Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 40, pp. 1–16.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu

DOI: 10.58997/ejde.2025.40

RANDOM ATTRACTORS AND THEIR STABILITY FOR NONCLASSICAL

DIFFUSION EQUATIONS DRIVEN BY ADDITIVE WHITE NOISE WITH

DELAY AND INTENSITY

WENHUI MA, QIAOZHEN MA

Abstract. In this article, we study the asymptotic behavior of solutions of nonclassical diffu-
sion equation driven by an additive noise with delay and intensity ϵ ∈ (0, 1] on Rn. We first

establish the existence and uniqueness of tempered pullback random attractors for the equations

in C([−ρ, 0], H1(Rn)), and then the upper semicontinuity of random attractors is also obtained
when the intensity of noise approaches zero. It’s worth mentioning that the Arzela-Ascoli theo-

rem, spectral decomposition, and uniform tail-estimates have been utilized to demonstrate the

asymptotic compactness of the solutions.

1. Introduction

In the real world applications, differential equations are influenced by stochastic perturbations,
stochastic environments and stochastic boundary conditions. Since these factors cannot be ignored,
we incorporate them in the corresponding deterministic models, so that stochastic differential
equations are used. We consider the following initial value problem for nonclassical diffusion
equation driven by the additive noise with delay and intensity ϵ on Rn:

ut −∆ut + λu−∆u = N(t, x, u(t, x)) + f(t, x, u(t− ρ, x)) + g(t, x) + ϵh(x)Ẇ ,

uτ (s, x) := u(τ + s, x) = ϕ(s, x), s ∈ [−ρ, 0], x ∈ Rn, t > τ.
(1.1)

Here λ > 0 is a constant, τ ∈ R, ϵ ∈ (0, 1], ρ > 0 is the delay time of the system, h ∈ H1(Rn) ∩
Lp(Rn) with p ⩾ 2, g ∈ L2

loc(R, L2(Rn)) is a non-autonomous deterministic forcing term, the
nonlinear functions N, f : R × Rn × R → R have polynomial growth of certain order, the initial
data ϕ ∈ C([−ρ, 0], H1(Rn)) and W is a two-side real-valued Wiener process on a probability
space.

Throughout this article, we assume that the nonlinearity N, f : R × Rn × R → R satisfy the
following conditions: for all t, u, u1, u2 ∈ R and x ∈ Rn,

N(t, x, u)u ≤ −α1|u|p + β1(t, x), β1 ∈ L1
loc(R, L1(Rn)), (1.2)

|N(t, x, u)| ≤ α2|u|p−1 + β2(t, x), β2 ∈ Lp1

loc(R, L
p1(Rn)), (1.3)

∂

∂u
N(t, x, u) ≤ −α3|u|p−2 + β3(t, x), β3 ∈ L∞

loc(R, L∞(Rn)), (1.4)

|f(t, x, u1)− f(t, x, u2)| ≤ ϖf (t, x)|u1 − u2|,
f(t, x, 0) = 0, ϖf ∈ L∞

loc(R, L∞(Rn)),
(1.5)

where α1, α2, α3, p are positive constants with 2 ≤ p <∞, and p1 = p
p−1 .

Problem (1.1), as a nonclassical diffusion equation, is well known for its mathematical and
physical significance in viscoelasticity and pressure of the medium. It is usually utilized in the

2020 Mathematics Subject Classification. 35B40, 35B41, 35R60, 37L55.
Key words and phrases. Pullback random attractors; nonclassical diffusion equation; nonlinear delay;

additive white noise.
©2025. This work is licensed under a CC BY 4.0 license.
Submitted October 2, 2024. Published April 11, 2025.

1



2 W. MA, Q. MA EJDE-2025/40

various fields, including non-Newtonian fluid mechanics, solid mechanics, and heat conduction
theory (see [1, 2, 4, 6, 10, 25]).

As ϵ tends to zero, it is easy to see that problem (1.1) becomes deterministic nonclassical
diffusion equation with delay. Of course, in this case, the change of the current state for the
system depends not only on its present state but also on its state at a certain time in the past.

About the deterministic case, the dynamics of nonclassical diffusion equation on bounded do-
mains or unbounded domains have been extensively studied by several authors in [3, 5, 7, 8, 9,
10, 11, 12, 13, 14, 22, 26, 27]. For instance, for problem ut − ∆ut − ∆u = f(t, u(t − ρ)) + g(t),
Hu and Wang [11] proposed a new method to test the asymptotic compactness of the solutions
and investigated the existence of pullback attractors in CH1

0 (Ω) and CH2(Ω)∩H1
0 (Ω), where ρ is a

delay function and f contains some memory effects in a fixed time interval. Harraga and Yebdri
[12] analyzed the existence of solutions for a nonclassical reaction-diffusion equation with critical
nonlinearity, a time-dependent force with exponential growth and delayed force term, where the
delay term can be entrained by a function under assumptions of measurability. They proved the
existence of the pullback D-attractors in H1

0 (Ω).
As far the stochastic case, Zhao and Song [27] verified the existence and the upper semi-

continuity of random attractors in H1(Rn) for ut − ∆ut − ∆u + u + f(x, u) = g(x) + ϵhẆ .
Later, Chen, Wang et al. [10] studied the long-time dynamics of fractional nonclassical diffusion
equations with nonlinear colored noise and delay on unbounded domains, and they proved the
existence and uniqueness of pullback random attractors in C([−ρ, 0], Hα(Rn)) (α ∈ (0, 1)), the
asymptotic compactness of the solutions was derived by virtue of the arguments of Arzela-Ascoli
theorem, spectral decomposition as well as uniform tail-estimates.

We note that the existence of random attractors for stochastic PDEs driven by additive or
linear multiplicative noise have been extensively studied in the recent years (see [9, 10, 17, 20,
19, 24, 25, 28, 29, 30]). Moreover, random attractors of stochastic equations driven by nonlinear
white noise have been investigated in [15, 19, 23]. However, as far as the author is aware, there
are still many problems to be be studied on random attractors of nonclassical diffusion equation;
so we are going to continue investigating this problem.

The first purpose of this paper is to establish the existence and uniqueness of pullback random
attractor for the nonclassical diffusion equation (1.1) with delay and intensity ϵ in C([−ρ, 0], H1(Rn)),
and then we are concerned with the upper semicontinuity of random attractors when the intensity
of noise approaches zero. Of course, we need to overcome the following two difficulties for solving
foregoing problem:

(1) Since equation (1.1) contains the term −∆ut, it is different from the usual reaction-
diffusion equation essentially. That is, the weakly dissipativeness of the nonclassical dif-
fusion equation, which implies that if the initial datum belongs to H1(Rn), the solution is
always in H1(Rn) and has no regularity at least higher than H1(Rn) available, which is
similar to the hyperbolic case.

(2) It is known that the existence of attractor depends on some compactness. The acquisition
of compactness on bounded domains can use a prior estimate along with Sobolev embed-
ding, while on unbounded domains, Sobolev embedding is non-compact, which is overcome
by the “tail” estimate of solutions or the energy equation approach. In this paper, we will
use the Arzela-Ascoli theorem, the uniform tail-estimates and the spectral decomposition
to prove the pullback asymptotic compactness of the solutions in C([−ρ, 0], H1(Rn)).

For convenience, we give some notation which will be used throughout this paper. Without loss
of generality, L2(Rn) is equipped with inner product (·, ·) and the norm ∥ · ∥. H1(Rn) is equipped
with the inner product (u, v)H1(Rn) = (u, v)L2(Rn) + (∇u,∇v)L2(Rn) and the norm ∥u∥H1(Rn) =
∥u∥L2(Rn) + ∥∇u∥L2(Rn). The norm of Lp(Rn) is denoted as ∥ · ∥p for p > 2. We denote by

C([−ρ, 0], H1(Rn)) with ρ > 0 the space of all continuous functions from [−ρ, 0] to H1(Rn) with
norm

∥u∥C([−ρ,0],H1(Rn)) = sup
s∈[−ρ,0]

∥u(s)∥H1(Rn), ∀u ∈ C([−ρ, 0], H1(Rn)).



EJDE-2025/40 NONCLASSICAL DIFFUSION EQUATIONS 3

We use the symbols c and ci to represent positive constants, whose values may vary from line to
line.

This article is organized as follows: In Section 2, we review some basic concepts on the pull-
back random attractor. In Section 3, we obtain the existence of a continuous cocycle. In Section
4, we establish the uniform estimates of solutions for (1.1). In Section 5, we obtained the exis-
tence and uniqueness of the pullback random attractor in C([−ρ, 0], H1(Rn)). Finally, the upper
semicontinuity of random attractors is also obtained when the intensity of noise approaches zero.

2. Preliminaries

In this section, we iterate some basic conclusions on pullback random attractor for nonau-
tonomous random dynamical system coming from [16, 17]. Let (Ω,F ,P, {θt}t∈R) be a metric
dynamical system, where Ω = {ω ∈ C(R,R) : ω(0) = 0} with the open compact topology, F is
the Borel σ-algebra of Ω, P represents the Wiener measure, and {θt}t∈R is the measure-preserving
transformation group on Ω given by

θtω(·) = ω(·+ t)− ω(t), ω ∈ Ω, t ∈ R.

Suppose W be a two-sided real-valued Wiener process on (Ω,F ,P), and define a random variable
y : Ω → R by

y(θtω) = −
∫ 0

−∞
es(θtω)(s)ds.

Then y is the unique stationary solution of the one-dimensional Ornstein-Uhlenbeck equation
dy + ydt = dW . Note that there exists a subset of full probability measure (still denoted by Ω)

such that for all ω ∈ Ω, y(θtω) is continuous in t ∈ R and limt→±∞
y(θtω)

t = 0.
Let (X, d) be a complete separable metric space with Borel σ-algebra B(X), the collection of

all subsets of X is denoted by 2X . Suppose D be a collection of some families of nonempty subsets
of X.

Definition 2.1. A mapping Φ : R+ × R × Ω × X → X is called a continuous non-autonomous
random dynamical system (continuous cocycle) on X over (Ω,F ,P, {θt}t∈R) if for all τ ∈ R, ω ∈ Ω
and t, s ∈ R+,

(i) Φ(·, τ, ·, ·) : R+ × Ω×X → X is (B(R+)×F × B(X),B(X))-measurable;
(ii) Φ(0, τ, ω, ·) is the identity on X;
(iii) Φ(t+ s, τ, ω, ·) = Φ(t, τ + s, θsω, ·) ◦ Φ(s, τ, ω, ·);
(iv) Φ(t, τ, ω, ·) : X → X is continuous.

A function Φ is said to be T -periodic if there exists a positive number T such that for every
t ∈ R+, τ ∈ R and ω ∈ Ω,

Φ(t, τ + T, ω, ·) = Φ(t, τ, ω, ·).

Definition 2.2. Let K = {K(τ, ω) : τ ∈ R, ω ∈ Ω} ∈ D be a family of nonempty closed subsets of
X, then K is called a D-pullback absorbing set for Φ if for all τ ∈ R, ω ∈ Ω and for every D ∈ D,
there exists T = T (D, τ, ω) > 0 such that

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

If K is measurable with respect to F in Ω, then K is called a closed measurable D-pullback
absorbing set of Φ.

Definition 2.3. A non-autonomous random dynamical system Φ is said to be D-pullback asymp-
totically compact in X if for all τ ∈ R, ω ∈ Ω, and any sequences tn → +∞, xn ∈ D(τ−tn, θ−tnω),
the sequence {Φ(tn, τ − tn, θ−tnω, xn)}∞n=1 has a convergent subsequence in X.

Definition 2.4. A family A = {A(τ, ω) : τ ∈ R, ω ∈ Ω} ∈ D is called a D-pullback attractor of Φ
if for all t ∈ R+, τ ∈ R and ω ∈ Ω, the following conditions hold:

(i) A is measurable with respect to F in Ω and A(τ, ω) is compact;
(ii) A is invariant: Φ(t, τ, ω,A(τ, ω)) = A(t+ τ, θtω);



4 W. MA, Q. MA EJDE-2025/40

(iii) A attracts every member D of D:

lim
t→+∞

d(Φ(t, τ − t, θ−tω,D(τ − t, θ−tω)),A(τ, ω)) = 0,

where d(·, ·) is the Hausdorff semi-distance in X.

A is called a periodic pullback attractor with period T if, in addition,

A(τ + T, ω) = A(τ, ω), for all τ ∈ R and ω ∈ Ω.

We have the following abstract result for the continuous non-autonomous random dynamical sys-
tem which can be found in [16, 17].

Proposition 2.5. Let D be an inclusion-closed collection of families of nonempty subsets of X,
and Φ be a continuous non-autonomous random dynamical system on X over (Ω,F ,P, {θt}t∈R).
Then Φ has a D-pullback attractor A in D if and only if

(i) Φ is D-pullback asymptotically compact in X;
(ii) Φ has a closed measurable D-pullback absorbing set K in D.

The attractor A is unique and given by the ω-limit of K,

A(τ, ω) = ∩r≥0∪t≥rΦ(t, τ − t, θ−tω,K(τ − t, θ−tω)).

If, in addition, both Φ and K are T -periodic, then so is the attractor A.

3. Existence of a continuous cocycle

In this section, we establish the existence of a continuous cocycle for (1.1) on the whole space
Rn. We will convert the nonclassical diffusion equations (1.1) driven by additive white noise with
intensity ϵ and delay into a deterministic one, and then obtain the existence of random attractor
for such deterministic system parametrized by ω ∈ Ω. For this purpose, we introduce the notation

(I −∆)z(θtω) = h(x)y(θtω), (3.1)

it is easy to show that

(I −∆)dz(θtω) + (I −∆)z(θtω)dt = h(x)dW. (3.2)

Given τ ∈ R, t ≥ τ, ω ∈ Ω and ϕ ∈ C([−ρ, 0], H1(Rn)), if u = u(t, τ, ω, ϕ) is a solution of (1.1),
then we introduce a new variable v = v(t, τ, ω, ψ) by

v(t, τ, ω, ψ) = u(t, τ, ω, ϕ)− ϵz(θtω), t ∈ R, ϵ ∈ (0, 1]. (3.3)

In terms of (1.1) and (3.3) we see that for t > τ ,

vt −∆vt + λv −∆v = N(t, x, v(t, x) + ϵz(θtω)) + f(t, x, v(t− ρ, x) + ϵz(θt−ρω))

+ g(t, x) + ϵ(1− λ)z(θtω)), x ∈ Rn, t > τ,
(3.4)

with initial condition

vτ (s, x) := v(τ + s, x) = ϕ(s, x)− ϵz(θτ+sω) := ψ(s, x), x ∈ Rn, s ∈ [−ρ, 0]. (3.5)

We will first prove the existence and uniqueness of solutions for problem (3.4)-(3.5), and then
obtain the solutions of (1.1) via the transform (3.3).

Definition 3.1. For τ ∈ R, ω ∈ Ω, s ∈ [−ρ, 0], ϵ ∈ (0, 1] and ψ ∈ C([−ρ, 0], H1(Rn)), a function
v(·, τ, ω, ψ) : [τ − ρ,∞) → H1(Rn) is called a solution of the nonclassical diffusion equations
(3.4)-(3.5) with intensity ϵ and delay if vτ (·, τ, ω, ψ) = ψ and

v(·, τ, ω, ψ) ∈ C([τ − ρ,∞), H1(Rn)) ∩ Lp(τ, τ + T ;Lp(Rn)),

dv(t, τ, ω, ψ)

dt
∈ L2(τ, τ + T ;H−1(Rn)) + Lp1(τ, τ + T ;Lp1(Rn)),



EJDE-2025/40 NONCLASSICAL DIFFUSION EQUATIONS 5

and v satisfies, for every ϑ ∈ H1(Rn) ∩ Lp(Rn) and ξ ∈ C∞
0 (τ, τ + T ),

−
∫ τ+T

τ

(v(t), ϑ)H1(Rn)ξ
′(t)dt+

∫ τ+T

τ

(∇v,∇ϑ)ξ(t)dt+ λ

∫ τ+T

τ

(v(t), ϑ)ξ(t)dt

=

∫ τ+T

τ

(g(t), ϑ)ξ(t)dt+

∫ τ+T

τ

∫
Rn

N(t, x, v(t, x) + ϵz(θtω))ϑξ(t)dxdt

+

∫ τ+T

τ

∫
Rn

f(t, x, v(t− ρ) + ϵz(θt−ρω))ϑξ(t)dxdt+ ϵ(1− λ)

∫ τ+T

τ

(z(θtω), ϑ)ξ(t)dt.

(3.6)

Under the assumptions (1.2)-(1.5), by using the standard Galerkin method as in [21] (see
also [20]), we can prove that for every τ ∈ R, ω ∈ Ω and ψ ∈ C([−ρ, 0], H1(Rn)), the nonclas-
sical diffusion equation (3.4)-(3.5) with intensity ϵ and delay has a unique continuous solution
v(·, τ, ω, ψ) : [τ − ρ,∞) → H1(Rn) in the sense of Definition 3.1 such that v(·, τ, ω, ψ) is continu-
ous in ψ and is (F ,B(C([−ρ, 0], H1(Rn)))-measurable in ω. Moreover, the solution v satisfies the
energy equation: for almost all t ≥ τ ,

1

2

d

dt
∥v(t, τ, ω, ψ)∥2H1(Rn) + λ∥v∥2 + ∥∇v∥2

= (g(t), v) +

∫
Rn

N(t, x, u(t))vdx+

∫
Rn

f(t, x, u(t− ρ))vdx+ ϵ(1− λ)(z(θtω), v).
(3.7)

Now by solution v of (3.4)-(3.5) and the transform (3.3), we obtain a solution u of the stochastic
equation (1.1) which is given by

u(t, τ, ω, ϕ) = v(t, τ, ω, ψ) + ϵz(θtω) with ϕ = ψ + ϵz(θτ+sω).

Therefore, we find that u(t, τ, ω, ϕ) is both continuous in t and in ϕ ∈ C([−ρ, 0], H1(Rn)).
Moreover, u(t, τ, ·, ϕ) : Ω → C([−ρ, 0], H1(Rn)) is measurable. Then we can define a contin-
uous cocycle in C([−ρ, 0], H1(Rn)) associated with the solutions of problem (1.1). Let Φϵ :
R+ × R× Ω× C([−ρ, 0], H1(Rn)) → C([−ρ, 0], H1(Rn)) be a mapping given as follows, for every
t ∈ R+, τ ∈ R, ω ∈ Ω and ϕ ∈ C([−ρ, 0], H1(Rn)),

Φϵ(t, τ, ω, ϕ) = ut+τ (·, τ, θ−τω, ϕ) = v(t+ τ + s, τ, θ−τω, ψ) + ϵz(θt+τ+sω). (3.8)

Let D = {D(τ, ω) ⊆ C([−ρ, 0], H1(Rn)) : τ ∈ R, ω ∈ Ω} be a family of bounded nonempty
subsets of C([−ρ, 0], H1(Rn)). A family D is called tempered if for every τ ∈ R, ω ∈ Ω,

lim
t→−∞

eγt∥D(τ + t, θtω)∥C([−ρ,0],H1(Rn)) = 0, ∀γ > 0, (3.9)

where ∥D∥C([−ρ,0],H1(Rn)) = supu∈D ∥u∥C([−ρ,0],H1(Rn)). From now on, we will use D to denote the

collection of all tempered families of bounded nonempty subsets of C([−ρ, 0], H1(Rn)):

D = {D = {D(τ, ω) ⊆ C([−ρ, 0], H1(Rn)) : τ ∈ R, ω ∈ Ω} : D satisfies (3.9)}.

Next, we show some uniform estimates to obtain the existence of a D-pullback absorbing set, and
then verify the asymptotic compactness of solutions. We suppose that

λ >
4
√
6

3
∥ϖf∥L∞(R,L∞(Rn)). (3.10)

Furthermore, we will assume that for every τ ∈ R,∫ 0

−∞
eµr(∥g(r + τ, ·)∥2 + ∥β1(r + τ, ·)∥L1(Rn) + ∥β2(r + τ, ·)∥p1

Lp1 (Rn))dr <∞. (3.11)

Sometimes, we also assume g, β1, β2 are tempered in the following sense: for every γ > 0,

lim
t→+∞

e−γt

∫ 0

−∞
eµr(∥g(r − t, ·)∥2 + ∥β1(r − t, ·)∥L1(Rn) + ∥β2(r − t, ·)∥p1

Lp1 (Rn))dr = 0. (3.12)

Note that these conditions do not require g to be bounded in C([−ρ, 0], H1(Rn)) when t→ +∞.



6 W. MA, Q. MA EJDE-2025/40

4. Uniform estimates of solutions

In this section, some uniform estimates of solutions for (1.1) are achieved, which are crucial
for constructing the D-pullback absorbing sets and D-pullback asymptotic compactness for the
continuous cocycle Φϵ defined by (3.8).

Lemma 4.1. Suppose (1.2)-(1.5), (3.11), (3.12) are satisfied. Let σ, τ ∈ R, ω ∈ Ω, s ∈ [−ρ, 0], ϵ ∈
(0, 1] and D = {D(τ, ω) : τ ∈ R, ω ∈ Ω} ∈ D. Then there exists T = T (τ, ω,D, σ) such that for all
t ≥ T , the solution of problem (3.4)-(3.5) satisfies

∥v(σ + s, τ − t, θ−τω, ψ)∥2C([−ρ,0],H1(Rn)) + α1

∫ σ

τ−t

eµ(r−σ)∥u(r)∥pLp(Rn)dr

≤ Q

∫ 0

−∞
eµr(∥g(r + τ, ·)∥2 + ∥β1(r + τ, ·)∥L1(Rn)

+ ϵ∥β2(r + τ, ·)∥p1

Lp1 (Rn) + ϵ|y(θrω)|p + 1)dr + sup
−ρ≤s≤0

|y(θsω)|2,

(4.1)

where Q > 0 is a constant independent of τ, ω,D and ψ ∈ D(τ − t, θ−tω).

Proof. We estimate all the terms on the right-hand side of energy equation (3.7). First, thanks to
(1.2), (1.3), (3.1) and Young’s inequality, we obtain

∫
Rn

N(t, x, u(t))vdx =

∫
Rn

N(t, x, u(t))u(t)dx− ϵ

∫
Rn

z(θtω)N(t, x, u(t))dx

≤ −α1∥u(t)∥pLp(Rn) + ∥β1(t)∥L1(Rn)

+ ϵα2

∫
Rn

|hy(θtω)| · |u(t)|p−1dx+ ϵ

∫
Rn

|hy(θtω)|β2(t, x)dx

≤ −α1

2
∥u(t)∥pLp(Rn) + ∥β1(t)∥L1(Rn) + ϵ∥β2(t)∥p1

Lp1 (Rn) + c1ϵ|y(θtω)|p.

(4.2)

Second, due to (1.5)-(3.1) and Young’s inequality, we have

(g(t), v) + ϵ(1− λ)(z(θtω), v) ≤
λ

16
∥v∥2 + 8

λ
∥g(t)∥2 + c2ϵ|y(θtω)|2, (4.3)∫

Rn

f(t, x, u(t− ρ))v(t)dx ≤ λ

4
∥v(t)∥2 +

∥ϖf (t)∥2L∞(Rn)

λ

∫
Rn

|v(t− ρ) + z(θt−ρω)|2dx

≤ λ

4
∥v(t)∥2 +

2∥ϖf (t)∥2L∞(Rn)

λ
∥v(t− ρ)∥2 + c3|y(θt−ρω)|2.

(4.4)

It follows from (4.2)-(4.4) with µ = min{1, λ} that

d

dt
∥v(t, τ, ω, ψ)∥2H1(Rn) + µ∥v(t)∥2H1(Rn) + α1∥u(t)∥pLp(Rn)

≤ −3λ

8
∥v(t)∥2 +

4∥ϖf (t)∥2L∞(Rn)

λ
∥v(t− ρ)∥2

+ c5(∥β1(t)∥L1(Rn) + ϵ∥β2(t)∥p1

Lp1 (Rn) + ϵ|y(θtω)|p + ∥g(t)∥2 + |y(θt−ρω)|2 + 1).

(4.5)



EJDE-2025/40 NONCLASSICAL DIFFUSION EQUATIONS 7

Multiplying (4.5) by eµt and then integrating the inequality on (τ − t, σ + s) with σ > τ − t+ ρ,
we obtain

eµ(σ+s)∥v(σ + s, ω)∥2H1(Rn) + α1

∫ σ+s

τ−t

eµr∥u(r, ω)∥pLp(Rn)dr

≤ eµ(τ−t)∥ψ∥2C([−ρ,0],H1(Rn)) −
3λ

8

∫ σ+s

τ−t

eµr∥v(r, ω)∥2dr

+
4∥ϖf∥2L∞(R,L∞(Rn))

λ

∫ σ+s

τ−t

eµr∥v(r − ρ, ω)∥2dr

+ c5

∫ σ+s

τ−t

eµr
(
∥β1(r)∥L1(Rn) + ϵ∥β2(r)∥p1

Lp1 (Rn) + ϵ|y(θrω)|p

+ ∥g(r)∥2 + |y(θr−ρω)|2 + 1
)
dr.

(4.6)

Replacing ω by θ−τω in the above leads to

eµ(σ+s)∥v(σ + s, θ−τω)∥2H1(Rn) + α1

∫ σ+s

τ−t

eµr∥u(r, θ−τω)∥pLp(Rn)dr

≤ eµ(τ−t)∥ψ∥2C([−ρ,0],H1(Rn)) −
3λ

8

∫ σ+s

τ−t

eµr∥v(r, θ−τω)∥2dr

+
4∥ϖf∥2L∞(R,L∞(Rn))

λ

∫ σ+s

τ−t

eµr∥v(r − ρ, θ−τω)∥2dr

+ c5

∫ σ+s

τ−t

eµr
(
∥β1(r)∥L1(Rn) + ϵ∥β2(r)∥p1

Lp1 (Rn) + ϵ|y(θr−τω)|p

+ ∥g(r)∥2 + |y(θr−τ−ρω)|2 + 1
)
dr.

(4.7)

We now deal with the third term on the right-hand side of (4.7),∫ σ+s

τ−t

eµr∥v(r − ρ, τ − t, θ−τω, ψ)∥2dr =
∫ σ+s−ρ

τ−t−ρ

eµ(r+ρ)∥v(r, τ − t, θ−τω, ψ)∥2dr

=

∫ τ−t

τ−t−ρ

eµ(r+ρ)∥v(r, τ − t, θ−τω, ψ)∥2dr +
∫ σ+s−ρ

τ−t

eµ(r+ρ)∥v(r, τ − t, θ−τω, ψ)∥2dr

≤ 1

µ
eµ(τ−t+ρ)∥ψ∥2C([−ρ,0],H1(Rn)) + eµρ

∫ σ+s

τ−t

eµr∥v(r, τ − t, θ−τω, ψ)∥2dr.

(4.8)

By (4.7), (4.8) and (3.10) we obtain

∥v(σ + s, θ−τω)∥2H1(Rn) + α1

∫ σ+s

τ−t

eµ(r−σ−s)∥u(r, θ−τω)∥pLp(Rn)dr

≤ c4e
µ(τ−t−σ−s+ρ)∥ψ∥2C([−ρ,0],H1(Rn))

+ (
4∥ϖf∥2L∞(R,L∞(Rn))

λ
eµρ − 3λ

8
)

∫ σ+s

τ−t

eµ(r−σ−s)∥v(r, θ−τω)∥2dr

+ c5

∫ σ+s

τ−t

eµ(r−σ−s)(∥β1(r)∥L1(Rn) + ϵ∥β2(r)∥p1

Lp1 (Rn) + ϵ|y(θr−τω)|p

+ ∥g(r)∥2 + |y(θr−τ−ρω)|2 + 1)dr

≤ c4e
µ(τ−t−σ−s+ρ)∥ψ∥2C([−ρ,0],H1(Rn)) + c5

∫ σ+s

τ−t

eµ(r−σ−s)
(
∥β1(r)∥L1(Rn)

+ ϵ∥β2(r)∥p1

Lp1 (Rn) + ϵ|y(θr−τω)|p + ∥g(r)∥2 + |y(θr−τ−ρω)|2 + 1
)
dr,

(4.9)



8 W. MA, Q. MA EJDE-2025/40

which combined with the fact that s ∈ [−ρ, 0] yields

∥v(σ + s, τ − t, θ−τω, ψ)∥2H1(Rn) + α1

∫ σ+s

τ−t

eµ(r−σ)∥u(r, τ − t, θ−τω, ϕ)∥pLp(Rn)dr

≤ c4e
µ(τ−t−σ)∥ψ∥2C([−ρ,0],H1(Rn)) + c5

∫ σ−τ

−t

eµ(r−σ+τ)(∥β1(r + τ)∥L1(Rn)

+ ϵ∥β2(r + τ)∥p1

Lp1 (Rn) + ϵ|y(θrω)|p + ∥g(r + τ)∥2 + |y(θr−ρω)|2 + 1)dr.

(4.10)

Note that g ∈ L2
loc(R, L2(Rn)), using (3.11) and the continuity of y(θtω), it is clear that for every

σ, τ ∈ R, ω ∈ Ω with σ > τ − t+ ρ,∫ σ−τ

−t

eµ(r−σ+τ)(∥g(r + τ)∥2 + ∥β1(r + τ)∥L1(Rn)

+ ϵ∥β2(r + τ)∥p1

Lp1 (Rn) + ϵ|y(θrω)|p + 1)dr

≤
∫ σ−τ

−∞
eµ(r−σ+τ)(∥g(r + τ)∥2 + ∥β1(r + τ)∥L1(Rn) + ϵ∥β2(r + τ)∥p1

Lp1 (Rn)

+ ϵ|y(θrω)|p + 1)dr <∞.

(4.11)

Furthermore, ψ ∈ D(τ − t, θ−tω) with D ∈ D, as t→ ∞,

eµ(τ−t−σ)∥ψ∥2C([−ρ,0],H1(Rn)) ≤ eµ(τ−t−σ)∥D(τ − t, θ−tω)∥2C([−ρ,0],H1(Rn)) → 0. (4.12)

According to (4.10)-(4.12), we find that there exists T = T (τ, ω,D, σ) such that for all t ≥ T ,

∥vσ(s, τ − t, θ−τω, ψ)∥2C([−ρ,0],H1(Rn)) + α1

∫ σ

τ−t

eµ(r−σ)∥u(r, τ − t, θ−τω, ϕ)∥pLp(Rn)dr

≤ c5

∫ σ−τ

−t

eµ(r−σ+τ)(∥β1(r + τ)∥L1(Rn)

+ ϵ∥β2(r + τ)∥p1

Lp1 (Rn) + ϵ|y(θrω)|p + ∥g(r + τ)∥2 + |y(θr−ρω)|2 + 1)dr

≤ c5

∫ 0

−∞
eµ(r)(∥β1(r + τ)∥L1(Rn) + ϵ∥β2(r + τ)∥p1

Lp1 (Rn)

+ ϵ|y(θrω)|p + ∥g(r + τ)∥2 + 1)dr + sup
−ρ≤s≤0

|y(θsω)|2,

(4.13)

which completes the proof. □

Lemma 4.2. Suppose (1.2)-(1.5) hold. Then for every τ ∈ R, ω ∈ Ω, ϵ ∈ (0, 1] and D = {D(τ, ω) :
τ ∈ R, ω ∈ Ω} ∈ D, the solution of problem (3.4)-(3.5) satisfies

∥ d
dt
v(t, τ − t, θ−τω, ψ)∥2H1(Rn) ≤ Q1(∥v(t, τ − t, θ−τω, ψ)∥2H1(Rn) + ∥g(t)∥2 + ϵ|y(θt−τω)|2

+ ∥u(t− ρ, τ − t, θ−τω, ϕ)∥2H1(Rn) + 1),
(4.14)

where Q1 > 0 is a constant independent of τ, ω,D and ψ ∈ D(τ − t, θ−tω).

Proof. Taking the inner product (3.4) with dv
dt , we find

∥ d
dt
v(t, τ − t, θ−τω, ψ)∥2H1(Rn) + λ(v, vt) + (∇v,∇vt)

= (N(t, x, u(t, x)), vt) + (f(t, x, u(t− ρ, x)), vt) + ϵ(1− λ)(z(θtω), vt) + (g(t), vt).
(4.15)

Next, we estimate the terms of (4.15). By (1.3) and Young’s inequality we have

(N(t, x, u(t, x)), vt) ≤
∫
Rn

|N(t, x, u(t, x)||vt|dx

≤
∫
Rn

(α2|u|p−1 + β2(t, x))|vt|dx

≤ 1

16
∥vt∥2 + c∥u(t)∥2p−2

L2p−2 + c∥β2(t)∥2.

(4.16)



EJDE-2025/40 NONCLASSICAL DIFFUSION EQUATIONS 9

From ϖf ∈ L∞
loc(R, L∞(Rn)) and Young’s inequality, we arrive at

(f(t, x, u(t− ρ, x)), vt) ≤
∫
Rn

|f(t, x, u(t− ρ, x))||vt|dx ≤
∫
Rn

ϖf (x)|u(t− ρ)||vt|dx

≤ 1

16
∥vt∥2 + c∥ϖf (t)∥L∞(Rn)∥u(t− ρ)∥2L2(Rn)

≤ 1

16
∥vt∥2 + c∥u(t− ρ)∥2H1(Rn).

(4.17)

Also from the Young’s inequality we have

− λ(v, vt)− (∇v,∇vt) + ϵ(1− λ)(z(θtω), vt) + (g(t), vt)

≤ 5

8
∥vt∥2 +

3

4
∥∇vt∥2 + c∥v(t)∥2H1(Rn) + c∥g(t)∥2 + cϵ|y(θtω)|2.

(4.18)

Therefore, from (4.16)-(4.18) by substituting τ and ω with τ − t and θ−τω it follows that

1

4
∥ d
dt
v(t, τ − t, θ−τω, ψ)∥2H1(Rn) ≤ c(∥v(t, τ − t, θ−τω, ψ)∥2H1(Rn) + ∥g(t)∥2 + ϵ|y(θt−τω)|2

+ ∥u(t− ρ, τ − t, θ−τω, ϕ)∥2H1(Rn) + 1),
(4.19)

Finally, it is easy to obtain the desired result (4.14). This proof is complete. □

Next, we derive the uniform tail-estimates of the solutions. For every x ∈ Rn, k ∈ N, let

ϱk(x) = ϱ( |x|k ), where ϱ ∈ C1(R+, [0, 1]) is an increasing smooth function satisfying

ϱ(s) ≡

{
0, ∀s ∈ [0, 12 ];

1, ∀s ∈ [1,∞).
(4.20)

We denote

Ok = {x ∈ Rn : |x| < k}, Oc
k = Rn −Ok. (4.21)

Lemma 4.3. Suppose (1.2)-(1.5) and (3.11) hold. Then for every τ ∈ R, ω ∈ Ω, s ∈ [−ρ, 0],
D = {D(τ, ω) : τ ∈ R, ω ∈ Ω} ∈ D and ψ ∈ D(τ − t, θ−tω), the solution of problem (3.4)-(3.5)
satisfies

lim
k,t→+∞

∫
Oc

k

∥v(τ + s, τ − t, θ−τω, ψ)∥2H1(Rn)dx = 0. (4.22)

Proof. With the help of smooth functions we prove this lemma. First of all, multiplying (3.4) by

ϱ( |x|k )v and integrating over Rn, we find that

1

2

d

dt

∫
Rn

ϱ(
|x|
k
)∥v(t, τ, ω, ψ)(x)∥2H1(Rn)dx+ µ

∫
Rn

ϱ(
|x|
k
)∥v(t, τ, ω, ψ)(x)∥2H1(Rn)dx

=

∫
Rn

ϱ(
|x|
k
)g(t, x)vdx+ ϵ(1− λ)

∫
Rn

ϱ(
|x|
k
)z(θtω)vdx

+

∫
Rn

ϱ(
|x|
k
)N(t, x, v(t) + ϵz(θtω))vdx+

∫
Rn

ϱ(
|x|
k
)f(t, x, v(t− ρ) + ϵz(θt−ρω))vdx.

(4.23)

Next, we now estimate all the terms on the right-hand side of (4.23). For the first term, we obtain
from Young’s inequality that∫

Rn

ϱ(
|x|
k
)g(t, x)vdx ≤ λ

32

∫
Rn

ϱ(
|x|
k
)|v|2dx+

8

λ

∫
Rn

ϱ(
|x|
k
)|g(t, x)|2dx. (4.24)

For the second term, by the continuity of z(θtω) and Young’s inequality we know that

ϵ(1− λ)

∫
Rn

ϱ(
|x|
k
)z(θtω)vdx ≤ c

∫
Rn

ϱ(
|x|
k
)|v|2dx+ cϵ2

∫
Rn

ϱ(
|x|
k
)|y(θtω)|2dx. (4.25)



10 W. MA, Q. MA EJDE-2025/40

For the third term, we conclude from (1.2)- (1.3) that∫
Rn

ϱ(
|x|
k
)N(t, x, v + ϵz(θtω))vdx

≤ −α1

∫
Rn

ϱ(
|x|
k
)|v + ϵz(θtω)|pdx+

∫
Rn

ϱ(
|x|
k
)|β1(t, x)|dx

+ α2|ϵz(θtω)|
∫
Rn

ϱ(
|x|
k
)|v + ϵz(θtω)|p−1dx+ |ϵz(θtω)|

∫
Rn

ϱ(
|x|
k
)|β2(t, x)|dx

≤ −α1

2

∫
Rn

ϱ(
|x|
k
)|v + ϵz(θtω)|pdx+ c

∫
Rn

ϱ(
|x|
k
)(|β1(t, x)|+ |β2(t, x)|p1)dx+ c.

(4.26)

For the last term, we deduce from (1.5) that∫
Rn

ϱ(
|x|
k
)f(t, x, v(t− ρ) + ϵz(θt−ρω))v(t)dx

≤ λ

4

∫
Rn

ϱ(
|x|
k
)|v|2dx+

∥ϖf (t)∥2L∞(Rn)

λ

∫
Rn

ϱ(
|x|
k
)|v(t− ρ) + ϵz(θt−ρω)|2dx

≤ λ

4

∫
Rn

ϱ(
|x|
k
)|v|2dx+

2∥ϖf (t)∥2L∞(Rn)

λ

∫
Rn

ϱ(
|x|
k
)(|v(t− ρ)|2|+ ϵ2|y(θt−ρω)|2)dx.

(4.27)

Substituting (4.24)-(4.27) into (4.23), we arrive at

d

dt

(∫
Rn

ϱ(
|x|
k
)∥v(t, τ, ω, ψ)∥2H1(Rn)dx

)
+ 2µ

(∫
Rn

ϱ(
|x|
k
)∥v(t, τ, ω, ψ)∥2H1(Rn)dx

)
≤ 9λ

16

∫
Rn

ϱ(
|x|
k
)|v|2dx+

4∥ϖf (t)∥2L∞(Rn)

λ

∫
Rn

ϱ(
|x|
k
)(|v(t− ρ)|2|+ ϵ2|y(θt−ρω)|2)dx

+ c

∫
|x|≥ k

2

ϱ(
|x|
k
)(|β1(t, x)|+ |β2(t, x)|p1 + |g(t, x)|2)dx.

(4.28)

Multiplying (4.28) by e2µt and integrating over (τ − t, τ + s) for any fixed s ∈ [−ρ, 0] with t > ρ,
we replace ω by θ−τω in the resulting inequality and by a similar calculations with (4.6)-(4.7), we
achieve from (3.10) and the properties of ϱk(x) that∫

Rn

ϱ(
|x|
k
)∥v(τ + s, τ − t, θ−τω, ψ)(x)∥2H1(Rn)dx

≤ e−2µ(τ+s)∥ψ∥2C([−ρ,0],H1(Rn))

+ c

∫ τ+s

τ−t

e2µ(r−τ−s)(∥v(r)∥2 + ∥v(r − ρ)∥2 + ϵ2|y(θr−ρω)|2)dr

+ c

∫ 0

−∞
e2µ(r−τ−s)

∫
|x|≥ k

2

(|β1(r + τ, x)|+ |β2(r + τ, x)|p1 + |g(r + τ, x)|2) dx dr.

(4.29)

Due to ψ ∈ D(τ − t, θ−tω) with D ∈ D, (3.10), the continuity of z(θtω) and Lemma 4.2, there
exists T = T (τ, ω,D) such that for all t ≥ T and s ∈ [−ρ, 0],∫

Rn

ϱ(
|x|
k
)∥v(τ + s, τ − t, θ−τω, ψ)(x)∥2H1(Rn)dx→ 0 as k → ∞,

which means that

lim
k,t→+∞

∫
Rn

ϱ(
|x|
k
)∥v(τ + s, τ − t, θ−τω, ψ)(x)∥2H1(Rn)dx = 0. (4.30)

Finally, by (4.20), (4.21) and (4.30) it is easy to obtain the desired result (4.22). □

To obtain the pullback asymptotic compactness of solutions in H1(Rn), we also need to de-
rive the uniform estimates of solutions on bounded domains. For every x ∈ Rn, k ∈ N, de-

note ǔ(t, τ, ω, ϕ̌)(x) = ξk(x)u(t, τ, ω, ϕ)(x), where ξk(x) = 1 − ϱ( |x|k ), then for k ∈ N, x ∈ Oc
k,



EJDE-2025/40 NONCLASSICAL DIFFUSION EQUATIONS 11

ǔ(t, τ, ω, ϕ̌)(x) = 0; for some constant c > 0 independent of k, ∥ǔ∥H1(Rn) ≤ c∥u∥H1(Rn), where
solution ǔ satisfies problem (1.1).

Consider the eigenvalue problem

−∆u = µu in Ok and u = 0in Oc
k. (4.31)

Apparently, this eigenvalue problem has a family of eigenfunctions {ej}∞j=1 such that {ej}∞j=1 form

an orthonormal basis of H = {u ∈ L2(Rn) : u = 0 on Oc
k}, the corresponding family of eigenvalues

{µj}∞j=1 satisfies

0 < µ1 ≤ µ2 ≤ · · · ≤ µj → ∞ as j → ∞.

Given n ∈ N, let Xn = span{ej : j = 1, . . . , n} and Πn : H → Xn be the canonical projection
operator. By[10, Lemma 4.4] or [20, Lemma 4.3], we have a certain estimate in H1(Rn), that is,
for every τ ∈ R, ω ∈ Ω, s ∈ [−ρ, 0], D = {D(τ, ω) : τ ∈ R, ω ∈ Ω} ∈ D and ψ ∈ D(τ − t, θ−tω), the
solution of problem (3.4)-(3.5) satisfies

lim
n→∞,t→∞

∥(I −Πn)ξkv(τ + s, τ − t, θ−τω, ψ)∥H1(Rn) = 0, for each k ∈ N. (4.32)

5. Existence of pullback random attractors

In this section, we first give some uniform estimates to obtain the existence of a D-pullback
absorbing set, and then establish the asymptotic compactness of solutions To that end, we prove
the existence of D-pullback random attractor of Φϵ generated by (3.8).

Lemma 5.1. Suppose (1.2)-(1.5) and (3.11)-(3.12) hold. Then the continuous cocycle Φϵ associ-
ated with (1.1) has a closed D-pullback absorbing set Bϵ ∈ D:

Bϵ(τ, ω) = {u ∈ C([−ρ, 0], H1(Rn)) : ∥u∥2C([−ρ,0],H1(Rn)) ≤ QRϵ(τ, ω), τ ∈ R, ω ∈ Ω},

where Q > 0 is a positive constant independent of τ, ω and D, Rϵ(τ, ω) is given by

Rϵ(τ, ω) = c

∫ 0

−∞
eµr(∥β1(r + τ)∥L1(Rn) + ϵ∥β2(r + τ)∥p1

Lp1 (Rn) + ϵ|y(θrω)|p

+ ∥g(r + τ)∥2 + 1)dr + sup
−ρ≤s≤0

|y(θsω)|2.
(5.1)

Proof. From (3.3), for all τ ∈ R, ω ∈ Ω, s ∈ [−ρ, 0], ϵ ∈ (0, 1] and σ > τ − t+ ρ, we obtain that

u(σ + s, τ − t, θ−τω, ϕ) = v(σ + s, τ − t, θ−τω, ψ) + ϵz(θσ+s−τω), (5.2)

where ϕ(r) = ψ(r) + ϵz(θr−τω). Hence, combining with the conclusion of Lemma 4.1 we know
that

∥uσ(s, τ − t, θ−τω, ψ)∥2C([−ρ,0],H1(Rn)) + α1

∫ σ

τ−t

eµ(r−σ)∥u(r, τ − t, θ−τω, ϕ)∥pLp(Rn)dr

≤ c5

∫ σ−τ

−∞
eµ(r+τ−σ)(∥β1(r + τ)∥L1(Rn) + ϵ∥β2(r + τ)∥p1

Lp1 (Rn) + ϵ|y(θrω)|p

+ ∥g(r + τ)∥2 + 1)dr + sup
−ρ≤s≤0

|y(θσ+s−τω)|2.

(5.3)

From (5.1) and (5.3) it follows that

∥u∥2C([−ρ,0],H1(Rn)) ≤MR(τ, ω). (5.4)

Therefore, in line with (3.8) and (5.4) we claim that for all t ≥ T ,

Φϵ(t, τ − t, θ−tω,D(τ − t, θ−tω)) = uτ (s, τ − t, θ−tω,D(τ − t, θ−tω)) ⊆ Bϵ(τ, ω),

which means that Bϵ is a pullback absorbing set. It remains to show Bϵ ∈ D, i.e., Bϵ is tempered,
which satisfies for given γ > 0,

lim
t→−∞

eγtRϵ(τ + t, θtω) = 0. (5.5)



12 W. MA, Q. MA EJDE-2025/40

In terms of (5.1) we have

Rϵ(τ + t, θtω) =

∫ 0

−∞
eµr(∥β1(r + τ + t)∥L1(Rn) + ϵ∥β2(r + τ + t)∥p1

Lp1 (Rn)

+ ϵ|y(θr+tω)|p + ∥g(r + τ + t)∥2 + 1)dr + sup
−ρ≤s≤0

|y(θs+tω)|2.
(5.6)

Let χ = min{µ, λ}, by a simple calculation we have∫ 0

−∞
ϵeµχ(|y(θr+tω)|p + 1) < +∞, (5.7)

which along with (3.12) implies that

lim
t→−∞

eγtRϵ(τ + t, θtω)

≤ eγτ lim
t→−∞

eγt
∫ 0

−∞
eµr(∥β1(r + t)∥L1(Rn)

+ ϵ∥β2(r + t)∥p1

Lp1 (Rn) + ϵ|y(θr+tω)|p + ∥g(r + t)∥2)dr

+ lim
t→−∞

∫ −t

−∞
ϵeµχ(|y(θrω)|p + 1)dr + lim

t→−∞
eγt sup

−ρ≤s≤0
|y(θs+tω)|2 = 0.

(5.8)

As a result, we obtain from (5.8) that for every γ > 0,

lim
t→−∞

eγt∥Bϵ(τ + t, θtω)∥C([−ρ,0],H1(Rn)) = lim
t→−∞

eγt/2
√
M lim

t→−∞
(eγtRϵ(τ + t, θtω))

1/2 = 0,

which implies Bϵ ∈ D. As we explain before, note that Rϵ(τ, ω) is measurable in ω ∈ Ω, and so is
Bϵ(τ, ω). □

In what follows, we use the Arzela-Ascoli theorem to prove the asymptotic compactness of the
continuous cocycle Φϵ in C([−ρ, 0], H1(Rn)).

Lemma 5.2. Suppose (1.2)-(1.5) and (3.11)-(3.12) hold. Then the continuous cocycle Φϵ associ-
ated with (1.1) is D-pullback asymptotically compact in C([−ρ, 0], H1(Rn)).

Proof. Given τ ∈ R, ω ∈ Ω, s ∈ [−ρ, 0] and D ∈ D, we need to prove that sequences

{Φϵ(tn, τ − tn, θ−tnω, ϕn)}∞n=1 = {u(τ + s, τ − tn, θ−τω, ϕn)}∞n=1

has a convergent subsequence in C([−ρ, 0], H1(Rn)) whenever tn → ∞ and ϕn ∈ D(τ − tn, θ−tnω).
Firstly, we claim that {uτ (·, τ−tn, θ−τω, ϕn)}∞n=1 is uniformly equicontinuous. In fact, it follows

from (3.3), Lemma 4.1, Lemma 4.2 and that g ∈ L2
loc(R, L2(Rn)) there exists T1 = T1(τ, ω, ϵ) ≥ 1

and Q2 = Q2(τ, ω, ϵ) > 0 such that for all n ≥ T1,∫ τ

τ−ρ

∥ d
dr
v(r, τ − tn, θ−τω, ψn)∥2H1(Rn)dr ≤ Q2. (5.9)

By (5.9) and Hölder inequality, we infer that for each n ≥ T1 and s1, s2 ∈ [−ρ, 0],

∥u(τ + s2, τ − tn, θ−τω, ϕn)− u(τ + s1, τ − tn, θ−τω, ϕn)∥H1(Rn)

= ∥
∫ τ+s2

τ+s1

d

dr
u(r, τ − tn, θ−τω, ϕn)dr∥H1(Rn)

≤ |s2 − s1|1/2(
∫ τ+s2

τ+s1

∥ d
dr
u(r, τ − tn, θ−τω, ϕn)∥2H1(Rn)dr)

1/2

≤ |s2 − s1|1/2(
∫ τ

τ−ρ

∥ d
dr
u(r, τ − tn, θ−τω, ϕn)∥2H1(Rn)dr)

1/2 ≤
√
Q2|s2 − s1|1/2.

(5.10)

As s2 − s1 tends to 0, (5.10) approaches 0, which means that {uτ (·, τ − tn, θ−τω, ϕn)}∞n=1 is
uniformly equicontinuous in C([−ρ, 0], H1(Rn)).



EJDE-2025/40 NONCLASSICAL DIFFUSION EQUATIONS 13

Next, we show that {u(τ + s, τ − tn, θ−τω, ϕn)}∞n=1 is precompact in H1(Rn) for every fixed
s ∈ [−ρ, 0]. Thanks to Lemma 4.3 and (3.3), there exist η > 0, T2 = T2(τ, ω,D, η) ≥ 1 and
k0 = k0(τ, η) such that for all n ≥ T2 and s ∈ [−ρ, 0],∫

Oc
k0

|uτ (s, τ − tn, θ−τω, ϕn)|2H1(Rn)dx <
η2

2
. (5.11)

Therefore, from (5.11) that for all n ≥ T2 and s ∈ [−ρ, 0], we have

∥uτ (s, τ − tn, θ−τω, ϕn)∥2H1(Oc
k0

)dx < ϵ. (5.12)

Moreover, for every s ∈ [−ρ, 0], the sequence {uτ (s, τ − tn, θ−τω, ϕn)}∞n=1 has a finite ϵ-net in
H1(Ok0

), which along with (5.12) shows that for every s ∈ [−ρ, 0], the sequence {u(τ + s, τ −
tn, θ−τω, ϕn)}∞n=1 has a finite 2ϵ-net in H1(Rn).

According to Arzela-Ascoli theorem, we conclude that the continuous cocycle Φϵ associated
with (1.1) is D-pullback asymptotically compact in C([−ρ, 0], H1(Rn)). □

Theorem 5.3. Suppose (1.2)-(1.5) and (3.11)-(3.12) hold. Then the continuous cocycle Φϵ as-
sociated with (1.1) has a unique D-pullback random attractor Aϵ = {Aϵ(τ, ω) : τ ∈ R, ω ∈
Ω} ∈ D in C([−ρ, 0], H1(Rn)). If, in addition, for each fixed x ∈ Rn and u ∈ R, all functions
N(t, x, u), f(t, x, u), g(t, x), β1(t, x) and β2(t, x) are T -periodic in t ∈ R, then so is the attractor
Aϵ, i.e., Aϵ(τ + T, ω) = Aϵ(τ, ω) for all τ ∈ R and ω ∈ Ω.

Proof. The existence and uniqueness of the D-pullback attractor Aϵ follows from proposition 2.5
immediately based on Lemmas 5.1 and 5.2. Note that N(t, x, u), f(t, x, u), g(t, x), β1(t, x) and
β2(t, x) are T -periodic in t ∈ R, in this case, the continuous cocycle Φϵ corresponding to the
solution operator of problem (1.1) is also T -periodic, i.e., Φϵ(t, τ + T, ω, ϕ) = Φϵ(t, τ, ω, ϕ) for
all t ∈ R+, τ ∈ R, ω ∈ Ω and ϕ ∈ C([−ρ, 0], H1(Rn)). Furthermore, by (5.1) we obtain that
Rϵ(τ +T, ω) = Rϵ(τ, ω) if g(t, x), β1(t, x) and β2(t, x) are T -periodic in t ∈ R, which together with
Lemma 5.1 implies that the absorbing set Bϵ is also T -periodic, i.e., Bϵ(τ + T, ω) = Bϵ(τ + T, ω)
for all τ ∈ R and ω ∈ Ω. Therefore, the T -periodicity of Aϵ follows from proposition 2.5 in terms
of the T -periodicity of Φϵ and Bϵ. □

6. Stability of attractors with respect to perturbation parameters

In this section, we consider the limiting behavior of the pullback random attractors Aϵ of
problem (1.1) as the intensity of noise ϵ → 0. Throughout the paper, we assume ϵ ∈ (0, 1], and
write the cocycle of problem (1.1) as Φϵ to indicate its dependence on ϵ. Then Φϵ has a tempered
pullback attractor Aϵ by Theorem 5.3, and has a tempered pullback absorbing set Bϵ by Lemma
5.1. Given τ ∈ R, ω ∈ Ω, let

R(τ, ω) = c

∫ 0

−∞
eµr(∥β1(r + τ)∥L1(Rn) + ∥β2(r + τ)∥p1

Lp1 (Rn) + |y(θrω)|p

+ ∥g(r + τ)∥2 + 1)dr + sup
−ρ≤s≤0

|y(θsω)|2

and

B(τ, ω) = {u ∈ C([−ρ, 0], H1(Rn)) : ∥u∥2C([−ρ,0],H1(Rn)) ≤ QR(τ, ω)}.
By Lemma 5.1, for all τ ∈ R, ω ∈ Ω, we have

∪0<ϵ<1Aϵ(τ, ω) ⊆ ∪0<ϵ<1B
ϵ(τ, ω) ⊆ B(τ, ω).

The limiting equation of (1.1) with ϵ = 0 is

ũt −∆ũt + λũ−∆ũ = N(t, x, ũ(t, x)) + f(t, x, ũ(t− ρ, x)) + g(t, x), t > τ, x ∈ Rn, (6.1)

with initial condition

ũτ (s, x) := ũ(τ + s, x) = ϕ̃(s, x), s ∈ [−ρ, 0], x ∈ Rn. (6.2)

Similar to problem (1.1), we can prove that problem (6.1)-(6.2) generates a continuous cocycle Φ0 in
C([−ρ, 0], H1(Rn)). Moreover, Φ0 has a unique tempered pullback attractor A0 = {A0(τ), τ ∈ R}



14 W. MA, Q. MA EJDE-2025/40

in C([−ρ, 0], H1(Rn)) and has a tempered pullback absorbing set B0 = {B0(τ) : τ ∈ R}, where
B0(τ) is given by

B0(τ) = {u ∈ C([−ρ, 0], H1(Rn)) : ∥u∥2C([−ρ,0],H1(Rn)) ≤ QR(τ)} (6.3)

and

R0(τ) = c

∫ 0

−∞
eµr(∥β1(r + τ)∥L1(Rn) + ∥β2(r + τ)∥p1

Lp1 (Rn) + ∥g(r + τ)∥2 + 1)dr. (6.4)

In terms of Lemma 5.1 and (6.3)-(6.4) we have that for all τ ∈ R, ω ∈ Ω,

lim sup
ϵ→0

∥Bϵ(τ, ω)∥ ≤ ∥B0(τ)∥. (6.5)

To obtain the upper semicontinuity of Aϵ, the convergence of solutions of (1.1) as ϵ → 0 is
necessary. To that end, we further assume the nonlinearity N satisfies: there exists β4 ∈
L∞
loc(R, L∞(Rn)) such that for all t, u ∈ R and x ∈ Rn,∣∣∂N

∂u
(t, x, u)

∣∣ ≤ β4(t, x)(1 + |u|p−2), (6.6)

where 2 ≤ p <∞.

Lemma 6.1. Suppose (1.2)-(1.5) and (6.6) hold. Let uϵ(t, τ, ω, uϵτ ) and ũ(t, τ, ũτ ) be the solu-
tions of (1.1) and (6.1)-(6.2) with initial data uϵτ and ũτ , respectively. If limϵ→0 u

ϵ
τ = ũτ in

C([−ρ, 0], H1(Rn)), then for any t ≥ τ, ω ∈ Ω,

lim
ϵ→0

uϵ(t, τ, ω, uϵτ ) = ũ(t, τ, ũτ ).

Proof. Let vϵ be the solution of (3.4)-(3.5) and ṽ = vϵ − ũ. Then from (3.4) and (6.1) we know
that

ṽt −∆ṽt + λṽ −∆ṽ = ϵ(1− λ)z(θtω),

which means
1

2

d

dt
∥ṽ∥2H1(Rn) + λ∥ṽ∥2 + ∥∇ṽ∥2 = (ϵ(1− λ)z(θtω), ṽ). (6.7)

For the right-hand side of (6.7), by ∥z(θtω)∥ ≤ c we have∫
Rn

(ϵ(1− λ)z(θtω)ṽdx ≤ ϵ(1− λ)∥z(θtω)∥ · ∥ṽ∥ ≤ c6∥ṽ∥2 + c7, (6.8)

where c7 is a positive constant dependent of ϵ and λ. Using (6.7)-(6.8) we obtain

d

dt
∥ṽ∥2H1(Rn) + c8∥ṽ∥2H1(Rn) ≤ c7, (6.9)

where c8 = min{2λ− 1, 2}. Integrating (6.9) over (τ, t) with t ∈ [τ, τ + T ] yields

∥ṽ(t)∥2H1(Rn) ≤ ec8(τ−t)∥ṽ(τ)∥2H1(Rn) + c7

∫ t

τ

ec8(s−t)ds. (6.10)

By (1.2), (3.7) and (4.10), this leads to

∥vϵ(t, τ, ω, vϵτ )∥2H1(Rn) + α1

∫ t

τ

eµ(r−t)∥uϵ(r)∥pLp(Rn)dr ≤ c4e
µ(τ−t)∥vϵτ∥2C([−ρ,0],H1(Rn)) + c5.

(6.11)
In the deterministic case, similar to the approach in proof (6.11), after simple calculations, we
obtain that for all t ∈ [τ, τ + T ],

∥ũ(t, τ, uτ )∥2H1(Rn) + α1

∫ t

τ

eµ(r−t)∥ũ(r)∥pLp(Rn)dr ≤ c9e
µ(τ−t)∥ũτ∥2C([−ρ,0],H1(Rn)) + c10. (6.12)

This and (6.10)-(6.12) imply that

∥vϵ(t, τ, ω, vϵτ )− ũ(t, τ, uτ )∥2H1(Rn)

≤ c11e
µ(τ−t)∥vϵτ − ũτ∥2C([−ρ,0],H1(Rn)) + c12ϵ+ c13ϵ(∥vϵτ∥2 + ∥ũτ∥2).

(6.13)



EJDE-2025/40 NONCLASSICAL DIFFUSION EQUATIONS 15

From vϵτ = uϵτ − ϵz(θτ+sω), (6.13) and limϵ→0 u
ϵ
τ = ũτ , it follows that for all t ∈ [τ, τ + T ],

lim
ϵ→0

vϵ(t, τ, ω, vϵτ ) = ũ(t, τ, uτ ),

which together with (3.3), (3.5) means limϵ→0 u
ϵ(t, τ, ω, uϵτ ) = ũ(t, τ, uτ ). □

Lemma 6.2. Suppose that (1.2)-(1.5),(3.11)-(3.12) and (6.6) hold. suppose τ ∈ R, ω ∈ Ω, ϵ ∈
(0, 1], if ϵn → 0 and un ∈ Aϵn(τ, ω), then the sequence {un}∞n=1 is precompact in C([−ρ, 0], H1(Rn)).

Proof. For every bounded sequence {u0,n}∞n=1, we need to prove the sequence

{u(t, τ, ω, u0,n)}∞n=1

has a convergent subsequence in C([−ρ, 0], H1(Rn)). This is done with the aid of the argument in
Lemma 5.2. □

Theorem 6.3. Suppose that (1.2)-(1.5),(3.11)-(3.12) and (6.6) hold. Then for every τ ∈ R, ω ∈ Ω,

lim
ϵ→0

distC([−ρ,0],H1(Rn))(Aϵ(τ, ω),A0(τ)) = 0.

Proof. This is an immediate consequence of [18, Theorem 3.2] based on (6.5), Lemma 6.1, and
Lemma 6.2. □

Conclusions. In this article, we prove the existence and uniqueness of pullback random attractor
for the nonclassical diffusion equation (1.1) with delay and intensity ϵ in C([−ρ, 0], H1(Rn)),
and then we obtain the upper semicontinuity of random attractors when the intensity of noise
approaches zero. It’s worth mentioning that the Arzela-Ascoli theorem, spectral decomposition,
and uniform tail-estimates have been utilized to demonstrate the asymptotic compactness of the
solutions. Furthermore, we will consider the case where time delay is replaced by state dependent
time delay in the near future.

Acknowledgment. The authors would like to thank the anonymous referees for carefully reading
of the manuscript and for the useful suggestions and comments. This work was partially supported
by the National Natural Science Foundation of China (No. 11961059, 12101502), and by the
Natural Science Foundation of Qinghai Province (No. 2024-ZJ-931).

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Wenhui Ma
College of Mathematics and Statistics, Northwest Normal University, Lanzhou, China

Email address: ma15193089786@163.com

Qiaozhen Ma

College of Mathematics and Statistics, Northwest Normal University, Lanzhou, China

Email address: maqzh@nwnu.edu.cn


	1. Introduction
	2. Preliminaries
	3. Existence of a continuous cocycle
	4. Uniform estimates of solutions
	5. Existence of pullback random attractors
	6. Stability of attractors with respect to perturbation parameters
	Conclusions
	Acknowledgment

	References