Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 22, pp. 1–27.

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

DOI: 10.58997/ejde.2024.22

ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO NONCLASSICAL

DIFFUSION EQUATIONS WITH DEGENERATE MEMORY AND

A TIME-DEPENDENT PERTURBED PARAMETER

JIANGWEI ZHANG, ZHE XIE, YONGQIN XIE

Abstract. This article concerns the asymptotic behavior of solutions for a
class of nonclassical diffusion equation with time-dependent perturbation coef-

ficient and degenerate memory. We prove the existence and uniqueness of time-

dependent global attractors in the family of time-dependent product spaces, by
applying the operator decomposition technique and the contractive function

method. Then we study the asymptotic structure of time-dependent global

attractors as t → ∞. It is worth noting that the memory kernel function sat-
isfies general assumption, and the nonlinearity f satisfies a polynomial growth

of arbitrary order.

1. Introduction

In this article, we discuss the long-term behavior of solutions of the perturbed
nonclassical diffusion equation with degenerate memory,

ut − ε(t)∆ut −∆u−
∫ ∞

0

k(s) div{a(x)∇u(t− s)}ds+ f(u) = g(x), (1.1)

with boundary condition
u(x, t)|∂Ω = 0, (1.2)

and initial conditions

u(x, τ) = uτ (x), u(x, τ − s) = uτ (x, τ − s), s ≥ 0, (1.3)

where (x, t) ∈ Ω× (τ,∞), and Ω ⊂ Rn (n ≥ 3) is a bounded domain with smooth
boundary ∂Ω, τ ∈ R is the initial time, g = g(x) ∈ L2(Ω) is the the external
force, a(x) satisfies conditions specified in (H3) below. The perturbation parameter
ε(t) ∈ C1(R) is assumed to be a decreasing bounded function satisfying

lim
t→+∞

ε(t) = 0, (1.4)

and there exists L > 0, such that

sup
t∈R

(|ε(t)|+ |ε′(t)|) ≤ L. (1.5)

We use the following hypotheses

2020 Mathematics Subject Classification. 35K57, 35B40, 35B41.
Key words and phrases. Nonclassical diffusion equation; time-dependent global attractor;
polynomial growth; contractive function; asymptotic structure.
©2024. This work is licensed under a CC BY 4.0 license.
Submitted November 9, 2023. Published March 12, 2024.

1



2 J. ZHANG, Z. XIE, Y. XIE EJDE-2024/22

(H1) k(s) =
∫∞
s
µ(r)dr > 0, where the integrand µ satisfies that for any interval

[0, T ] with T > 0,

µ ∈ L1(R+) is a decreasing piecewise absolutely continuous function, (1.6)

and that there exists δ > 0, such that

k(s) ≤ δµ(s), ∀s ∈ R+. (1.7)

As described in [13], inequality (1.7) is equivalent to the requirement that

µ(t+ s) ≤ Ke−δtµ(s), (1.8)

for some K ≥ 1, δ > 0, any t ≥ 0, and almost every s > 0. Moreover, if µ belongs
to C1(R+), then µ′(s) + δµ(s) ≤ 0 can derive (1.8) when K = 1, which shows that
the same condition for K ≥ 1 is more general.

For simplicity, we let

m0 := k(0) =

∫ ∞

0

µ(s)ds <∞.

(H2) The nonlinearity f satisfies f ∈ C1, f(0) = 0, and the arbitrary order
polynomial growth restriction

α1|s|p − β1 ≤ f(s)s ≤ α2|s|p + β2, ∀s ∈ R, p ≥ 2, (1.9)

and the dissipative condition

f ′(s) ≥ −l, (1.10)

where αi, βi (i = 1, 2) and l are positive constants.

Denoting F (s) =
∫ s

τ
f(σ)dσ we can confirm that there exist α̃i, β̃i > 0 (i = 1, 2)

from (1.9), such that

α̃1|s|p − β̃1 ≤ F (s) ≤ α̃2|s|p + β̃2, , ∀s ∈ R. (1.11)

(H3) a(x) ∈ C∞(Ω) ∩ C0(Ω) is non-negative, and there exists a connected set
A ⊂⊂ Ω such that

a(x) = 0 ⇐⇒ x ∈ A.

Following Dafermos [14] we introducing an additional variable ηt, which is the
history of u, i.e.

ηt = ηt(x, s) :=

∫ s

0

u(x, t− r) dr, s ∈ R+. (1.12)

Let ηtt =
∂
∂tη

t and ηts =
∂
∂sη

t, it follows that

ηtt = −ηts + u. (1.13)

By (H1), (1.12) and (1.13) yields∫ ∞

0

k(s) div{a(x)∇u(t− s)}ds =
∫ ∞

0

µ(s) div{a(x)∇ηt(s)}ds. (1.14)

Thus, system (1.1)-(1.3) can be rewritten as

ut − ε(t)∆ut −∆u−
∫ ∞

0

µ(s) div{a(x)∇ηt(s)}ds+ f(u) = g(x),

ηtt = −ηts + u.

(1.15)



EJDE-2024/22 NONCLASSICAL DIFFUSION EQUATIONS 3

with the initial and boundary conditions

u(x, t)|∂Ω = 0, ηt(x, s)|∂Ω×R+ = 0, t ∈ (τ,∞),

u(x, τ) = uτ (x), ητ (x, s) =

∫ s

0

uτ (x, τ − r)dr, (x, s) ∈ Ω× R+.
(1.16)

Throughout this article, unless otherwise stated, we assume that z(t) = (u(t), ηt)
is the solution of system (1.15)-(1.16) with initial value zτ = (uτ , η

τ ), and let
Rτ = [τ,∞), R+ = [0,∞).

Equation (1.1), as a nonclassical diffusion equation, is well known for its mathe-
matical and physical significance. For instance, it is usually utilized in the various
fields, including fluid mechanics, solid mechanics, and heat conduction theory, see
[8, 2, 21]. In contrast to the classical reaction-diffusion equation, it mainly considers
viscous factor and historical influence of u and this historical influence (i.e., mem-
ory term

∫∞
0
µ(s) div{a(x)∇ηt(s)}ds) is degenerate. Specifically, the degeneracy is

reflected in the sense that the function a(x) ≥ 0 in
∫∞
0
µ(s) div{a(x)∇ηt(s)}ds is

allowed to vanish in some positive measure subset ω0 of Ω.
When ε(t) is zero (or a positive constant) and memory term is non-degenerate, it

is easy to show that the equation (1.1) becomes the usual reaction-diffusion equation
(or nonclassical diffusion equation) with memory, under these circumstances, the
asymptotic behavior of solutions has been researched by many scholars in recent
years (see [9, 17, 18, 19, 31, 35, 36, 37]). Especially to deserve to be mentioned, more
recently, the authors considered the existence, regularity and upper semicontinuity
of global and uniform attractors for autonomous and non-autonomous nonclassical
diffusion equation lacking instantaneous damping −∆u in bounded and unbounded
domain when the nonlinearity satisfies critical exponential growth and polynomial
growth of arbitrary order respectively, see [10, 29, 32, 33, 36, 38, 39].

Nevertheless, for equation (1.1) with time-dependent parameter, the current
studies focus on the nonclassical diffusion equation without memory (i.e., k(s) = 0
in (1.1)), see [23, 24, 30, 40] and the references therein. In [23, 24, 40], the authors
proved the existence of time-dependent global attractors in Ht when the nonlin-
earity f satisfies the subcritical exponential growth, critical exponential growth,
and polynomial growth of arbitrary p − 1 (p ≥ 2) order respectively. Particularly,
when f meets polynomial growth of arbitrary order, the authors of [30] proved
the existence, regularity and the asymptotic structure of the time-dependent global
attractors for the equation

ut − ε(t)∆ut −∆u+ λu+ f(u) = g(x). (1.17)

To sum up, we try to consider the long-term behavior of equation (1.17) with
memory (i.e., the case of non-degenerate) and without linear damping. At this
point, if the memory term satisfies classical assumption µ(s) + δµ′(s) ≤ 0 (see [31,
35]), then the results we obtain are perfectly predictable. This allows us to think of
the asymptotic behavior of problem (1.1) under the premises that the (degenerate)
memory term satisfies the weaker assumption (1.7) and the nonlinearity fulfills
polynomial growth of arbitrary order?

So why would we consider the system (1.1) with the degenerated memory? In [4],
the authors completed a pioneering work, namely, the uniform decay of solutions
was obtained for degenerate problem

ut − (a(x)ux)x + b(x)u = 0,



4 J. ZHANG, Z. XIE, Y. XIE EJDE-2024/22

where a(x) ∈ C1([0, 1]) satisfies a(x) > 0 and a(x)|x=0,1 = 0, and b(x) ∈ C([0, 1])
meets b(x) ≥ 0. Whereafter, some authors investigated the asymptotic behavior
and stability of solutions for above problem under suitable assumptions, see [16, 1]
and the references therein. In addition, many scholars considered wave equation
with degenerate memory, see [5, 6, 7, 27] and the references therein. In should be
emphasized that the authors in [27] obtained regularity of global attractors for the
following wave equation with degenerate memory

utt −∆u+

∫ ∞

0

g(s)div[a(x)∇u(t− s)]ds+ b(x)ut + f(u) = g(x).

Furthermore, Faria et al. showed the existence of global attractors for the following
heat equation with degenerate memory

θt − k0∆θ −
∫ t

−∞
k(t− s)div[a(x)∇θ(s)]ds+ f(θ) = g

in recent a study [15] when the nonlinear term f(θ) fulfills critical exponential
growth and the memory kernel function satisfies the weaker conditions (see (H1)).

The ideas in [15, 39] inspires us to take into account the existence and uniqueness
of time-dependent global attractors for equation (1.1) when the assumptions (H1)
and (H2) hold, which answers the question we posed earlier. Moreover, in this
article, we also incidentally consider asymptotic structure of time-dependent global
attractors based on existing studies in [11, 12, 30]. Thus, it is a comprehensive
and innovative problem for us to think about the existence, uniqueness and the
asymptotic structure of time-dependent global attractors for the equation (1.1),
and this article improves the existing work in [15, 32, 33, 37].

Of course, we need to overcome the following two difficulties for solving foregoing
problem:

(i) On the one hand, because the nonlinearity f has polynomial growth of
arbitrary order and equation (1.1) includes the memory term, we cannot
use Sobolev compact embedding to verify asymptotic compactness of the
solution process generated by equation (1.1) as [15].

(ii) On the other hand, since the memory term is degenerate and the memory
kernel function k(s) satisfies the weaker condition (H1), which does not
allow us to the classical estimation methods from [18, 31, 35, 37] in our
problem.

To solve the above difficulties, we use some ingenious analytical techniques, and
the operator decomposition method is adopted to obtain constructive function.
Then we can verify that the process {U(t, τ)}t≥τ generated by equation (1.1) is
pullback asymptotically compact. Meanwhile, the asymptotic regularity of solu-
tions for equation (1.1) is also obtained, and it follows that we can construct the
contractive function and further show pullback asymptotical compactness of the
corresponding process {U(t, τ)}t≥τ associated with the equation (4.1). In addi-
tion, the study of asymptotic structures shows that the limit relation between the
time-dependent attractors for the problem (1.1) and the global attractor for the
reaction-diffusion equation with degenerate memory of [15] with the same condi-
tions by using the method in [11].

This article is organized as follows: In Section 2, we recall some basic concepts
with respect to the time-dependent global attractors and other some useful results
that will be used later. In Section 3, we first prove pullback asymptotic compactness



EJDE-2024/22 NONCLASSICAL DIFFUSION EQUATIONS 5

of the process {U(t, τ)}t≥τ generated by problem (1.15) by establishing contractive
function, and then the existence and uniqueness of time-dependent global attractors
are attained for system (1.15)-(1.16). In Section 4, we obtain the limit relation be-
tween the time-dependent attractors for the equation (1.1) and the global attractor
for the equation (1.1) in [15] satisfying the same conditions.

2. Preliminaries

In this section, we first give some notation used later, and then describe some
basic concepts and theories of the existence of time-dependent global attractors,
for details see [12, 11].

Basic concepts and notation. Hereafter let |u| be the modular (or absolute
value) of u and | · |p be the norm of Lp(Ω)(p ≥ 1), and (·, ·) be the inner product of
L2(Ω). Let (∇·,∇·), (∆·,∆·) and |∇·|22, |∆·|22 be the inner products and the equiva-
lent norms of H1

0 (Ω) and D(A) = H1
0 (Ω)∩H2(Ω) respectively. The time-dependent

spaces Ht := H1
0 (Ω) and H1

t := D(A) are endowed with the corresponding norms

∥ · ∥2Ht
= | · |22 + ε(t)|∇ · |22, ∥ · ∥2H1

t
= |∇ · |22 + ε(t)|∆ · |22.

As in [7, 6], we assume that there is a Hilbert space

H1
a(Ω) =

{
u ∈ L2(Ω) : |

√
a∇u| ∈ L2(Ω), u|∂Ω = 0

}
with inner-product

(u, v)H1
a
= (u, v) + (

√
a∇u,

√
a∇v).

We define the weight space

Va := L2
µ(R+;H1

a) =
{
ηt : R+ → H1

a ,

∫ ∞

0

µ(s)∥ηt∥2H1
a
<∞

}
with inner product and norm

⟨ψ, η⟩µ,1 =

∫ ∞

0

µ(s)(ψ, η)H1
a
ds, ∥ηt∥2µ,1 =

∫ ∞

0

µ(s)∥ηt∥2H1
a
ds.

As in [27] we define the regular Hilbert space

H2
a(Ω) =

{
u ∈ H1

0 (Ω),
√
a∆u ∈ L2(Ω)

}
,

with inner-product

(u, v)H2
a
= (∇u,∇v) + (

√
a∆u,

√
a∆v).

We define the weight space

V1
a := L2

µ(R+;H2
a) =

{
ηt : R+ → H2

a ,

∫ ∞

0

µ(s)∥ηt∥2H2
a
<∞

}
with inner product and norm

⟨ψ, η⟩µ,2 =

∫ ∞

0

µ(s)(ψ, η)H2
a
ds, ∥ηt∥µ,2 =

∫ ∞

0

µ(s)∥ηt∥2H2
a
ds.

Additionally, we denote V0 = L2
µ(R+;L2(Ω)) and its inner product and norm

⟨ψ, η⟩µ,0 =

∫ ∞

0

µ(s)(ψ, η)ds, ∥ηt∥2µ,0 =

∫ ∞

0

µ(s)|ηt|22ds.

Then phase space of equation (1.1) is

Mr
t = Hr

t × Vr
a (r = 0, 1).



6 J. ZHANG, Z. XIE, Y. XIE EJDE-2024/22

and ∥ · ∥2Mr
t
= ∥ · ∥2Hr

t
+ ∥ · ∥2µ,r+1, where {Mt}t∈R represents a family of time-

dependent normed spaces, and it should be noted that the superscript is omitted
when r = 0.

Remark 2.1. As stated in [7, 27, 5], there exists reasonable continuous embedding
H2

a(Ω) ↪→ H1
0 (Ω) ↪→ H1

a(Ω) ↪→ L2(Ω); in particular, H2
a(Ω) ↪→ H1

a(Ω) is compact,
but D(A) ↪→ H2

a(Ω) does not hold.

Next, we introduce some common notation based on processes of time-dependent
space (see [23, 12, 11]).

Let {Mt}t∈R be a family of normed spaces. Note that the ball of radius R in
Mt is

Bt(R) = {w ∈ Mt : ∥w∥Mt ≤ R}.
For any given ε > 0, we define the ε neighborhood of set B ⊂ Mt as

Oε
t (B) = ∪x∈B{y ∈ Mt : ∥x− y∥Mt

< ε} = ∪x∈B{x+ Bt(ε)}.

In particular, the Hausdorff semidistance of between two nonempty sets A,B ⊂ Mt

is defined as

distMt
(A,B) = sup

x∈A
inf
y∈B

∥x− y∥Mt
.

Definition 2.2. Let {Mt}t∈R be a family of normed spaces. A two-parameter
family of operators U(t, τ) : Mτ → Mt is called a process if it satisfies the following
properties:

(i) U(τ, τ) = Id for τ ∈ R (Identity operator);
(ii) U(t, s)U(s, τ) = U(t, τ) for t ≥ s ≥ τ ∈ R.

Definition 2.3. A family of sets C̃ = {Ct ⊂ Mt : Ct is bounded}t∈R is called
uniformly bounded if there exists a constant R > 0, such that Ct ⊂ Bt(R) for all
t ∈ R.

Definition 2.4. A family of sets B̃ = {Bt}t∈R is called pullback absorbing if

B̃ = {Bt}t∈R is uniformly bounded and for all R > 0, there exists a constant
t0 = t0(t, R) ≤ t such that U(t, τ)Bτ (R) ⊂ Bt for any τ ≤ t0.

The process {U(t, τ)}t≥τ is called dissipative whenever it enters a pullback ab-

sorbing family B̃0 = {B0
t }t∈R.

Definition 2.5. A time-dependent absorbing set for the process U(t, τ) is a uni-

formly bounded family B̃ = {Bt}t∈R with the following characteristic: for any
R > 0, there exists t0 = t0(t, R) ≥ 0, such that

U(t, τ)Bτ (R) ⊂ Bt for all τ ≤ t− t0.

Definition 2.6. The process U(t, τ) is called pullback asymptotic compact if for
any t ∈ R, any bounded sequence {zn}∞n=1 ⊂ Mτn and {τn}∞n=1 ⊂ (−∞, t] with
τn → −∞ as n → ∞, the sequence {U(t, τn)zn}∞n=1 has a convergent subsequence
in Mt.

Definition 2.7. A time-dependent global attractor of the process U(t, τ) is the

smallest family à = {At}t∈R such that

(i) for every t ∈ R, At is compact in Mt;



EJDE-2024/22 NONCLASSICAL DIFFUSION EQUATIONS 7

(ii) Ã is pullback attracting, namely, Ã is uniformly bounded and

lim
τ→−∞

distMt(U(t, τ)Cτ ,At) = 0

holds for all uniformly bounded family C̃ = {Cτ}τ∈R and every fixed t ∈ R
and τ ≤ t.

Remark 2.8. The pullback attracting essence can be equivalently described in the
light of pullback absorbing: a (uniformly bounded) family K = {Kt}t∈R is said to
be pullback attracting if for all ε > 0 the family {Oε

t (Kt)}t∈R is pullback absorbing.

Theorem 2.9. A time-dependent global attractor à exists and it is unique if and
only if the process U(t, τ) is asymptotically compact, i.e., the set

K = {K = {Kt}t∈R : Kt ⊂ Mt is compact, and K is pullback attracting}
is non-empty.

It can be seen from Definition 2.7 that the time-dependent global attractor does
not have to be invariant, which is because the process does not require to meet
some continuity. If the process U(t, τ) satisfies appropriate continuity, then the

invariance of time-dependent global attractor à can be obtained.

Definition 2.10. The time-dependent global attractor à = {At}t∈R is said to be
invariant if

U(t, τ)Aτ = At, t ≥ τ ∈ R.

Lemma 2.11. If the time-dependent global attractor à exists and the process
U(t, τ) is a strongly continuous process, then à is invariant.

Next, we will state the definitions of contractive function and Mt-contractive
process, which will be utilized to prove asymptotic compactness of a family of
process {U(t, τ)}t≥τ (see [22, 25, 28, 34, 35]).

Definition 2.12. Let {Mt}t∈R be a family of Banach spaces and B̃ = {Bt ⊂
Mt}t∈R be a family of uniformly bounded subset. We call function φ(·, ·), defined on
Mτ×Mτ , to be a contractive function on Bτ×Bτ if for any sequence {zn}∞n=1 ⊂ Bτ ,
there exists a subsequence {znk

}∞k=1 ⊂ {zn}∞n=1 such that

lim
k→∞

lim
l→∞

φt
τ (znk

, znl
) = 0, ∀ t ≥ τ ∈ R.

We use E(Bτ ) to denote the set all contractive function on Bτ ×Bτ .

Definition 2.13. Assume that U(t, τ) is a process on {Mt}t∈R and it has a pull-

back bounded absorbing set B̃ = {Bt}t∈R. U(t, τ) is called Mt-contractive process
if for any given ε > 0, there exist T = T (ε) and φt

T (·, ·) ∈ E(BT ) such that

∥U(t, T )z1 − U(t, T )z2∥Mt
≤ ε+ φt

T (z1, z2), ∀ zi ∈ BT (i = 1, 2).

where φt
T depends on T .

Next, we give the method to prove the existence of time-dependent global at-
tractors for evolution equations, which will be used in the later discussion.

Theorem 2.14 ([25]). Let {Mt}t∈R be a family of Banach spaces, then U(t, τ) has
a time-dependent global attractor in {Mt}t∈R, if the following conditions hold

(i) U(t, τ) has a pullback absorbing set B̃ = {Bt}t∈R in {Mt}t∈R;



8 J. ZHANG, Z. XIE, Y. XIE EJDE-2024/22

(ii) U(t, τ) is a Mt-contractive process.

In what follows, we give two Lemmas, which shall be used to prove the compact-
ness of solution process.

Lemma 2.15 ([26]). Let X ⊂⊂ H ⊂ Y be Banach spaces, with X reflexive and
T, τ be two constants with τ ≤ T . Suppose that un is a sequence that is uniformly
bounded in L2(τ, T ;X) and dun/dt is uniformly bounded in Lp(τ, T ;Y ), for some
p > 1. Then there is a subsequence of un that converges strongly in L2(τ, T ;H).

Lemma 2.16 ([3]). Suppose that the nonnegative function µ ∈ L1(R+) is decreasing
piecewise absolutely continuous, and it satisfies that if there exists s0 ∈ R+ such that
µ(s0) = 0, then µ(s) = 0 is true for any s ≥ s0. Furthermore, let B0 ↪→↪→ B1 ↪→ B2

be Banach spaces, where B0, B1 are reflexive. If C ⊂ L2
µ(R+, B1) and satisfies

(i) C in L2
µ(R+, B0) ∩H1

µ(R+, B2);

(ii) supϑ∈C ∥ϑ∥2B1
≤ h(s), for all s ∈ R+, h(s) ∈ L2

µ(R+); then C is rela-

tively compact in L2
µ(R+, B1), where H1

µ(R+, B2) = {f : f(s), ∂sf(s) ∈
L2
µ(R+, B2)}.

3. Existence and uniqueness of a time-dependent global attractor

It is easy to know that the key for existence and uniqueness is to verify the
pullback asymptotic compactness of process generated by (1.1). To do this, we
first prove the asymptotic regularity of solutions, then the contractive function can
be constructed by it, which can ensure the pullback asymptotic compactness of
corresponding process.

3.1. Well-posedness. We now describe the well-posedness for the equation (1.1),
which can be obtained by using standard Faedo-Galerkin method (see e.g., [30, 26,
29]). For simplicity, we only give the final conclusion.

Lemma 3.1. Let Ω be a bounded domain of Rn (n ≥ 3) with smooth boundary ∂Ω,
zτ = (uτ , η

τ ) ∈ Mτ , and (H1)–(H3) be satisfied. Then for any T > τ , the system
(1.15)-(1.16) possesses a unique weak solution z(t) = (u(t), ηt) satisfying

u ∈ C(τ, T ;Ht) ∩ Lp(τ, T ;Lp(Ω)), ηt ∈ C(τ, T ;Va). (3.1)

In addition, if zi = U(t, τ)ziτ = (ui(t), η
t
i) (i = 1, 2) are two weak solutions of

(1.15)-(1.16), then for any t ≥ τ , it is easy to obtain a positive constant Cℜ inde-
pendent of t, such that

∥U(t, τ)z1τ − U(t, τ)z2τ∥Mt
≤ Cℜ∥z1τ − z2τ∥Mτ

. (3.2)

Remark 3.2. By Lemma 3.1, the following solution process can be defined on the
family of time-dependent spaces {Mt}t∈R

U(t, τ) : Mτ → Mt, U(t, τ)zτ = z(t), ∀t ≥ τ. (3.3)

In particular, from (3.2), it is easy to find the process U(t, τ) is Lipschitz continuous.
That is to say, U(t, τ) is a strongly continuous process over the family of time-
dependent phase space {Mt}t∈R.



EJDE-2024/22 NONCLASSICAL DIFFUSION EQUATIONS 9

3.2. Time-dependent absorbing sets. In this subsection, we shall study the
dissipative feature for the process {U(t, τ)}t≥τ . To this end, we need a series of
prior estimates. Throughout this subsection and subsequent sections, we always
assume that Ω ⊂ Rn(n ≥ 3) is a bounded domain with smooth boundary, the
initial value zτ = (uτ , η

τ ) ∈ B(R) ⊂ Mτ , ε(t) satisfies (1.4)-(1.5), g ∈ L2(Ω) and
the assumptions (H1)–(H3) hold.

Lemma 3.3. Assume that z(t) = (u(t), ηt) is sufficiently regular solution of (1.15)-
(1.16). Let

L(t) =

∫ ∞

0

k(s)|
√
a∇ηt(s)|22ds,

then L(t) satisfies the differential inequality

d

dt
L(t) ≤ 2δ2m0|a|∞|∇u|22 −

1

2

∫ ∞

0

µ(s)|
√
a∇ηt(s)|22ds, (3.4)

and

|L(t)| ≤ δE1(t), (3.5)

where E1(t) = ∥u∥2Ht
+
∫∞
0
µ(s)|

√
a∇ηt(s)|22ds.

Proof. Combining with Hölder inequality, Young inequality and assumption (H3),
we have

d

dt
L(t) =

∫ ∞

0

k(s)
d

dt
(
√
a∇ηt,

√
a∇ηt)ds

= 2

∫ ∞

0

k(s)(
√
a∇ηtt ,

√
a∇ηt)ds

≤ 2

∫ ∞

0

k(s)(
√
a∇u,

√
a∇ηt)ds− 2

∫ ∞

0

k(s)(
√
a∇ηts,

√
a∇ηt)ds

≤ 2
(∫ ∞

0

k(s)|
√
a∇u|22ds

)1/2(∫ ∞

0

k(s)|
√
a∇ηt|22ds

)1/2

−
∫ ∞

0

k(s)
d

ds
(
√
a∇ηt,

√
a∇ηt)ds

≤ 2δm
1/2
0 |

√
a∇u|2

(∫ ∞

0

k(s)|
√
a∇ηt|22ds

)1/2

−
∫ ∞

0

µ(s)|
√
a∇ηt(s)|22ds

≤ 2δ2m0|
√
a∇u|22 −

1

2

∫ ∞

0

µ(s)|
√
a∇ηt(s)|22ds.

(3.6)

In addition,

|
√
a∇u|22 =

∫
Ω

|a||∇u|2dx ≤ |a|∞|∇u|22. (3.7)

By (3.6) and (3.7), one can obtain (3.4). Then it is easy to obtain

|L(t)| ≤
∫ ∞

0

k(s)|
√
a∇ηt(s)|22ds ≤ δ

∫ ∞

0

µ(s)|
√
a∇ηt(s)|22ds ≤ δE1(t). (3.8)

The proof is complete. □



10 J. ZHANG, Z. XIE, Y. XIE EJDE-2024/22

Lemma 3.4. For each positive constant R, let zτ = (uτ , η
τ ) ∈ Bτ (R) ⊂ Mτ .

Then there exist κ0 > 0 and T0 = T0(R) ≥ 0, such that

∥u∥2Ht
+

∫ ∞

0

µ(s)|
√
a∇ηt(s)|22ds ≤ κ0, ∀t ≥ τ + T0.

Proof. Multiplying (1.15) by u, and integrating over Ω yields

1

2

d

dt
∥u∥2Ht

− 1

2
ε′(t)|∇u|22 + |∇u|22 +

∫ ∞

0

µ(s)(a(x)∇ηt,∇u)ds+ ⟨f(u), u⟩

= ⟨g, u⟩.
(3.9)

In addition, we have∫ ∞

0

µ(s)(a(x)∇ηt,∇u)ds

=

∫ ∞

0

µ(s)(
√
a∇ηt,

√
a∇ηtt)ds+

∫ ∞

0

µ(s)(
√
a∇ηt,

√
a∇ηts)ds

=
1

2

d

dt

∫ ∞

0

µ(s)|
√
a∇ηt|22ds−

∫ ∞

0

µ′(s)|
√
a∇ηt|22ds

(3.10)

and

⟨g, u⟩ ≤ 2

λ1
|g|22 +

λ1
8
|u|22. (3.11)

Combining this, (3.9)-(3.10), the Poincaré inequality, (1.4) and (H2), we obtain

1

2

d

dt
E1(t) +

3

4
|∇u|22 +

λ1
8
|u|22 + α1|u|pp ≤ β1|Ω|+

2

λ1
|g|22, (3.12)

where |Ω| denotes Lebesgue measure of domain Ω.
Next, we define functional

S(t) = E1(t) + 2γL(t).

Then by (3.5), it is easy to obtain

(1− 2γδ)E1(t) ≤ S(t) ≤ (1 + 2γδ)E1(t), (3.13)

where γ is small enough to guarantee 1− 2γδ > 0.
Furthermore, from (3.4) and (3.12), it follows that

1

2

d

dt
S(t) +

ε(t)

2L
|∇u|22 + (

1

4
− 2γδ2m0|a|∞)|∇u|22

+ γ

∫ ∞

0

µ(s)|
√
a∇ηt(s)|22ds+

λ1
8
|u|22 + α1|u|pp

≤ β1|Ω|+
2

λ1
|g|22,

(3.14)

similarly, we can choose suitable γ such that 1
4 − 2γδ2m0|a|∞ ≥ 0.

In conclusion, letting γ = min{ 1
2δ ,

1
8δ2m0|a|∞ } > 0, and c1 = min{ 1

2L , γ,
λ1

8 },
inequality (3.14) becomes

d

dt
S(t) + 2c1E1(t) + 2α1|u|pp ≤ 2β1|Ω|+

4

λ1
|g|22, (3.15)

Combining this, (3.13), and (3.15), we have

d

dt
S(t) + c2S(t) ≤ 2β1|Ω|+

4

λ1
|g|22, (3.16)



EJDE-2024/22 NONCLASSICAL DIFFUSION EQUATIONS 11

where c2 = 2c1
1+2γδ . Then Gronwall’s inequality yields

S(t) ≤ e−c2(t−τ)S(τ) +
1

c2
(2β1|Ω|+

4

λ1
|g|22). (3.17)

From (3.13) and (3.17) we have

E1(t) ≤
1 + 2γδ

1− 2γδ
e−c2(t−τ)E1(τ) +

1

c2(1− 2γδ)
(2β1|Ω|+

4

λ1
|g|22)

≤ 1 + 2γδ

1− 2γδ
Re−c2(t−τ) +

1

c2(1− 2γδ)
(2β1|Ω|+

4

λ1
|g|22).

(3.18)

Thus, in light of (3.18), there exists T0 = T0(R) > 0, such that

∥u∥2Ht
+

∫ ∞

0

µ(s)|
√
a∇ηt(s)|22ds ≤ κ0

for all t − τ ≥ T0, where κ0 = 2
c2(1−2γδ) (2β1|Ω| +

4
λ1
|g|22). This completes the

proof. □

From (3.13) and (3.17) (or just use (3.18)), we have the following result.

Corollary 3.5. Let zτ = (uτ , η
τ ) ∈ Bτ (R) ⊂ Mτ . Then there exists K0 = K0(R)

such that for all t ≥ τ ,

∥u∥2Ht
+

∫ ∞

0

µ(s)|
√
a∇ηt(s)|22ds ≤ K0 .

Proof. By (3.18) by letting

K0 =
1 + 2γδ

1− 2γδ
R+

1

c2(1− 2γδ)
(2β1|Ω|+

4

λ1
|g|22)

the conclusion holds. □

The following Lemma can be obtained from Lemma 3.3. we omit its proof.

Lemma 3.6. Under the assumption of Lemma 3.3. Let

B(t) =

∫ ∞

0

k(s)|ηt(s)|22ds.

Then B(t) satisfies the differential inequality

d

dt
B(t) ≤ 2δ2m0|u|22 −

1

2

∫ ∞

0

µ(s)|ηt(s)|22ds, (3.19)

and the estimate

|B(t)| ≤ δE2(t), (3.20)

where E2(t) =
∫∞
0
µ(s)|ηt(s)|22ds.

To obtain the bounded absorbing set of Mt, we need the following result.

Lemma 3.7. Under the assumption of Lemma 3.4, for each t − τ ≥ T1 = T (R),
there exists κ1 > 0, such that∫ ∞

0

µ(s)|ηt(s)|22ds ≤ κ1 .



12 J. ZHANG, Z. XIE, Y. XIE EJDE-2024/22

Proof. Taking the inner product of ηt and the second equation of (1.15), we have

1

2

d

dt

∫ ∞

0

µ(s)|ηt|22ds−
∫ ∞

0

µ′(s)|ηt|22ds =
∫ ∞

0

µ(s)(u, ηt)ds. (3.21)

For the right-hand side we have∫ ∞

0

µ(s)(u, ηt)ds ≤ m0

γ1
|u|22 +

γ1
4

∫ ∞

0

µ(s)|ηt|22ds. (3.22)

From (3.21) and (3.22) it follows that

1

2

d

dt

∫ ∞

0

µ(s)|ηt|22ds ≤
m0

γ1
|u|22 +

γ1
4

∫ ∞

0

µ(s)|ηt|22ds. (3.23)

Furthermore, letting

S1(t) = E2(t) + 2γ1B(t),

by (3.17), we have

(1− 2γ1δ)E2(t) ≤ S1(t) ≤ (1 + 2γ1δ)E2(t), (3.24)

where γ1 is appropriately small to ensure 1− 2γ1δ > 0.
By combining with (3.19) and (3.23), we obtain

d

dt
S1(t) +

γ

2
E2(t) ≤ 4δ2m0γ|u|22 +

2m0

γ1
|u|22. (3.25)

Then by Corollary 3.5 and (3.24)-(3.25), we have

d

dt
S1(t) + c3S1(t) ≤ (4δ2m0γ1 +

2m0

γ1
)K0, (3.26)

where c3 = γ1

2(1+2γ1δ)
.

Applying Gronwall’s Lemma to (3.26), we have

S1(t) ≤ ec3(t−τ)S1(τ) +
2K0

c3
(2δ2m0γ1 +

m0

γ1
). (3.27)

From this, (3.24), and (3.27), there exists T1 = T1(R)(> 0), such that∫ ∞

0

µ(s)|ηt(s)|22ds ≤ κ1,

where κ1 = 4K0

c3
(2δ2m0γ1 +

m0

γ1
). □

Next, we show the existence of a bounded absorbing set.

Theorem 3.8. Let zτ = (uτ , η
τ ) ∈ Bτ (R) ⊂ Mτ . Then, for any given R ∈ R+,

there exists ρ0 > 0 such that the process U(t, τ) generated by (1.15) possesses a
time-dependent bounded absorbing set B̄0 = {B0

t }t∈R (:= {Bt(ρ0)}t∈R), i.e.,

B̄0 =
{
z = (u, ηt) ∈ Mt : ∥u∥2Ht

+ ∥ηt∥2µ,1 ≤ ρ0,∀ t ∈ R
}
; (3.28)

this is, there exists T0 = max{T0, T1} ≥ 0, such that

U(t, τ)Bτ (R) ⊂ B0
t , ∀τ ≤ t− T0.

Proof. In Lemmas 3.4 and 3.7, just take ρ0 = κ0 + κ1 and T0 = max{T0, T1} ≥ 0.
Then the above conclusion of the theorem follows. □



EJDE-2024/22 NONCLASSICAL DIFFUSION EQUATIONS 13

3.3. Time-dependent global attractors. Now we prove the existence of time-
dependent global attractors in {Mt}t∈R for the process defined by (3.3). This will
be Theorem 3.17, but we first give the following lemmas.

Lemma 3.9. Under the assumption of Lemma 3.4, there exists positive constant
ρ1 = ρ1(ρ0) such that∫ t+1

t

(|∇u(r)|22 + |u(r)|pp)dr +
∫ t+1

t

∫ ∞

0

µ(s)|
√
aηr(s)| ds dr ≤ ρ1, ∀t− τ ≥ T0 .

Proof. According to (3.12), by letting c4 = min{ 3
4 , 2α1}, we have

d

dt
E1(t) + c4(|∇u|22 + |u|pp) ≤ 2β1|Ω|+

4

λ1
|g|22. (3.29)

Integrating (3.29) on [t, t+ 1], and using Theorem 3.8, we obtain∫ t+1

t

(|∇u(r)|22 + |u(r)|pp)dr ≤ κ3, ∀t− τ ≥ T0, (3.30)

where κ3 = 1
c4
(ρ0 + 2β1|Ω|+ 4

λ1
|g|22). Similarly, by (3.15), we have

d

dt
S(t) + c1

∫ ∞

0

µ(s)|
√
a∇ηt(s)|22ds ≤ 2β1|Ω|+

4

λ1
|g|22. (3.31)

Then integrating (3.31) from t to t+ 1 about t yields

c1

∫ t+1

t

∫ ∞

0

µ(s)|
√
a∇ηr(s)|22 ds dr ≤ S(t+ 1) + 2β1|Ω|+

4

λ1
|g|22. (3.32)

Combining with this, (3.13), and Theorem 3.8, we have∫ t+1

t

∫ ∞

0

µ(s)|
√
a∇ηr(s)|22 ds dr ≤ κ4, ∀t− τ ≥ T0, (3.33)

where

κ4 =
1

c1
[(1 + 2γδ)ρ0 + 2β1|Ω|+

4

λ1
|g|22].

Therefore, from (3.30) and (3.33), by letting ρ1 = κ3 +κ4, it is easy to see that the
aforementioned result is true. □

Corollary 3.10. Under the assumptions of Lemma 3.4, there exists a positive
constant K1, such that ∫ t+1

t

(
|∇u(r)|22 + |u(r)|pp

)
dr ≤ K1

for all t ≥ τ .

Proof. Integrating (3.29) on [t, t+ 1], then using Corollary 3.5, we have∫ t+1

t

(|∇u(r)|22 + |u(r)|pp)dr ≤ K1, ∀t ≥ τ,

where K1 = 1
c4
(K0 + 2β1|Ω|+ 4

λ1
|g|22). □

Lemma 3.11. Under the assumptions of Lemma 3.4, there exists a positive con-
stant ρ2 = ρ2(R) such that∫ t+1

t

(
ε(s)|ut(s)|22 + ε2(s)|∇ut(s)|20

)
ds ≤ ρ2, ∀t ≥ τ.



14 J. ZHANG, Z. XIE, Y. XIE EJDE-2024/22

Proof. Using ε(t)ut to make inner product with the first equation of (1.16) in L2(Ω),
and combining with (1.4)-(1.5), we have

d

dt

[ε(t)
2

|∇u|22 + ε(t)

∫
Ω

F (u)dx
]
+
ε(t)

2
|ut|22 + ε2(t)|∇ut|22

≤ −
∫ ∞

0

µ(s)(a(x)∇ηt, ε(t)∇ut)ds+
L

2
|g|22 + Lβ̃1|Ω|.

(3.34)

Estimating the first term at the right-hand side hand of (3.34), we obtain∣∣− ∫ ∞

0

µ(s)(a(x)∇ηt, ε(t)∇ut)ds
∣∣

≤ m0

2
|a|∞

∫ ∞

0

µ(s)|
√
a∇ηt|22ds+

1

2
ε2(t)|∇ut|22.

(3.35)

By (3.34)-(3.35) we have

d

dt

[
ε(t)|∇u|22 + 2ε(t)

∫
Ω

F (u)dx
]
+ ε(t)|ut|22 + ε2(t)|∇ut|22

≤ m0|a|∞
∫ ∞

0

µ(s)|
√
a∇ηt|22ds+ L|g|22 + 2Lβ̃1|Ω|.

(3.36)

Integrating (3.36) about t from s to t + 1, (t ≤ s ≤ t + 1); then by Corollary 3.5,
we have

ε(t+1)

∫
Ω

F (u(t+1))dx ≤ m0

2
|a|∞K0+

L

2
|g|22+Lβ̃1|Ω|+ε(s)

∫
Ω

F (u(s))+L|∇u(s)|22,

from (1.11), we know that the above inequality can be turned into

ε(t+ 1)|u(t+ 1)|pp

≤ m0

2α̃1
|a|∞K0 +

L

2α̃1
|g|22 +

L

α̃1
|Ω|(2β̃1 + β̃2) +

L

α̃1
(α̃2|u(s)|pp + |∇u(s)|22).

(3.37)

Then integrating (3.37) over [t, t + 1], and combining with Corollary 3.10, we
have

ε(t+ 1)|u(t+ 1)|pp ≤ c5, ∀t ≥ τ, (3.38)

where c5 = m0

2α̃1
|a|∞K0 +

L
2α̃1

|g|22 + L
α̃1

|Ω|(2β̃1 + β̃2) +
L
α̃1

K1(α̃2 + 1).

From (1.11) and (3.5), integrating (3.36) over [t, t+ 1], we have∫ t+1

t

(ε(r)|ut(r)|22 + ε2(s)|∇ut(r)|22)dr

≤ m0|a|∞
∫ t+1

t

∫ ∞

0

µ(s)|
√
a∇ηr|22 ds dr + 2α̃2ε(t)|u(t)|pp

+ L|g|22 + 2L|Ω|(2β̃1 + β̃2).

(3.39)

Thus, by Corollary 3.5 and (3.38), we have∫ t+1

t

(ε(r)|ut(r)|22 + ε2(r)|∇ut(r)|22)dr ≤ ρ2, (3.40)

where ρ2 = m0|a|∞K0+2α̃2c5+L|g|22+2L|Ω|(2β̃1+β̃2). This proof iscomplete. □



EJDE-2024/22 NONCLASSICAL DIFFUSION EQUATIONS 15

To demonstrate asymptotic properties of solutions corresponding to the process
{U(t, τ)}t≥τ , we decompose the solution z = (u, ηt) of problem (1.15)-(1.16) with
initial data zτ = (uτ , η

τ ) ∈ Mτ into the sum

U(t, τ)zτ = U1(t, τ)zτ + U2(t, τ)zτ , (3.41)

where z1 = U1(t, τ)zτ = (v(t), ζt) and z2 = U2(t, τ)zτ = (w(t), θt) are two solutions
of the following systems respectively:

vt − ε(t)∆vt −∆v −
∫ ∞

0

µ(s)div{a(x)∇ζt(s)}ds+ f(u)− f(w) + µ̃v = 0,

ζtt = v − ζts,

(3.42)

with initial-boundary conditions

v(x, t)|∂Ω = 0, ζt(x, s)|∂Ω×R+ = 0, t ≥ τ,

v(x, τ) = uτ (x), ζ
τ (x, s) =

∫ s

0

uτ (x, τ − r)dr, (x, s) ∈ Ω× R+,
(3.43)

where µ̃ > l (from (1.10)) is a constant, and

ωt − ε(t)∆ωt −∆ω −
∫ ∞

0

div{a(x)∇θt(s)}ds+ f(ω)− µ̃v = g,

θtt = ω − θts.

(3.44)

with initial-boundary conditions

ω(x, t)|∂Ω = 0, θt(x, s)|∂Ω×R+ = 0, t ≥ τ,

ω(x, τ) = 0, θτ (x, s) = 0, (x, s) ∈ Ω× R+.
(3.45)

Next, we show that z1 has Mt-decay, but not necessarily exponential decay.

Lemma 3.12. Assume that ∥zτ∥Mτ
≤ R for any given R > 0, then the solution

z1 = U1(t, τ)zτ = (v(t), ζt) of problem (3.42)-(3.43) satisfies the decaying property

lim
τ→−∞

(∥v∥Ht
+ ∥ζt∥2µ,1) = 0 (3.46)

for τ ≤ t.

Proof. We divide the proof into two steps.

Step 1. Multiplying the first equation of (3.42) by v, then integrating over Ω, we
obtain

d

dt
E1 + |∇v|22 + (µ̃− l)|v|22 ≤ 0, (3.47)

where E1(t) =
1
2 (|v|

2
2 + ε(t)|∇v|22 +

∫∞
0
µ(s)|

√
a∇ζt|22ds). By assuming

N1(t) =

∫ +∞

0

k(s)|
√
a∇ζt(s)|22ds,

then as in the proof of Lemma 3.3, we obtain that N1(t) satisfies the differential
inequality

d

dt
N1(t) ≤ 2δ2m0|a|∞|∇v|22 −

1

2

∫ ∞

0

µ(s)|
√
a∇ζt(s)|22ds, (3.48)

and the estimate

|N1(t)| ≤ 2δE1(t), (3.49)



16 J. ZHANG, Z. XIE, Y. XIE EJDE-2024/22

Next, let
S1(t) = E1(t) + πN1(t).

Then for suitably small π = min{ 1
2δ ,

1
2δ2m0|a|∞ } > 0, we have

(1− 2πδ)E1(t) ≤ S1(t) ≤ (1 + 2πδ)E1(t). (3.50)

Combining (3.47)-(3.48) and (3.50), we obtain

d

dt
S1(t) + γ̃1S1(t) ≤ 0, (3.51)

where

γ̃1 =
1

1 + 2πδ
min

{2− 4πδ2m0|a|∞
L

, π, 2(µ̃− l)
}
.

Applying Gronwall’s Lemma for (3.51), and taking limit about τ , we have

lim
τ→−∞

S1(t, τ) = 0.

Thus, from (3.50) it follows that

0 ≤ lim
τ→−∞

E1(t, τ) ≤
1

1− 2πδ
lim

τ→−∞
S1(t, τ) = 0. (3.52)

Then
lim

τ→−∞
E1(t, τ) = 0. (3.53)

Step 2. As in Lemma 3.6, suppose that

N2(t) =

∫ ∞

0

k(s)|ζt(s)|22ds.

Then we have that N2(t) satisfies the inequality

d

dt
N2(t) ≤ 2δ2m0|v|22 −

1

2

∫ ∞

0

µ(s)|ζt(s)|22ds, (3.54)

and the estimate
|N2(t)| ≤ 2δE2(t), (3.55)

where E2(t) =
1
2

∫∞
0
µ(s)|ζt(s)|22ds.

Taking the inner product of v and the second equation of (3.42) on V0, we obtain

d

dt
E2(t) ≤

γ̃2
4

∫ ∞

0

µ(s)|ζt(s)|22ds+
m0

γ̃2
|v|22, (3.56)

where γ̃2 is a constant to be defined later. Let

S2(t) = E2(t) + γ̃2N2(t).

Then we have
(1− 2γ̃2δ)E2(t) ≤ S2(t) ≤ (1 + 2γ̃2δ)E2(t), (3.57)

with γ̃2 ∈ (0, 1
2δ ). Combining (3.54) and (3.56), we have

d

dt
S2(t) + γ̃3S2(t) ≤ (2γ̃2δ

2m0 +
m0

γ̃2
)|v|22. (3.58)

where γ̃3 = γ̃2

2(1+2γ̃2δ)
.

Applying Gronwall’s lemma to (3.58) we obtain

S2(t) ≤ (1 + 2γ̃2δ)e
−γ̃3(t−τ)R+ (2γ̃2δ

2m0 +
m0

γ̃2
)e−γ̃3t

∫ t

τ

eγ̃3r|v(r)|22dr. (3.59)



EJDE-2024/22 NONCLASSICAL DIFFUSION EQUATIONS 17

However, from (3.53), we obtain that |v|22 → 0 as τ → −∞, which implies that∫ t+1

t
|v(r)|22dr → 0 as τ → −∞. Hence

e−γ̃3t

∫ t

τ

eγ̃3r|v(r)|22dr = e−γ̃3t(

∫ t

t−1

+

∫ t−1

t−2

+ · · · )eγ̃3r|v(r)|22dr

≤ 1

1− e−γ̃3

∫ t+1

t

|v(r)|22dr.
(3.60)

Combining (3.59) and (3.60), we have

lim
τ→−∞

S2(t) = 0;

so

lim
τ→−∞

E2(t) = 0. (3.61)

In conclusion, by (3.53) and (3.61), it follows that (3.46) holds. The proof is
complete. □

Lemma 3.13. For each R > 0, let ∥zτ∥Mτ
≤ R, and let z2 = U2(t, τ)zτ = (v(t), ζt)

be the solution of (3.44)-(3.45). Then there exist constants K2, ρ2 > 0, such that

∥ω(t)∥2Ht
+

∫ ∞

0

µ(s)|
√
aθt(s)|22ds+ ε(t)|ω(t)|pp ≤ K2,∫ t+1

t

(ε(s)|ωt(s)|22 + ε2(s)|∇ωt(s)|22 + |ω(s)|pp)ds ≤ ρ3,

for all t ≥ τ .

The proof of the above lemma is similar to the proof of Corollaries 3.5 and 3.10,
Lemma 3.11, and (3.38) word by word. So we omit it here. Next we show that, for
all time, the component z2 belongs to a subset of M1

t , uniformly as the initial data
zτ belongs to the absorbing set B̄0, given by (3.28).

Lemma 3.14. For each R > 0, let ∥zτ∥Mτ ≤ R. Then there exists a constant
K3 = K3(R, t, τ) > 0, such that the solution z2 = (v(t), ζt) of the problem (3.44)-
(3.45) satisfies

∥ω(t)∥2H1
t
+ ∥θt∥2µ,2 ≤ K3, ∀t ≥ τ.

Proof. Firstly, for degenerate memory term, we have

−
∫ ∞

0

µ(s) div{a(x)∇θt(s)}ds

= −
∫ ∞

0

µ(s)∇a(x)∇θt(s)ds−
∫ ∞

0

µ(s)a(x)∆θt(s)ds.

(3.62)

Then using −∆ω to make inner product over L2(Ω) for (3.62). At this time, we
only need to deal with the right-hand side of (3.62); that is∣∣ ∫ ∞

0

µ(s)(∇a(x)∇θt(s),−∆ω)ds
∣∣ ≤ ∫ ∞

0

µ(s)|∇a(x)∇θt(s)|2|∆ω|2ds

≤ m0

2
|∇a|2∞

∫ ∞

0

µ(s)|∇θt(s)|22ds+
1

2
|∆ω|22.



18 J. ZHANG, Z. XIE, Y. XIE EJDE-2024/22

In addition, by combining with the second equation of (3.44), we obtain

−
∫ ∞

0

µ(s)(a(x)∆θt(s),−∆ω)ds

=
1

2

d

dt

∫ ∞

0

µ(s)|
√
a∆θt|22ds−

∫ ∞

0

µ′(s)|
√
a∆θt|22ds.

(3.63)

Next, applying −∆ω on the first equation of (3.44) on L2(Ω), then combining
with Hölder inequality, Young inequality, and (3.62)-(3.63), we have

1

2

d

dt

(
|∇ω|22 + ε(t)|∆ω|22 +

∫ ∞

0

µ(s)|
√
a∆θt|22ds

)
+

1

4
|∆ω|22 + (µ̃− l)|∇ω|22

≤ m0

2
|∇a|2∞

∫ ∞

0

µ(s)|∇θt(s)|22ds+ 2µ̃2|u|22 + 2|g|22.
(3.64)

Furthermore, using −∆θt to make inner product on V0, we obtain

1

2

d

dt

∫ ∞

0

µ(s)|∇θt(s)|22ds ≤
µ̃− l

2
|∇ω|22 +

m0

2(µ̃− l)

∫ ∞

0

µ(s)|∇θt(s)|22ds. (3.65)

By (3.64) and (3.65), it is easily to obtain

1

2

d

dt
(∥ω(t)∥2H1

t
+∥θt∥2µ,2) ≤

m0

2
(|∇a|2∞+

1

µ̃− l
)

∫ ∞

0

µ(s)|∇θt(s)|22ds+2µ̃2|u|22+2|g|22.

Then according to Corollary 3.5, we have

d

dt
(∥ω(t)∥2H1

t
+ ∥θt∥2µ,2) ≤ γ̃4(∥ω(t)∥2H1

t
+ ∥θt∥2µ,2) + 4µ̃2K0 + 4|g|22, (3.66)

where γ̃4 = 1 +m0(|∇a|2∞ + 1
µ̃−l ).

Applying Gronwall’s lemma to (3.66), we have

∥ω(t)∥2H1
t
+ ∥θt∥2µ,2 ≤ eγ̃4(t−τ)(4µ̃2K0 + 4|g|22), ∀t ≥ τ.

Letting K3 = eγ̃4(t−τ)(4µ̃2K0 + 4|g|22), the conclusion follows. □

The following result shall be used in the proof of asymptotic structure of time-
dependent global attractors.

Lemma 3.15. For each t > τ , let Ct := PU2(t, τ)B
0
τ , where P : Ht × Va → Va.

Then there exists a constant C∗ = C∗(∥B0
t ∥Mt

) > 0, such that

(1) Ct is bounded in L2
µ(R+;H2

a(Ω)) ∩H1
µ(R+;L2(Ω));

(2) supθt∈Ct
∥θt(s)∥2µ,1 ≤ C∗.

Therefore, Ct is relatively compact in L2
µ(R+;H1

a(Ω)).

Proof. From Lemma 3.14, it is easy to obtain that Ct is bounded in L2
µ(R+;H2

a(Ω)).

Also we know that θt(s) =
∫ t

t−s
ω(y)dy, so ∂sθ

t(s) = ω(t − s). Thus, from

and Lemma 3.7 and Lemma 3.13, it follows that θt and ω(t) are bounded in
L2
µ(R+;L2(Ω)), which shows that Ct is bounded in H1

µ(R+;L2(Ω)). Additionally,
by Lemma 3.7 and Lemma 3.14, we obtain

∥θt(s)∥2µ,1 =

∫ ∞

0

µ(s)(|θt(s)|22 + |
√
a∇∂sθt(s)|22)ds ≤ C∗

0 ,

where C∗
0 ∈ L1

µ(R+). The above formula indicates that supθt∈Ct
∥θt(s)∥2µ,1 ≤ C∗ ∈

L1
µ(R+).



EJDE-2024/22 NONCLASSICAL DIFFUSION EQUATIONS 19

From the above arguments, Lemma 2.16, and (H2
a(Ω) ↪→↪→ H1

a(Ω) ↪→ L2(Ω)),
one can infer that Ct is relatively compact in L2

µ(R+;H1
a(Ω)) for any t > τ . □

After the above preparations, we can prove the existence and uniqueness of
time-dependent global attractors. The key to achieve this goal is to demonstrate
the pullback asymptotic compactness of process U(t, τ). We know from [20] that
the method of the standard Kuratowski measure of non-compactness may be useful
for verifying the asymptotic compactness of solution process generated by equation
(1.15). But the contractive function method seems to be more concise for our
problem, which is mainly based on our previous research [32, 33, 36]. Thus, we
only need to prove that U(t, τ) is a Mt-contractive process by Theorem 2.14.

Theorem 3.16. The family of process {U(t, τ)}t≥τ generated by (1.15) with initial-
boundary value coditions (1.16) is a family of Mt-contractive process on B0

T ∈ B̄0.

Proof. Let zi(t) = (ui(t), ξ
t
i) = U(t, τ)ziτ (i = 1, 2) be the solutions of problem

(1.15)-(1.16) with initial data ziτ ∈ B0
τ ∈ B̄0 (i = 1, 2) (B̄0 from (3.28)) respectively.

By (3.41), we can establish the decomposition

zi(t) = U(t, τ)ziτ = U1(t, τ)z
i
τ + U2(t, τ)z

i
τ = (vi(t), ζ

t
i ) + (ωi(t), θ

t
i).

It yields

∥U(t, τ)z1τ − U(t, τ)z2τ∥2Mt

≤ 2∥U1(t, τ)z
1
τ − U1(t, τ)z

2
τ∥2Mt

+ 2(∥ω1(t)− ω2(t)∥2Ht
+ ∥θt1 − θt2∥2µ,1),

(3.67)

and by Lemma 3.12, we have

lim
τ→−∞

∥U1(t, τ)z
1
τ−U1(t, τ)z

2
τ∥2Mt

≤ 2 lim
τ→−∞

(∥U1(t, τ)z
1
τ∥2Mt

+∥U1(t, τ)z
2
τ∥2Mt

) = 0.

Hence, for each ε > 0, there exists δ = ε
4 such that

2∥U(t, T )z1T − U(t, T )z2T ∥2Mt
< ε, (3.68)

holds for any t ≥ T = T (ε) fixed.
Moreover, it is easy to check that (ψ(t), ξt) = (ω1(t) − ω2(t), θ

t
1 − θt2) is the

solution of the system

ψt − ε(t)∆ψt −∆ψ −
∫ ∞

0

µ(s) div{a(x)∇ξt(s)}ds+ f(ω1)− f(ω2) + µ̃ψ

= µ̃(u1 − u2),

ξtt = −ξts + ψ.

(3.69)

with initial-boundary value conditions

ψ(x, t)|∂Ω = 0, ξt(x, s)|∂Ω×R+ = 0, t ≥ τ,

ψ(x, τ) = 0, ξτ (x, s) = 0, (x, s) ∈ Ω× R+.
(3.70)

Taking the inner product of ψ and ξt on L2(Ω) and V0 for the first and second
equations of (3.69) respectively, by Hölder inequality, we have

d

dt

(
∥ψ∥2Ht

+

∫ ∞

0

µ(s)|
√
a∇ξ|22ds

)
+2|∇ψ|22+

3(µ̃− l)

2
|ψ|22 ≤ 2µ̃2

µ̃− l
|u1−u2|22, (3.71)

and
d

dt

∫ ∞

0

µ(s)|ξ|22ds ≤
2m0

ϵ̃
|ψ|22 +

ϵ̃

2

∫ ∞

0

µ(s)|ξ|22ds. (3.72)



20 J. ZHANG, Z. XIE, Y. XIE EJDE-2024/22

As in Lemma 3.6, we obtain

d

dt

∫ ∞

0

k(s)|ξ|22ds ≤ 2δ2m0|ψ|22 −
1

2

∫ ∞

0

µ(s)|ξ|22ds. (3.73)

Subsequently, let

G (t) =

∫ ∞

0

µ(s)|ξ|22ds+ ϵ̃

∫ ∞

0

k(s)|ξ|22ds.

Taking ϵ̃ ∈ (0, 1/δ), such that

(1− δϵ̃)

∫ ∞

0

µ(s)|ξ|22ds ≤ G (t) ≤ (1 + δϵ̃)

∫ ∞

0

µ(s)|ξ|22ds. (3.74)

by (3.72)-(3.74), we obtain
d

dt
G (t) ≤ γ̃5|ψ|22, (3.75)

where γ̃5 = 2m0

ϵ̃ + 2ϵ̃δ2m0.
Now integrating (3.71) and (3.75) from τ to t (t ≥ τ ≥ T ) respectively, and

combining with (3.74), it follows that

∥ω1(t)− ω2(t)∥2Ht
+

∫ ∞

0

µ(s)|
√
a∇ξ|22ds ≤ C

∫ t

T

|u1(s)− u2(s)|22ds, (3.76)∫ ∞

0

µ(s)|ξ|22ds ≤ C

∫ t

T

|ω1(s)− ω2(s)|22ds, (3.77)

where C = max{ 2µ̃2

µ̃−l ,
γ̃5

1−δϵ̃}.
From (3.76) and (3.77), we have

∥ω1(t)− ω2(t)∥2Ht
+ ∥θt1 − θt2∥2µ,1

≤ C

∫ t

T

|u1(s)− u2(s)|22ds+ C

∫ t

T

|ω1(s)− ω2(s)|22ds.
(3.78)

Then we let

φt
T (z1, z2) = C

∫ t

T

|u1(s)− u2(s)|22ds+ C

∫ t

T

|ω1(s)− ω2(s)|22ds

:= φ̃t
T (z1, z2) + φ̄t

T (z1, z2).

(3.79)

Combining this, Corollary 3.5, and Lemma 3.11, and applying Lemma 2.15, there
exists a subsequence of {un(s)}∞n=1 that converges strongly in L2(T, t;L2(Ω)). In
other words, for any sequences {znT = (unT , η

T
n )} ⊂ B0

T ∈ B̄0, {zn(t) = (un(t), η
t
n)},

as the solution of problem (1.15) with the initial data {znT = (unT , η
T
n )}, includes

a subsequence {znk
} satisfying:

lim
k→∞

lim
l→∞

φ̃t
T (znk

, znl
) = C lim

k→∞
lim
l→∞

∫ t

T

|unk
(s)− unl

(s)|22ds = 0. (3.80)

Similarly, through Lemmas 3.13 and 3.14, it is easy to obtain that the set

{ω(t) = Π1U2(t, τ)zτ : zτ ∈ B0
τ ∈ B̄0}

is bounded in H1
0 (Ω); therefore, {ω(t) = Π1U2(t, T )zT : T ≤ τ, zT ∈ B0

T ∈ B̄0} is
compact in L2(T, t;L2(Ω)), where Π is the projection from X × Y to X. That is,
we have

lim
k→∞

lim
l→∞

φ̄t
T (znk

, znl
) = C lim

k→∞
lim
l→∞

∫ t

T

|ωnk
(s)− ωnl

(s)|22ds = 0. (3.81)



EJDE-2024/22 NONCLASSICAL DIFFUSION EQUATIONS 21

By (3.80) and (3.81), we have

lim
k→∞

lim
l→∞

φt
T (znk

, znl
) = 0. (3.82)

This implies that φt
T ∈ E(BT ). Combining (3.67)-(3.67) and (3.71)-(3.80), one

obtain
∥U(t, T )x− U(t, T )y∥2Mt

≤ ε+ φt
T (x, y).

From Definitions 2.12 and 2.13, we know that φt
T is contractive function in B0

T .
Therefore, it’s easy to obtain that the process U(t, τ) is a Mt−contractive process
on B0

T ∈ B̄0. □

Theorem 3.17. The process U(t, τ) defined by (3.3) possesses a time-dependent
global attractor A = {A(t)}t∈R in {Mt}t∈R, and A is non-empty, compact, invari-
ant in {Mt}t∈R and pullback attracting every bounded set in {Mt}t∈R.

Proof. By Theorems 3.8 and 3.16, the existence and uniqueness of time-dependent
global attractor A for the process U(t, τ) generated by equation (1.1) in time-
dependent product spaces {Mt}t∈R. In addition, from Lemma 2.11 and (3.2) of
Lemma 3.1, we can obtain the invariance of time-dependent global attractor A . □

Remark 3.18. (i) By fully utilizing the method in this paper, if the condition of the
nonlinearity f(u) in [15] can be weakened to polynomial growth of arbitrary order
(such as (1.9)), then the existence of global attractors for the reaction-diffusion
equation with degenerate memory (i.e., the equation (4.1)) can still be obtained.

(ii) Similar to the study of [12], for sufficiently small δ, we can also obtain
the regularity of time-dependent attractor A by Lemmas 3.12 and 3.14, i.e., A ⊂
{M1

t}t∈R. This is so because we can obtain the uniform boundedness of ∥u(t, τ)zτ∥2M1
t

with respect to τ in Lemma 3.14 by introducing the functional of Lemma 3.3 when
δ is sufficiently small.

4. Asymptotic structure of the attractor

Following the idea in [11], we study the relationship between the time-dependent
attractor for problem (1.1)-(1.3) and the global attractor for the following limit
system (4.1) with initial-boundary (1.2)-(1.3) as t→ ∞ (i.e., ε(t) = 0):

ūt −∆ū−
∫ ∞

0

k(s) div{a(x)∇ū(t− s)}ds+ f(ū) = g(x), (4.1)

where
∫∞
0
k(s) div{a(x)∇ū(t − s)}ds =

∫∞
0
µ(s) div{a(x)∇η̄t(s)}ds. For this pur-

pose, we introduce the following conclusions about the completely bounded trajec-
tories (CBT); for more details see [11].

Definition 4.1. A function z : t 7→ z(t) ∈ Xt is a CBT of U(t, τ) if and only if

(i) supt∈R ∥z(t)∥Xt
<∞, and

(ii) z(t) = U(t, τ)z(τ), for all t ≥ τ ∈ R.

Theorem 4.2. Let à = {At}t∈R be the time-dependent global attractor of U(t, τ),

if à is invariant, then

At = {z(t) ∈ Xt : z is CBT of U(t, τ)}.
Consequently, we can write

At = {z : t 7→ z(t) ∈ Xt with zCBT of U(t, τ)}.



22 J. ZHANG, Z. XIE, Y. XIE EJDE-2024/22

Theorem 4.3. Suppose that, for any sequence zn = (xn, yn) of CBT of the process
U(t, τ) and any tn → ∞, there is a CBT w of the semigroup S(t) and any s ∈ R
for which

lim
n→∞

∥xn(s+ tn)− w(s)∥X = 0,

up to a subsequence. Then

lim
t→∞

distX(ΠtAt,A∞) = 0,

where Πt : Xt → X is the projection on the first component of Xt, i.e., for (u, η) ∈
Zt, Πt(u, η) = u. Accordingly, if Yt ∈ Xt, then ΠtYt = {u ∈ X : (u, η) ∈ Yt}.
Specially, if Y = {Yt}t∈R, denote ΠY = {ΠtYt}t∈R.

Let (H1)–(H3) hold, then the unique solution z̄(t) = (ū(t), η̄t) can be obtained
from [15]. In particular, one knows from Remark 3.18 that the semigroup {S(t)}t≥0

generated by (4.1) possesses global attractors on M0 = L2(Ω) × Va from Remark
3.18, and z̄(t) = S(t)z̄τ . It should be pointed out that {S(t)}t≥0 has a global
attractor A∞ in M0. Additionally, for fixed s ∈ R, there is

A∞ = {z̄(s) : R → M0 with z̄ CBT of S(t)}.

Next, we establish the asymptotic closeness of the time-dependent global attrac-
tor A = {A(t)}t∈R of the process {U(t, τ)}t≥τ generated by (1.1) and the global
attractor A∞ of the semigroup {S(t)}t≥0. For this purpose, we first give the fol-
lowing Lemma.

Lemma 4.4. Let g ∈ L2(Ω), ε(t) satisfy (1.4)-(1.5), and the conditions (H1)−(H3)
hold. Then, for any zτ = (uτ , η

τ ) ∈ Bτ (R) ⊂ Mτ , there exist positive constant
Ki (i = 4, 5) such that

sup
zτ∈A(τ)

sup
t≥τ

∥u∥2H1
t
≤ K3, (4.2)

and ∫ ∞

τ

|ut(r)|22dr ≤ K4. (4.3)

Proof. By Lemma 3.17, we can obtain (4.2). For the conclusion (4.3), as in the proof
of Lemma 3.11, we only need to use the inner product of ut and (1.15) on L2(Ω).
Then (4.2) and (4.3) can be obtained by using slightly different estimates. □

Remark 4.5. From Theorem 3.17, we know that A is invariant. Thus, by Lemma
4.2, we have

A = {z : t 7→ z(t) = (u(t), ηt) ∈ Mt with z CBT of U(t, τ)},

Lemma 4.6. Under the assumptions of Lemma 4.4, for every sequence zn =
(un, η

t
n) of CBT for the process U(t, τ) generated by (1.1) and any tn → ∞ as

n → ∞, there exists a CBT z̄ = (ū, η̄t) of the semigroup S(t) generated by (4.1)
such that, for each T > 0, up to a subsequence,

lim
n→∞

sup
t∈[−T ,T ]

|un(t+ tn)− ū|22 = 0,

lim
n→∞

sup
t∈[−T ,T ]

∥ηt+tn
n (s)− η̄t(s)∥2µ,1 = 0.

(4.4)



EJDE-2024/22 NONCLASSICAL DIFFUSION EQUATIONS 23

Proof. This lemma will be proved exactly as in [11, Lemma 6.2]. Combining (4.2)
and (4.3) in Lemma 4.4, for any T > 0, we know that

un(·+ tn) is bounded in L∞([−T , T ], H1
0 (Ω)),

∂tun(·+ tn) is bounded in L2([−T , T ], L2(Ω)).

Thus, combining this with Lemma 2.15, it yields, up to a subsequence, that

un(·+ tn) is relatively compact in C([−T , T ], L2(Ω)).

Then there exists a function ū of L2(Ω), such that un(· + tn) → ū(·) in the sense
of (4.4). Particularly, ū ∈ C([−T , T ], L2(Ω)). Also from (4.2), there exists K5 > 0
such that

sup
t∈R

|∇ū|22 ≤ K5. (4.5)

On the other hand, for any z ∈ A(t), by Theorem 3.17 we can obtain that

sup
t≥τ

(∥u∥H1
t
+ ∥ηt∥2µ,2 +

∫ t

τ

|∆u(s)|22ds) ≤ C. (4.6)

By Lemma 3.15 and (4.6), we obtain that the sequence η·+tn
t (s) is bounded in

L∞([−T , T ];L2
µ(R+;H2

a(Ω)) ∩H1
µ(R+;L2(Ω))),

which indicates that

η·+tn
t (s) is relatively compact in C([−T , T ], L2

µ(R+;H1
a(Ω)).

Then there exists a function η̄· ∈ L2
µ(R+;H1

a(Ω), such that, up to a subsequence,

η·+tn
t (s) → η·+tn

t (s).
Next we need to verify that z̄ = (ū, η̄t) solves (4.1). For this purpose, let

υn(t) = un(t+ tn), εn(t) = ε(t+ tn), ηt+tn
n (s) = θtn(s).

Then (1.1) can be rewritten as

∂tυn = εn(t)∆∂tυn +∆υn +

∫ ∞

0

k(s) div{a(x)∇υn(t− s)}ds− f(υn) + g,

or

∂tυn = εn(t)∆∂tυn +∆υn +

∫ ∞

0

µ(s) div{a(x)∇θtn(s)}ds− f(υn) + g.

Next, we handle the first term at the right end of the above formula. For a fixed
T > 0 and every smooth function ϕ with L2-value and supported on (−T , T ),
through the processing method in [30, 11, section 6], we can obtain that there
exists positive constant χ0 fixed later, such that∣∣ ∫ T

−T
(εn(t)∆∂tυn(t), ϕ(t))dt

∣∣ ≤ χ0T sup
t∈[−T ,T ]

√
εn(t) + χ0(

√
εn(T )−

√
εn(−T )).

(4.7)
By (4.2), we have∣∣ ∫ T

−T
(εn(t)∆∂tυn(t), ϕ(t))dt

∣∣
=

∣∣∣− ∫ T

−T
εn(t)(∆υ(t), ϕ

′(t))dt−
∫ T

−T
ε′n(t)(∆υn(t), ϕ(t))dt

∣∣∣



24 J. ZHANG, Z. XIE, Y. XIE EJDE-2024/22

≤ Q(|ϕ′|2)
∫ T

−T
εn|∆υ|2dt+Q1(|ϕ|2)

∫ T

−T
ε′n|∆υn(t)|2dt

≤ Q(|ϕ′|2)
(∫ T

−T
εn(t)|∆υ|22dt

)1/2(∫ T

−T
εn(t)dt

)1/2

+Q(|ϕ|2)1
∫ T

−T

ε′n(t)√
εn(t)

√
εn(t)|∆υn(t)|2dt

≤ Q(
√

K3, |ϕ′|2,
√
T ) sup

t∈[−T ,T ]

√
εn(t) +Q1(

√
K3, |ϕ′|2)(

√
εn(T )−

√
εn(−T )),

where Q1(·, ·) and Q(·, ·, ·) denote positive constants associated with some certain
parameters. Let

χ0 = χ0(K3, |ϕ|2, |ϕ′|2,
√
T ) = max{Q(

√
K3, |ϕ′|2,

√
T ),Q1(

√
K3, |ϕ′|2)}.

Then (4.7) holds.
Because

lim
n→∞

( sup
t∈[−T ,T ]

εn(t)) = 0,

we have

lim
n→∞

( sup
t∈[−T ,T ]

√
εn(t)) = 0.

Therefore, from (4.8), we have

lim
n→∞

∣∣ ∫ T

−T
(εn(t)∆∂tυn(t), ϕ(t))dt

∣∣ = 0. (4.8)

Moreover, by the continuity of f (see (H2)), and υn → ū is almost every conver-
gent in Ω as n→ ∞, this yields (up to a subsequence)

f(υn) → f(ū) and ∆υn → ∆ū (4.9)

in L∞([−T , T ];H−1(Ω)) for any T > 0. In particular, we can also have∫ ∞

0

µ(s) div{a(x)∇θtn(s)}ds→
∫ ∞

0

µ(s) div{a(x)∇η̄t(s)}ds (4.10)

in the sense of distributions. In fact, for any ϕ̄ ∈ C∞
0 (Ω), when n→ ∞, we have∣∣∣ ∫ T

−T
(

∫ ∞

0

µ(s) div{a(x)∇θtn(s)}ds−
∫ ∞

0

µ(s) div{a(x)∇η̄t(s)}ds, ϕ̄)dt
∣∣∣

=
∣∣∣ ∫ T

−T
(

∫ ∞

0

µ(s)(a(x)∇θtn(s)− a(x)∇η̄t(s))ds,∇ϕ̄)dt
∣∣∣

≤
∣∣∣ ∫ T

−T

∫ ∞

0

µ(s)|a(x)∇θtn(s)− a(x)∇η̄t(s)ds|2|∇ϕ̄|2dt
∣∣∣

≤ |∇ϕ̄|2
√
m0|a(x)|∞

∣∣∣ ∫ T

−T

(∫ ∞

0

µ(s)|∇θtn(s)−∇η̄t(s)|22ds
)1/2

dt
∣∣∣ → 0.

(4.11)

From the definition of θtn we obtain that

∂sθ
t
n =

{
un(t− s), τ < s ≤ t

0, s > t,
(4.12)



EJDE-2024/22 NONCLASSICAL DIFFUSION EQUATIONS 25

which together with Theorem 3.8, Lemma 3.12, Lemma 3.14, Theorem 3.17 and
Lemma 2.16 implies that θtn(s) is relatively compact in L2

µ(R+;H1
0 (Ω)) (the proof

is similar to Lemma 3.15). Thus, we have∫ ∞

0

µ(s)|∇θtn(s)−∇η̄t(s)|22ds→ 0, as n→ ∞,

this implies that (4.11) holds.
Meanwhile, in the distributional sense, there exists a subsequence such that

∂tυn → ūt.

In conclusion, we can obtain the equality

ūt −∆ū−
∫ ∞

0

µ(s) div{a(x)∇η̄t(s)}ds+ f(ū) = g,

i.e., in the sense of Dafermos’ transformation,

ūt −∆ū−
∫ ∞

0

k(s) div{a(x)∇ū(t− s)}ds+ f(ū) = g,

which implies that z̄ = (ū, η̄t) is solution of (4.1). Combining (1.12) and (4.5), it is
clear that z̄ is a CBT for the semigroup {S(t)}t≥0. □

By Lemma 4.6 and Theorem 4.3, the following conclusion can be obtained at
once.

Theorem 4.7. If A := {At}t∈R = {A(t)}t∈R and A∞ is time-dependent global
attractors and global attractors of {U(t, τ)}t≥τ and {S(t)}t≥0 generated by (1.1)
and (4.1) respectively. Then there is

lim
t→∞

distM0(ΠtAt,A∞) = 0.

Acknowledgments. This research was supported by the Key Technologies R&D
Program of CNBM (No. 2021HX1617), and by the General Project of Education
Department of Hunan Province (Nos. 21C0660).

References

[1] F. Alabau-Boussouira, P. Cannarsa, G. Fragnelli; Carleman estimates for degenerate para-
bolic operators with applications to null controllability, J. Evol. Equ., 6 (2006), 161-204.

[2] G. Barenblatt, Y. Zheltov, I. Kochina; Basic concepts in the theory of seepage of homogeneous
liquids in fissured rocks, J. Appl. Math. Mec., 24 (1960), 1286-1803.

[3] S. Borini, V. Pata; Uniform attractors for a strongly damped wave equations with linear

memory, Asymptotic Anal., 20 (1999), 263-277.
[4] M. Campiti, G. Metafune, D. Pallara; Degenerate self-adjoint evolution equations on the unit

interval, Semigroup Forum, 57 (1998), 1-36.
[5] P. Cannarsa, D. Rocchetti, J. Vancostenoble; Generation of analytic semi-groups in L2 for a

class of second order degenerate elliptic operators, Control Cybern., 37 (2008), 831-878.
[6] M. M. Cavalcanti, V. N. Domingos Cavalcanti, M. A. Jorge Silva, A. Y. Souza Franco;

Exponential stability for the wave model with localized memory in a past history framework,
J. Differential Equations, 264 (2018), 6535-6584.

[7] M. M. Cavalcanti, L. H. Fatori, T. F. Ma; Attractors for wave equations with degenerate
memory, J. Differential Equations, 260 (2016), 56-83.

[8] P. J. Chen, M. E. Gurtin; On a theory of heat conduction involving two temperatures, Z.
Angew. Math. Phys., 19 (1968), 614-627.

[9] V. V. Chepyzhov, A. Miranville; On trajectory and global attractors for semilinear heat

equations with fading memory, Indiana U. Math. J., 55 (2006), 119-167.



26 J. ZHANG, Z. XIE, Y. XIE EJDE-2024/22

[10] M. Conti, F. Dell’Oro, V. Pata; Nonclassical diffusion with memory lacking instantaneous

damping, Commun. Pur. Appl. Anal., 19 (2020), 2035-2050.

[11] M. Conti, V. Pata; Asymptotic structure of the attractor for processes on time-dependent
spaces, Nonlinear Anal. Real World Appl., 19 (2014), 1-10.

[12] M. Conti, V. Pata, R. Temam; Attractors for process on time-dependent spaces: Applications

to wave equations, J. Differential Equations, 255 (2013), 1254-1277.
[13] M. Conti, E. M. Marchini, V. Pata; Nonclassical diffusion with memory, Math. Method. Appl.

Sci., 38 (2015), 948-958.

[14] C. M. Dafermos; Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal., 37 (1970),
297-308.

[15] J. C. O. Faria, C. M. Webler; Existence of a global attractor for the heat equation with

degenerate memory, J. Dynam. Differential Equations, 35 (2023), 845-864.
[16] H. Gao, L. Li, Z. Liu; Stability of degenerate heat equation in non-cylindrical/cylindrical

domain, Z. Angew. Math. Phys., 70 (2019), 1-17.
[17] S. Gatti, M. Grasselli, V. Pata; Lyapunov functionals for reaction-diffusion equations with

memory, Math. Method. Appl. Sci., 28 (2005), 1725-1735.

[18] C. Giorgi, M. G. Naso, V. Pata; Exponential stability in linear heat conduction with memory:
a semigroup approach, Commun. Appl. Anal., 5 (2001), 121-133.

[19] C. Giorgi, V. Pata, A. Marzocchi; Asymptotic behavior of a semilinear problem in heat

conduction with memory, NoDEA-Nonlinear Diff. Equ. Appl., 5 (1998), 333-354.
[20] M. Grasselli, V. Pata; Uniform attractors for nonautonomous dynamical systems with mem-

ory, Progress Nonlinear Differ. Equ. Appl., 50 (2002), 155-178.

[21] J. Jackle; Heat conduction and relaxation in liquids of high viscosity, Physica A, 162 (1990),
377-404.

[22] P. Kloeden, T. Lorenz; Pullback incremental attraction, Nonauton. Dynam. Syst., 1 (2004),

53-60.
[23] Y. Liu; Time-dependent global attractor for the nonclassical diffusion equation, Appl. Anal.,

94 (2015), 1439-1449.
[24] Q. Ma, X. Wang, L. Xu; Existence and regularity of time-dependent global attractors for

the nonclassical reaction-diffusion equations with lower forcing term, Bound. Value Probl., 1

(2016), 1-11.
[25] F. Meng, M. Yang, C. Zhong; Attractors for wave equation with nonlinear damping on time-

dependent space, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 205-225.

[26] J. C. Robinson; Infinite-Dimensional Dynamical Dystems, Cambridge: Cambridge University
Press, 2001.

[27] J. L. Shomberg; Regular global attractors for wave equations with degenerate memory, Ural

Math. J., 5 (2019), 59-82.
[28] C. Sun, D. Cao, J. Duan; Non-autonomous dynamics of wave equations with nonlinear damp-

ing and critical nonlinearity, Nonlinearity, 19 (2006), 2645-2665.

[29] N. D. Toan; Uniform attractors of nonclassical diffusion equations lacking instantaneous
damping on RN with Memory, Acta Appl. Math., 170 (2020), 789-822.

[30] J. Wang, Q. Ma; Asymptotic dynamic of the nonclassical diffusion equation with time-
dependent coefficient, J. Appl. Anal. Comput., 11 (2020), 445-463.

[31] X. Wang, L. Yang, C. Zhong; Attractors for the nonclassical diffusion equations with fading

memory, J. Math. Anal. Appl., 362 (2010), 327-337.
[32] Y. Xie, D. Liu, J. Zhang, X. Liu; Uniform attractors for nonclassical diffusion equations with

perturbed parameter and memory, J. Math. Phys., 64 (2023), 022701.
[33] Z. Xie, J. Zhang, Y. Xie; Asymptotic behavior of quasi-linear evolution equations on time-

dependent product spaces, Discrete Contin. Dyn. Syst. Ser. B, 28 (2023), 2316-2334.

[34] Y. Xie, Q. Li, K. Zhu; Attractors for nonclassical diffusion equations with arbitrary polyno-

mial growth nonlinearity, Nonlinear Anal. Real World Appl., 31 (2016), 23-37.
[35] Y. Xie, Y. Li, Y. Zeng; Uniform attractors for nonclassical diffusion equations with memory,

J. Funct. Space, 3-4 (2016), 1-11.
[36] J. Yuan, S. Zhang, Y. Xie, J. Zhang; Attractors for a class of perturbed nonclassical diffusion

equations with memory, Discrete Contin. Dyn. Syst. Ser. B, 27 (2022), 4995-5007.

[37] J. Zhang, Y. Xie, Q. Luo, Z. Tang; Asymptotic behavior for the semilinear reaction-diffusion
equations with memory, Adv. Differ. Equations, 1 (2019), 510.



EJDE-2024/22 NONCLASSICAL DIFFUSION EQUATIONS 27

[38] J. Zhang, Z. Liu, J. Huang; Upper semicontinuity of pullback D-attractors for nonlinear

parabolic equation with nonstandard growth condition, Math. Nachr., 296 (2023), 5593-5616.

[39] J. Zhang, Z. Xie, Y. Xie; Long-time behavior of nonclassical diffusion equations with memory
on time-dependent spaces, Asymptotic Anal., (2023), doi: 10.3233/ASY-231887.

[40] K. Zhu, Y. Xie, F. Zhou; Attractors for the nonclassical reaction-diffusion equations on time-

dependent spaces, Bound. Value Probl., 95 (2020), 1-14.

Jiangwei Zhang
College of Science, National University of Defense Technology, Changsha 410073,

China
Email address: zjwmath@163.com

Zhe Xie

Research and Development Center, Sinoma Wind Power Blade Co. Ltd, Beijing 100192,
China

Email address: xiezhe890917@126.com

Yongqin Xie

School of Mathematics and Statistics, Changsha University of Science and Technology,

Changsha 410114, China
Email address: xieyqmath@126.com


	1. Introduction
	2. Preliminaries
	Basic concepts and notation

	3. Existence and uniqueness of a time-dependent global attractor
	3.1. Well-posedness
	3.2. Time-dependent absorbing sets
	3.3. Time-dependent global attractors

	4. Asymptotic structure of the attractor
	Acknowledgments

	References