} (\cdot)$ is an equivalent norm of $H^{β} (R^{d})$ , we can deduce from (3.60) that
$\underset{m \to \infty}{lim sup} I_{2} (k, m) ⩽ - \int_{- k}^{0} e^{λ s} φ (ϕ_{k} (s + ι)) d s .$
(4.14)
Using (4.10), we have
$lim_{m \to \infty} I_{3} (k, m) = 2 \int_{- k}^{0} e^{λ s} ⟨ g (s + ι, y), ϕ_{k} (s + ι) ⟩ d s .$
(4.15)
Furthermore, we find that for $n \in N$ ,
$\begin{aligned} I_{4} (k, m) = 2 \int_{- k}^{0} \int_{| y | ⩾ n} e^{λ s} (( & f (s + ι, y, ϕ_{k} (s + ι)) ϕ_{k} (s + ι)) \\ + Z_{δ} (θ_{t} ω) ℏ (s + ι, y, ϕ_{k} (s + ι)) ϕ_{k} (s + ι)) d y d s \\ + 2 \int_{- k}^{0} \int_{O_{n}} e^{λ s} (( & f (s + ι, y, ϕ_{k} (s + ι)) ϕ_{k} (s + ι)) \\ + Z_{δ} (θ_{t} ω) ℏ (s + ι, y, ϕ_{k} (s + ι)) ϕ_{k} (s + ι)) d y d s \\ ≜ I_{41} (k, m) + I_{42} (k & , m) . \end{aligned}$
By (3.3), (3.5), $ψ_{1} \in L_{loc}^{1} (R, L^{1} (R^{d}))$ , $ψ_{3} \in L^{\infty} (R, L^{q^{'}} (R^{d}))$ , and $ψ_{4} \in L^{\infty} (R$ , $L^{p^{'}} (R^{d}))$ , we find that for every $ϱ > 0$ , there exists $K_{3} = K_{3} (ϱ) ⩾ 1$ such that for $n ⩾ K_{3}$ ,
$\underset{n \to \infty}{lim sup} I_{41} (k, m) ⩽ ϱ .$
(4.16)
Similar to (3.76), by a diagonal process, one can verify that, up to a subsequence, for every $n \in N$ ,
$\begin{aligned} \underset{m \to \infty}{lim sup} I_{42} (k, m) ⩽ 2 \int_{- k}^{0} \int_{O_{n}} e^{λ s} (( & f (s + ι, y, ϕ_{k} (s + ι)) ϕ_{k} (s + ι)) \\ + Z_{δ} (θ_{t} ω) ℏ (s + ι, y, ϕ_{k} (s + ι)) ϕ_{k} (s + ι)) d y d s, \end{aligned}$
(4.17)
which along with (4.16) shows that for $n ⩾ K_{3}$ ,
$\begin{aligned} \underset{m \to \infty}{lim sup} I_{4} (k, m) ⩽ 2 \int_{- k}^{0} \int_{O_{n}} e^{λ s} (( & f (s + ι, y, ϕ_{k} (s + ι)) ϕ_{k} (s + ι)) \\ + Z_{δ} (θ_{t} ω) ℏ (s + ι, y, ϕ_{k} (s + ι)) ϕ_{k} (s + ι)) d y d s + ϱ . \end{aligned}$
(4.18)
First letting $n \to \infty$ , and then letting $ϱ \to 0$ , we get
$\begin{aligned} \underset{n \to \infty}{lim sup} I_{4} (k, m) ⩽ 2 \int_{- k}^{0} \int_{R^{d}} e^{λ s} (( & f (s + ι, y, ϕ_{k} (s + ι)) ϕ_{k} (s + ι)) \\ + Z_{δ} (θ_{t} ω) ℏ (s + ι, y, ϕ_{k} (s + ι)) ϕ_{k} (s + ι)) d y d s . \end{aligned}$
(4.19)
Inserting the estimates (4.13)–(4.15) and (4.19) into (4.12), yields
$\begin{aligned} \underset{m \to \infty}{lim sup} ‖ ϕ (ι, ι - t_{m}, θ_{- ι} ω, ϕ_{ι - t_{m}}) ‖^{2} \\ ⩽ & c_{1} \int_{- \infty}^{- k} e^{λ s} (‖ g (s + ι) ‖^{2} + ‖ ψ_{1} (s + ι) ‖_{L^{1}} + | Z_{δ} (θ_{s} ω) |^{q^{'}} + Z_{δ} (θ_{s} ω) |^{p^{'}}) d s - \int_{- k}^{0} e^{λ s} φ (ϕ_{k} (s + ι)) d s \\ + 2 \int_{- k}^{0} e^{λ s} ⟨ g (s + ι, y), ϕ_{k} (s + ι) ⟩ d s + 2 \int_{- k}^{0} e^{λ s} (⟨ f (s + ι, y, ϕ_{k} (s + ι)), ϕ_{k} (s + ι) ⟩ d s \\ + Z_{δ} (θ_{t} ω) ⟨ ℏ (s + ι, y, ϕ_{k} (s + ι)), ϕ_{k} (s + ι) ⟩) d s . \end{aligned}$
(4.20)
Appling the energy equation (3.69) to $\bar{ϕ} = ϕ_{k} (ι, ι - k, θ_{- ι} ω, {\bar{ϕ}}_{k})$ , we deduce that
$\begin{aligned} ‖ \bar{ϕ} ‖^{2} = & e^{- λ k} ‖ {\bar{ϕ}}_{k} ‖^{2} - \int_{ι - k}^{ι} e^{λ (s - ι)} φ (ϕ_{k} (s)) d s \\ + 2 \int_{s - ι}^{ι} e^{λ (s - ι)} ⟨ g (s, y), ϕ_{k} (s) ⟩ d s \\ + 2 \int_{s - ι}^{ι} e^{λ (s - ι)} ⟨ f (s, y, ϕ_{k} (s)), ϕ_{k} (s) ⟩ d s \\ + 2 \int_{s - ι}^{ι} e^{λ (s - ι)} Z_{δ} (θ_{t} ω) ⟨ ℏ (s, y, ϕ_{k} (s)), ϕ_{k} (s) ⟩ d s \\ = & e^{- λ k} ‖ {\bar{ϕ}}_{k} ‖^{2} - \int_{- k}^{0} e^{λ s} φ (ϕ_{k} (s + ι)) d s \\ + 2 \int_{- k}^{0} e^{λ s} ⟨ g (s + ι, y), ϕ_{k} (s + ι) ⟩ d s \\ + 2 \int_{- k}^{0} e^{λ s} ⟨ f (s + ι, y, ϕ_{k} (s + ι)), ϕ_{k} (s + ι) ⟩ d s \\ + 2 \int_{- k}^{0} e^{λ s} Z_{δ} (θ_{t} ω) ⟨ ℏ (s + ι, y, ϕ_{k} (s + ι)), ϕ_{k} (s + ι) ⟩ d s . \end{aligned}$
(4.21)
Combining with (4.20) and (4.21), we conclude that
$\begin{aligned} \underset{m \to \infty}{lim sup} ‖ ϕ (ι, ι - t_{m}, θ_{- ι} ω, ϕ_{ι - t_{m}}) ‖^{2} \\ ⩽ ‖ \bar{ϕ} ‖^{2} + c_{1} \int_{- \infty}^{- k} e^{λ s} (‖ g (s + ι) ‖^{2} + ‖ ψ_{1} (s + ι) ‖_{L^{1}} + | Z_{δ} (θ_{s} ω) |^{q^{'}} + | Z_{δ} (θ_{s} ω) |^{p^{'}}) d s . \end{aligned}$
Letting $k \to \infty$ , we get
$\underset{m \to \infty}{lim sup} ‖ ϕ (ι, ι - t_{m}, θ_{- ι} ω, ϕ_{ι - t_{m}}) ‖^{2} ⩽ ‖ \bar{ϕ} ‖^{2} .$
The proof is complete.

As an immediate consequence of Lemma 4.2, we have the $D$ -pullback asymptotic compactness of $Ψ$ .
Lemma 4.3
Under the assumptions of Lemma 4.2, the multivalued cocycle $Ψ$ is $D$ -pullback asymptotically compact.
Proof.
Given $ι \in R, D = {D = {D (ι, ω) : ι \in R, ω \in Ω}}$ , $t_{m} \to \infty$ and ${\tilde{ϕ}}_{m} \in Ψ (t_{m}, ι - t_{m}, D (ι - t_{m}, θ_{- t_{m}} ω))$ . Then, there exists $ϕ_{ι - t_{m}} \in D (ι - t_{m}, θ_{- t_{m}} ω)$ and a sequence ${ϕ (\cdot, ι - t_{m}, θ_{- t_{m}} ω, ϕ_{ι - t_{m}})}$ of solutions of problems (3.1) and (3.2) such that
${\tilde{ϕ}}_{m} = ϕ (ι, ι - t_{m}, θ_{- ι} ω, ϕ_{ι - t_{m}}) .$
(4.22)
By Lemma 4.1, the sequence ${ϕ (ι, ι - t_{m}, θ_{- t_{m}} ω, ϕ_{ι - t_{m}})}$ has a convergent subsequence in $L^{2} (R^{d})$ . Therefore, by (4.22), we conclude that ${{\tilde{ϕ}}_{m}}$ has a convergent subsequence in $L^{2} (R^{d})$ . The proof is complete.
4.3. Pullback Random Attractors

Corollary 4.4

Suppose that assumption (3.4). Then the multivalued non-autonomous random dynamical system $Ψ$ has a closed $D$ -pullback absorbing set the family $K \in D$ defined by the following equation:

K (ι, ω) = {ϕ \in L^{2} (R^{d}) : ‖ ϕ ‖^{2} ⩽ R (ι, ω), \forall ι \in R, ω \in Ω},

(4.23)

where

R (ι, ω)

is given by (4.3).

Proof.

By Lemma 4.1 and Gu & Wang (2020: Lemma 4.2) we can verify that $K$ is a closed $D$ -pullback absorbing set of $Ψ$ .

To establish the existence of

D

-pullback random attractors for problem (1.1), the following weakly upper semicontinuity of

Ψ

is required.

Lemma 4.5

Let $K$ be the closed $D$ -pullback absorbing set given by (4.23). Then the mapping $Ψ (t, ι, \cdot, K (ι, \cdot)) : Ω_{n} \to 2^{L^{2} (R^{d})}$ is weakly upper semicontinuous for all $n \in N, t \in R^{+}$ and $ι \in R$ .

Proof.

The proof follows a similar approach to that of Gu & Wang (2020) by employing Lemmas 2.7 and 3.6, and hence we do not repeat that.

We are now in a position to establish and prove the main result on the existence of a $D$ -pullback random attractor in $L^{2} (R^{d})$ .

Theorem 4.6

Suppose that assumptions (3.3)–(3.5) hold. Then there exist the unique $D$ -pullback random attractor $A = {A (ι, ω) : ι \in R$ , $ω \in Ω}$ for the multivalued cocycle $Ψ$ .

Proof.

Since the multivalued cocycle $Ψ$ is $D$ -pullback asymptotically compact by Lemma 4.3, upper semicontinuous by Lemma 3.7 and has a closed $D$ -pullback absorbing set $K \in D$ by Corollary 4.4, and $Ψ (t, ι, \cdot, K (ι, \cdot)) : Ω_{n} \to 2^{L^{2} (R^{d})}$ is weakly upper semicontinuous by Lemma 4.5. Consequently, we derive the existence and uniqueness of the $D$ -pullback random attractor following by Theorem 2.6. The proof is complete.

Remark 4.7

The above result also applies to problems (3.1) and (3.2) with $\vec{α} = \vec{0}$ . Thus, the $D$ -pullback random attractor for the cocycle generated by (3.1) and (3.2) with $\vec{α} = \vec{0}$ exists and it is the unique $D$ -pullback random attractor for the following non-autonomous fractional reaction–diffusion equation driven by superlinear colored noise defined on $R^{d}$ :

\begin{aligned} {\begin{cases} ϕ_{t} + (- Δ)^{β} ϕ + λ ϕ = f (t, y, ϕ) + g (t, y) + ℏ (t, y, ϕ) Z_{δ} (θ_{t} ω), y \in R^{d}, t > ι, \\ ϕ (ι, y) = ϕ_{ι} (y) . \end{cases} \end{aligned}

Remark 4.8

The above result also applies to problems (3.1) and (3.2) when $β = 1$ or when the fractional Laplacian operator $(- Δ)^{β} ϕ$ is replaced by a wide class of non-local fractional Laplacian operators defined by the following equation:

L_{K^{β}} ϕ (y) = P.V. \int_{R^{d}} (ϕ (y) - ϕ (x)) K^{β} (y, x) d x, y \in R^{d},

where

K^{β} : R^{d} \times R^{d} \to R^{+}

with

β \in (0, 1)

is a kernel function such that

Singularity: $K^{β} (y, x)$ has singularity at $y = x$ .

Symmetry: $K^{β} (y, x) = K^{β} (x, y)$ for all $y, x \in R^{d}$ with $y \neq x$ .

Translation invariance: $K^{β} (y + z, x + z) = K^{β} (y, x)$ for any $y, x, z \in R^{d}$ and $y \neq x$ .

Growth property: there exists a constant $C_{K^{β}} > 0$ such that for all $y, x \in R^{d}$ and $y \neq x$ ,

C_{K^{β}}^{- 1} | y - x |^{- d - β p} ⩽ K^{β} (y, x) ⩽ C_{K^{β}} | y - x |^{- d - β p} .

Continuity: $y \to K^{β} (y, x)$ is continuous in $R^{d} ∖ {x}$ for any $x \in R^{d}$ .

Footnotes

Acknowledgements

This work was supported by National Natural Science Foundation of China (No. 12301299), the Guiyang City Science and Technology Plan Project (No. [2024]2-17), Natural Science Research Project of Guizhou Provincial Department of Education (Nos. QJJ[2023]011; QJJ[2024]322), the Young Elite Scientist Sponsorship Program by GAST(GASTYESS202403), the Foundation of National Key Laboratory of Computational Physics (SYSQN2024-18), and the research fund of Qiankeheping-tairencai-YSZ[2022]002.

Declaration of Conflicting Interests

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

Funding

The author(s) received no financial support for the research, authorship and/or publication of this article.

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