Long-time behavior of nonclassical diffusion equations with memory on time-dependent spaces

Abstract

This paper aims to study the long-time behavior of nonclassical diffusion equation with memory and disturbance parameters on time-dependent space. By using the contractive process method on the family of time-dependent spaces and operator decomposition technique, the existence of pullback attractors is first proved. Then the upper semi-continuity of pullback attractors with respect to perturbation parameter ν in $M_{t}$ is obtained. It’s remarkable that the nonlinearity f satisfies the polynomial growth of arbitrary order.

Keywords

Nonclassical diffusion equation arbitrary polynomial growth pullback attractor memory upper semi-continuity

1. Introduction

In this paper, the asymptotic behavior of the following nonclassical diffusion equation with two perturbation parameters and memory is considered $\begin{matrix} (1) & u_{t} - ε (t) Δ u_{t} - ν Δ u - \int_{0}^{\infty} k (s) Δ u (t - s) d s + f (u) = g, (x, t) \in Ω \times (τ, \infty), \end{matrix}$ with the boundary condition $\begin{matrix} (2) & u (x, t) |_{\partial Ω} = 0, for all t ⩾ τ, \end{matrix}$ and the initial conditions $\begin{matrix} (3) & u (x, t) |_{t = τ} = u_{τ} (x), u (x, τ - s) |_{s ⩾ 0} = ϕ_{τ} (x, s), for all x \in Ω, τ \in R, \end{matrix}$ where Ω is a bounded smooth domain in $R^{n} (n ⩾ 3)$ , $t > τ$ , $τ \in R$ is initial time, the perturbation parameter $ν \in [0, \infty)$ does not depend on time, the forcing term $g = g (x) \in L^{2} (Ω)$ and $ϕ_{τ} (x, s)$ are given, and the time-dependent perturbation parameter $ε (t) \in C^{1} (R)$ is a decreasing bounded function satisfying $\begin{matrix} (4) & lim_{t \to \infty} ε (t) = 0, \end{matrix}$ and there exists $L > 0$ , such that $\begin{matrix} (5) & sup_{t \in R} (| ε (t) | + | ε^{'} (t) |) ⩽ L . \end{matrix}$

In addition, we establish the following hypotheses for the kernel function $k (s)$ and the nonlinearity f: $(H_{1})$ The hypothesis for the kernel function $k (s)$ : let $μ (s) = - k^{'} (s)$ , and assume $\begin{matrix} (6) & μ \in C^{1} (R^{+}) \cap L^{1} (R^{+}), μ (s) ⩾ 0, μ^{'} (s) ⩽ 0, \forall s \in R^{+}, \end{matrix}$ and there exists a positive constant δ such that $\begin{matrix} (7) & μ^{'} (s) + δ μ (s) ⩽ 0, \forall s \in R^{+} . \end{matrix}$

In particular, for simplicity, let $\begin{matrix} k (0) = \int_{0}^{\infty} μ (s) d s = 1 . \end{matrix}$

$(H_{2})$ The hypothesis for the nonlinearity f: it fulfills $f \in C^{1} (R)$ , $f (0) = 0$ , along with arbitrary order polynomial growth restriction $\begin{matrix} (8) & α_{1} | s |^{p} - β_{1} ⩽ f (s) s ⩽ α_{2} | s |^{p} + β_{2}, \forall s \in R, p ⩾ 2 \end{matrix}$ and dissipative condition $\begin{matrix} (9) & f^{'} (s) ⩾ - l, \end{matrix}$ where $α_{i}, β_{i} (i = 1, 2)$ and l are normal constants.

Let $F (s) = \int_{τ}^{s} f (σ) d σ$ , we can get that there exist $\tilde{α_{i}}, \tilde{β_{i}} > 0 (i = 1, 2)$ from (8), such that $\begin{matrix} (10) & \tilde{α_{1}} | s |^{p} - \tilde{β_{1}} ⩽ F (s) ⩽ \tilde{α_{2}} | s |^{p} + \tilde{β_{2}}, \forall s \in R . \end{matrix}$

Following the Dafermos’ idea of introducing an additional variable $η^{t}$ in [9], we can give the past history of u, i.e., $\begin{matrix} (11) & η^{t} = η^{t} (x, s) : = \int_{0}^{s} u (x, t - r) d r, s \in R^{+} . \end{matrix}$

Provided that $η_{t}^{t} = \frac{\partial}{\partial t} η^{t}$ , $η_{s}^{t} = \frac{\partial}{\partial s} η^{t}$ , then it’s easy to obtain that $\begin{matrix} (12) & η_{t}^{t} = - η_{s}^{t} + u . \end{matrix}$ By $(H_{1})$ and (11)–(12), we have $\begin{matrix} (13) & \int_{0}^{\infty} k (s) Δ u (t - s) d s = \int_{0}^{\infty} μ (s) Δ η^{t} (s) d s . \end{matrix}$ Thus, the system (1) can be rewrite as $\begin{aligned} (14) & \{\begin{array}{l} u_{t} - ε (t) Δ u_{t} - ν Δ u - \int_{0}^{\infty} μ (s) Δ η^{t} (s) d s + f (u) = g, \\ η_{t}^{t} = - η_{s}^{t} + u \end{array} \end{aligned}$ with initial-boundary conditions $\begin{array}{r} (15) & \{\begin{array}{l} u (x, t) |_{\partial Ω} = 0, η^{t} (x, s) |_{\partial Ω \times R^{+}} = 0, t ⩾ τ, \\ u (x, τ) = u_{τ} (x), η^{τ} (x, s) = \int_{0}^{s} u_{τ} (x, τ - r) d r, (x, s) \in Ω \times R^{+} . \end{array} \end{array}$

In the whole paper, unless otherwise stated, we always suppose that $z (t) = (u (t), η^{t})$ is the solution of system (14)–(15) with initial value $z_{τ} = (u_{τ}, η^{τ})$ .

The system (1), as a mathematical and physical model, is usually used in the fields of fluid mechanics, solid mechanics and heat conduction theory [2,4]. It mainly considers two aspects: one is viscosity, the other is the historical influence of u (such as polymer, high viscous liquid, etc. [13]). An energy equation is formed to reveal the whole process of diffusion when these two factors are considered, that is, the above development equation (1) with memory.

The asymptotic behavior of solutions has been studied by many scholars when $k (s) = 0$ and $ε (t)$ is a constant (or small disturbance parameter) of the equation (1) (see e.g., [1,5,26,27,29] and the references therein). In particular, if $ε (t) = 0$ , then the equation (1) becomes usual reaction-diffusion equation with memory, and this equation has been also researched by many researchers (see e.g., [9–12,32] and the references therein). Furthermore, for the equation (1), the current research focuses on the nonclassical diffusion equation with memory when $ε (t)$ is a positive constant (see e.g., [25,30] and the references therein).

Recently, for the case of $k (s) = 0$ in (1), in [16,17], the authors proved the existence and regularity of the time-dependent global attractor on $H_{t}$ when the nonlinearity satisfies the growth of subcritical and critical exponents respectively, and Wang and Ma studied the existence, regularity and asymptotic structure of the time-dependent global attractor in [24] for the following equation $\begin{aligned} (16) & u_{t} - ε (t) Δ u_{t} - Δ u + λ u + f (u) = g (x), \end{aligned}$ where f meets polynomial growth of arbitrary order.

Based on the above-mentioned researches, there is no relevant research for the problem (1) with memory. Thus, we can study the asymptotic behavior of solutions to equation (1). Particularly, when the linear damping $λ u$ and instantaneous damping $- Δ u$ in equation (16) are replaced by memory term $\int_{0}^{\infty} k (x) Δ u (t - s) d s$ (or it is said to be that $ν = 0$ in equation (1)), the corresponding equation will become a class of equation with perturbed parameters and memory lacking instantaneous damping, similar situations have been considered in [6,23]. In addition, we know that the relationship of attractors to two kinds of equations can usually be depicted by upper semi-continuity. This inspires us to further consider the upper semi-continuity (w.r.t. ν) of the pullback attractor for the equation (1) in the time-dependent space as $ν \to 0$ . It is worth emphasizing that the nonlinearity f satisfies the exponential growth of arbitrary order rather than critical growth in [6,23].

In order to study the above mentioned problem, the analytical techniques, the contractive function method and the operator decomposition technique are all applied to solve the above problem. And especially, we shall encounter the following some difficulties in this paper for tackling the corresponding problems:

$(i)$ under the case of time-dependent space, the verification of asymptotic compactness of the corresponding process ${U (t, τ)}_{t ⩾ τ}$ associated with the equation (1) is complex. Specifically, we cannot directly construct the contractive function by using methods in [28,29].

$(ii)$ since the perturbation parameter ν is before $- Δ u$ instead of $- Δ u_{t}$ , which causes that the (1) inherently lacks an instantaneous damping term. Thus, usually analysis methods are no longer applicable when we consider existence, regularity the semicontinuity of pullback attractors over time-dependent space.

Accordingly, we use the basic ideas in [29,31] to overcome the above difficulties, combining with the analytical techniques, so as to obtain the existence of time-dependent global attractors in the product space $M_{t}$ . In addition, the operator decomposition method based on the idea in [21,28] are used to obtain the asymptotic regularity of solutions. These results improves the existing conclusions.

For conveniences, hereafter let $| \cdot |_{p}$ be the norm of $L^{p} (Ω) (p ⩾ 1)$ . Let $(\cdot, \cdot)$ , ${⟨ \cdot, \cdot ⟩}_{H_{0}^{1}} = (\nabla \cdot, \nabla \cdot)$ and ${⟨ \cdot, \cdot ⟩}_{D (A)} = (Δ \cdot, Δ \cdot)$ be the inner product of $L^{2} (Ω)$ , $H_{0}^{1} (Ω)$ and $D (A) = H_{0}^{1} (Ω) \cap H^{2} (Ω)$ with the norms $| \cdot |_{2}$ , $‖ \cdot ‖_{0} = | \nabla \cdot |_{2}$ and $‖ \cdot ‖_{1} = | Δ \cdot |_{2}$ respectively.

The time-dependent space $H_{t} : = H_{0}^{1} (Ω)$ and $H_{t}^{1} : = D (A)$ are equipped with the norms: $\begin{matrix} ‖ \cdot ‖_{H_{t}}^{2} = | \cdot |_{2}^{2} + ε (t) ‖ \cdot ‖_{0}^{2} and ‖ \cdot ‖_{H_{t}^{1}}^{2} = ‖ \cdot ‖_{0}^{2} + ε (t) ‖ \cdot ‖_{1}^{2} . \end{matrix}$ Denote the weight space $V = L_{μ}^{2} (R^{+}; H_{0}^{1} (Ω))$ and $V_{1} = L_{μ}^{2} (R^{+}; D (A))$ , and their inner product and norm are $\begin{array}{c} {⟨ ψ, ϕ ⟩}_{V} = \int_{0}^{\infty} μ (s) (\nabla ψ, \nabla ϕ) d s, ‖ ϕ ‖_{μ, 0}^{2} = \int_{0}^{\infty} μ (s) ‖ ϕ ‖_{0}^{2} d s, \forall ψ, ϕ \in V; \\ {⟨ ψ, ϕ ⟩}_{V_{1}} = \int_{0}^{\infty} μ (s) (Δ ψ, Δ ϕ) d s, ‖ ϕ ‖_{μ, 1}^{2} = \int_{0}^{\infty} μ (s) ‖ ϕ ‖_{1}^{2} d s, \forall ψ, ϕ \in V_{1} . \end{array}$ Then, by the assumptions of symbols above, the norm of time-dependent product space $M_{t}$ and $M_{t}^{1}$ corresponding to the equation (1) can be defined as $\begin{matrix} M_{t} = H_{t} \times V and M_{t}^{1} = H_{t}^{1} \times V_{1} \end{matrix}$ and they are equipped norms as follows $\begin{matrix} ‖ \cdot ‖_{M_{t}}^{2} = ‖ \cdot ‖_{H_{t}}^{2} + ‖ \cdot ‖_{μ, 0}^{2} and ‖ \cdot ‖_{M_{t}^{1}}^{2} = ‖ \cdot ‖_{H_{t}^{1}}^{2} + ‖ \cdot ‖_{μ, 1}^{2} \end{matrix}$ respectively.

Particularly, we also use ${M_{t}}_{t \in R}$ to denote a family of normed time-dependent product spaces. Moreover, we introduce some common notations based on processes of time-dependent space (see e.g., [3,7,8]).

Let ${X_{t}}_{t \in R}$ be a family of normed space. Note that the ball with radius of R in $X_{t}$ is $\begin{matrix} B_{t} (R) = {w \in X_{t} : ‖ w ‖_{X_{t}} ⩽ R} . \end{matrix}$ For any given $ε > 0$ , we define the ε neighborhood of set $B \subset X_{t}$ as follows $\begin{matrix} O_{t}^{ε} (B) = ⋃_{x \in B} {x \in X_{t} : ‖ x - y ‖_{X_{t}} < ε} = ⋃_{x \in B} {y + B_{t} (ε)} . \end{matrix}$ Hausdorff semidistance of between two nonempty sets $A, B \subset X_{t}$ is defined as $\begin{aligned} (17) & {dist}_{X_{t}} (A, B) = sup_{x \in A} inf_{y \in B} ‖ x - y ‖_{X_{t}} . \end{aligned}$

Remark 1.1.
$X_{t}$ , as mentioned above or as will appear later, can be time-dependent product space. In particular, unless otherwise specified in this article, $X_{t}$ denotes time-dependent product space.

The plan of this paper is as follows. In Section 2, we recall some basic concepts as to time-dependent pullback attractors and useful results that will be used later. In Section 3, we obtain the existence of time-dependent pullback attractors and the asymptotic regularity of solutions of problem (1). In Section 4, we prove the upper semi-continuity of time-dependent pullback attractors in $M_{t}$ .
2. Preliminaries

In this section, we review some concepts of time-dependent global attractors and restate the results of the existence of time-dependent global attractors (see [7,8] for more details).

Definition 2.1.
Let ${X_{t}}_{t ⩾ τ}$ be a family of normed spaces. A two-parameter operator ${U (t, τ)}_{t ⩾ τ}$ defined on ${X_{t}}_{t ⩾ τ}$ is called a processes, if a family of mappings ${U (t, τ) : X_{τ} \to X_{t}, t ⩾ τ \in R}$ satisfies the following properties:
$U (τ, τ) = Id$ , $τ \in R$ (Identity operator);

$U (t, s) U (s, τ) = U (t, τ)$ , $\forall t ⩾ s ⩾ τ$ , $τ \in R$ .

Definition 2.2.
A family of sets $\tilde{C} = {C_{t} : C_{t} \subset X_{t} is bounded}_{t \in R}$ is called uniformly bounded if there exists a constant $R > 0$ , such that $C_{t} \subset B_{t} (R)$ for any $t \in R$ .
Definition 2.3.
A family of sets $\tilde{B} = {B_{t}}_{t \in R}$ is called a family of pullback absorbing sets if $\tilde{B} = {B_{t}}_{t \in R}$ is uniformly bounded and for all $R > 0$ , there exists a constant $t_{0} = t_{0} (t, R) ⩽ t$ such that $U (t, τ) B_{τ} (R) \subset B_{t}$ for any $τ ⩽ t_{0}$ .

The process ${U (t, τ)}_{t ⩾ τ}$ is called dissipative whenever it enters a pullback absorbing family.
Definition 2.4.
A time-dependent absorbing set for the process ${U (t, τ)}_{t ⩾ τ}$ is a uniformly bounded family $\tilde{B} = {B_{t}}_{t \in R}$ with the following characteristics: for any $R > 0$ , there exists a $t_{0} : = t_{0} (R) ⩾ 0$ , such that $\begin{matrix} U (t, τ) B_{τ} (R) \subset B_{t} for all τ ⩽ t - t_{0} \end{matrix}$
Remark 2.5.
The pullback attracting can be equivalently described in the light of pullback absorbing: a (uniformly bounded) family $K = {K_{t}}_{t \in R}$ is said to be pullback attracting if for any $ε > 0$ the family ${O_{t}^{ε} (K_{t})}_{t \in R}$ is pullback absorbing.
Definition 2.6.
$\tilde{A} = {A_{t}}_{t \in R}$ is said to be invariant, if $\begin{matrix} U (t, τ) A_{τ} = A_{t}, \forall τ ⩽ t . \end{matrix}$

Setting the collection $\begin{matrix} K = {K = {K_{t}}_{t \in R} : K_{t} \subset M_{t} is compact, K is pullback attracting} . \end{matrix}$
Definition 2.7.
We call a time-dependent global attractor the smallest element of $K$ , i.e., the family $\tilde{A} = {A_{t}}_{t \in R} \in K$ such that $\begin{matrix} A_{t} \subset K_{t}, \forall t \in R, \end{matrix}$ for any element $K = {K_{t}}_{t \in R} \in K$ .
Definition 2.8.
A time-dependent global attractor of the process ${U (t, τ)}_{t ⩾ τ}$ is a smallest family $\tilde{A} = {A_{t}}_{t \in R}$ such that
$\tilde{A}$ is compact in $X_{t}$ ;

$\tilde{A}$ is pullback attracting, namely, $\tilde{A}$ is uniformly bounded and $\begin{matrix} lim_{τ \to - \infty} {dist}_{X_{t}} (U (t, τ) C_{τ}, A_{t}) = 0; \end{matrix}$
hold for any uniformly bounded family $\tilde{C} = {C_{τ}}_{τ \in R}$ and every fixed $t \in R$ and $τ ⩽ t$ . It should be noted that if $U (t, τ)$ satisfies some continuity, then it’s easy know that $\tilde{A}$ is invariant.
Theorem 2.9.
A time-dependent global attractor $\tilde{A}$ exists and it is unique if and only if the process ${U (t, τ)}_{t ⩾ τ}$ is asymptotically compact, i.e., the set $K$ is non-empty.
Remark 2.10.
If we know that $K$ is an invariant pullback attracting family of compact sets, it is clear that $K$ is the smallest element of $K$ , hence it coincides with the time-dependent attractor $\tilde{A}$ .

The main results of this paper (the existence of time-dependent global attractors) can be obtained by the following definitions and theorem. Next, let’s talk about it (it’s similar to [14,18,20,29,30,32]).
Definition 2.11.
The process ${U (t, τ)}_{t ⩾ τ}$ is called pullback asymptotically compact if for any fixed $t \in R$ , bounded sequence ${z_{n}}_{n = 1}^{\infty} \subset X_{τ_{n}}$ and any ${τ_{n}}_{n = 1}^{\infty} \subset R^{- t}$ with $τ_{n} \to - \infty$ as $n \to \infty$ , the sequence ${U (t, τ_{n}) z_{n}}_{n = 1}^{\infty}$ is precompact in $X_{t}$ , where $R^{- t} = {τ : τ \in R, τ ⩽ t}$ .

From Theorem 2.9, we know that the existence of time-dependent global attractors can be obtained by verifying the asymptotic compactness of the corresponding process. Thus, from [18], it follows that
Lemma 2.12.
If $U (\cdot, \cdot)$ is asymptotically compact, then it is pullback asymptotically compact.
Definition 2.13.
Let ${M_{t}}_{t \in R}$ be a family of Banach spaces and $\tilde{B} = {B_{t}}_{t \in R}$ be a family of uniformly bounded subset of ${M_{t}}_{t \in R}$ . A function $φ_{τ}^{t} (\cdot, \cdot)$ defined on $M_{t} \times M_{t}$ is called the contractive function on $B_{τ} \times B_{τ}$ , if for any fixed $t \in R$ and any sequence ${z_{n} = (x_{n}, σ_{n})}_{n = 1}^{\infty} \subset B_{τ} \subset M_{τ}$ , there exists a subsequence ${z_{n_{k}} = (x_{n_{k}}, σ_{n_{k}})}_{k = 1}^{\infty} \subset {z_{n} = (x_{n}, σ_{n})}_{n = 1}^{\infty}$ such that $\begin{matrix} lim_{k \to \infty} lim_{l \to \infty} φ_{τ}^{t} (z_{n_{k}}, z_{n_{l}}) = 0 for all τ ⩽ t . \end{matrix}$

We use $E (B_{τ})$ to denote the set all contractive function on $B_{τ} \times B_{τ}$ .
Lemma 2.14.
Let ${U (t, τ)}_{t ⩾ τ}$ be a process on ${M_{t}}_{t \in R}$ and have a pullback bounded absorbing set ${\tilde{B}}_{0} = {B_{t}^{0}}_{t \in R}$ . ${U (t, τ)}_{t ⩾ τ}$ is called $M_{t}$ -contractive process if for any $ε > 0$ , there exist $T (ε) ⩽ t$ and $φ_{T}^{t} (\cdot, \cdot) \in E (B_{T}^{0})$ such that $\begin{matrix} {‖ U (t, T) z_{1} - U (t, T) z_{2} ‖}_{M_{t}} ⩽ ε + φ_{T}^{t} (z_{1}, z_{2}), for all z_{i} = (x_{i}, σ_{i}) \in B_{T}^{0} (i = 1, 2) \end{matrix}$ holds for any fixed $t \in R$ , where $φ_{T}^{t}$ depends on T. Then ${U (t, τ)}_{t ⩾ τ}$ is pullback asymptotically compact
Proof.
This conclusion can be proved by imitating theorem 4.2 in [18] word by word, and hence is omitted. □

Next, we will give a method to prove the existence of time-dependent global attractors for evolution equations, which will be used in the later discussion (see Theorem 3.3 of [18] for more details). Theorem 2.15.
Let ${M_{t}}_{t \in R}$ be a family of Banach spaces, then ${U (t, τ)}_{t ⩾ τ}$ has a time-dependent global attractor in ${M_{t}}_{t \in R}$ for any $t \in R$ , if the following conditions hold

(i) ${U (t, τ)}_{t ⩾ τ}$ has a pullback absorbing set ${\tilde{B}}_{0} = {B_{t}^{0}}_{t \in R}$ in ${M_{t}}_{t ⩾ τ}$ ;

(ii) ${U (t, τ)}_{t ⩾ τ}$ is a $M_{t}$ -contractive process.
Lemma 2.16 ([19]).

Let $X \subset \subset H \subset Y$ be Banach spaces, with X reflexive and $t \in R$ be fixed. Suppose that $u_{n}$ is a sequence that is uniformly bounded in $L^{2} (T, t; X)$ and $d u_{n} / d t$ is uniformly bounded in $L^{p} (T, t; Y)$ , for some $p > 1$ . Then there is a subsequence of $u_{n}$ that converges strongly in $L^{2} (T, t; H)$ for any $T ⩽ t$ .

3. The pullback attractors in time-dependent $M_{t}$

Firstly, the well-posedness for the equation (1) can be obtained by using Faedo–Galerkin method (see e.g., [22,33]). In the following, we only give result.

Lemma 3.1.
Let $Ω \subset R^{n} (n ⩾ 3)$ be bounded smooth domain, for any $T ⩾ τ$ and $z_{τ} = (u_{τ}, η^{τ}) \in M_{τ}$ . The system ( 1 ) has unique weak solution $\begin{matrix} z = (u (x, t), η^{t}) \end{matrix}$ satisfying $\begin{aligned} u \in C (τ, T; H_{t}) \cap L^{p} (τ, T; L^{p} (Ω)) \cap L^{2} (τ, T; H_{0}^{1} (Ω)); \\ η^{t} \in C (τ, T; V) \cap L^{2} (τ, T; V) . \end{aligned}$ In addition, it’s easy to get that there exists a constant $C_{ℜ}$ independent of t, such that the process ${U^{ν} (t, τ)}_{t ⩾ τ}$ is Lipschitz continuous $\begin{matrix} (18) & {‖ U^{ν} (t, τ) z_{τ}^{1} - U^{ν} (t, τ) z_{τ}^{2} ‖}_{M_{t}} ⩽ C_{ℜ} {‖ z_{τ}^{1} - z_{τ}^{2} ‖}_{M_{τ}}, \forall t ⩾ τ . \end{matrix}$

Through Lemma 3.1, we know that the solution process on time-dependent space $M_{t}$ can be defined as $\begin{matrix} (19) & U^{ν} (t, τ) : M_{τ} \to M_{t}, U (t, τ) z_{τ} = z (t), \forall t ⩾ τ \in R \end{matrix}$ and the process ${U^{ν} (t, τ)}_{t ⩾ τ}$ is a strongly continuous process on the time-dependent phase space $M_{t}$ .
3.1. Time-dependent absorbing sets

Next, we will deal with the dissipative feature of the process ${U^{ν} (t, τ)}_{t ⩾ τ}$ . For arriving at this purpose, we firstly give the following technical lemma, and then prove the existence of time-dependent absorbing sets.

Lemma 3.2.
Assume that $z (t) = (u (t), η^{t})$ is a sufficiently regular solution to ( 14 )–( 15 ). Then, the functional $\begin{matrix} N_{0} (t) = - {⟨ ε (t) u (t), η^{t} ⟩}_{V}, \end{matrix}$ satisfies the following differential inequality $\begin{aligned} \frac{d}{d t} N_{0} (t) + \frac{ε (t)}{2} {‖ u (t) ‖}_{0}^{2} \\ (20) & ⩽ ε (t) {‖ u_{t} (t) ‖}_{0} {‖ η^{t} ‖}_{μ, 0} - ε^{'} (t) {‖ u (t) ‖}_{0} {‖ η^{t} ‖}_{μ, 0} - \frac{μ (0) ε (t)}{2} \int_{0}^{\infty} μ^{'} (s) {‖ η^{t} (s) ‖}_{0}^{2} d s . \end{aligned}$ Besides, we have the control $\begin{matrix} (21) & | N_{0} (t) | ⩽ ϱ_{0} {‖ (u (t), η^{t}) ‖}_{M_{t}}^{2} = ϱ_{0} [{| u (t) |}_{2}^{2} + ε (t) {‖ u (t) ‖}_{0}^{2} + {‖ η^{t} ‖}_{μ, 0}^{2}], \end{matrix}$ where $ϱ_{0} = max {1, \frac{L}{2}}$ .
Proof.
Similar to the proof of Lemma 3.3 in [6], this result can be easily proved. □
Remark 3.3.
In fact, if $ν > 0$ in (1), then the equation (1) becomes the usual nonclassical diffusion equation with memory, at this moment, we can prove the existence of pullback attractors by using the method in [33], thus we don’t need Lemma 3.2. However, if $ν = 0$ in (1), then the equation (1) degenerates to the nonclassical diffusion equation with memory lacking instantaneous damping, at this moment, the equation lacks dissipation, so it is necessary to construct the extra functional $N_{0}$ (i.e., Lemma 3.2) as mentioned in [6,23]. In this paper, we will unify the two situations, therefore, the introduction of the Lemma 3.2 is necessary.
Theorem 3.4.
Let f satisfy ( 8 )–( 9 ), $ε (t)$ satisfy ( 4 )–( 5 ), the kernel function μ satisfy $(H_{1})$ , $g \in L^{2} (Ω)$ and $z_{τ} = (u_{τ}, η^{τ}) \in B_{τ} (R) \subset M_{τ}$ , then there exists $ρ_{0} > 0$ such that, the solution process ${U^{ν} (t, τ)}_{t ⩾ τ}$ corresponding to the equation ( 14 ) possesses a time-dependent bounded absorbing set ${\tilde{B}}_{0} = {B_{t}^{0} = B_{t} (ρ_{0})}_{t \in R}$ , i.e., $\begin{matrix} (22) & B_{t}^{0} = {(u, η^{t}) \in H_{t} \times L_{μ}^{2} (R^{+}; H_{0}^{1} (Ω)) : | u |_{2}^{2} + ε (t) ‖ u ‖_{0}^{2} + {‖ η^{t} ‖}_{μ, 0}^{2} ⩽ ρ_{0}} . \end{matrix}$ In addition, the uniformly bounded family ${\tilde{B}}_{0}$ has the following property: for any $ρ_{0} > 0$ , there exists $t_{0} = t_{0} (R) ⩾ 0$ , such that $\begin{matrix} U (t, τ) B_{τ} (ρ_{0}) \subset B_{t}^{0}, τ ⩽ t - t_{0} . \end{matrix}$
Proof.
Multiplying the first equation of (14) by u and integrating over Ω, by (8) and Hölder inequality, we get $\begin{matrix} (23) & \frac{d}{d t} E (t) + \frac{α_{1}}{2} | u |_{p}^{p} + (ν - ε^{'} (t)) ‖ u ‖_{0}^{2} + \frac{1}{2} \int_{0}^{\infty} (- μ^{'} (s)) {‖ η^{t} (s) ‖}_{0}^{2} d s ⩽ R_{1}, \end{matrix}$ where both $E (t) = \frac{1}{2} [| u |_{2}^{2} + ε (t) ‖ u ‖_{0}^{2} + ‖ η^{t} ‖_{μ, 0}^{2}]$ and $\begin{array}{r} R_{1} = \{\begin{array}{ll} \frac{1}{2} | g |_{2}^{2} + [β_{1} + \frac{p - 2}{p} {(\frac{4}{α_{1} p})}^{\frac{2}{p - 2}}] | Ω |, & p > 2, \\ \frac{1}{2 α_{1}} | g |_{2}^{2} + β_{1} | Ω |, & p = 2, \end{array} \end{array}$ are independent of ν.

From (23), for any $τ < t ⩽ τ + 1$ , we get that $\begin{matrix} (24) & {| u (t) |}_{2}^{2} + ε (t) {‖ u (t) ‖}_{0}^{2} + {‖ η^{t} ‖}_{μ, 0}^{2} + \int_{τ}^{t} {| u (s) |}_{p}^{p} d s ⩽ \frac{2 (R_{1} + ‖ z_{τ} ‖_{M_{τ}})}{γ_{0}}, \end{matrix}$ where $γ_{0} = min {1, α_{1}}$ . It follows for a.e. $t \in (τ, τ + 1]$ that $\begin{matrix} {| u (t) |}_{p}^{p} ⩽ \frac{2 (R_{1} + ‖ z_{τ} ‖_{M_{τ}})}{γ_{0}} . \end{matrix}$ Therefore, there exists the following result by removing a zero measure set E from $(τ, τ + 1]$ $\begin{matrix} inf_{E \subset (τ, τ + 1], | E | = 0} (sup_{(τ, τ + 1] / E} {| u (t) |}_{p}^{p}) ⩽ \frac{2 (R_{1} + ‖ z_{τ} ‖_{M_{τ}})}{γ_{0}} < \infty . \end{matrix}$ Furthermore, in order to get $L^{2} \to L^{p}$ smoothing property for u, we will add the definition of the $L^{p}$ -value of u at τ, i.e., $\begin{matrix} (25) & {| u (τ) |}_{p}^{p} ⩽ \frac{2 (R_{1} + ‖ z_{τ} ‖_{M_{τ}})}{γ_{0}} . \end{matrix}$

Next, multiplying the first equation of (14) by $ε (t) u_{t}$ on $L^{2} (Ω)$ and combining with (5) and (10) we get $\begin{aligned} \frac{d}{d t} [⟨ ε (t) F (u), 1 ⟩ + \frac{ν ε (t)}{2} ‖ u ‖_{0}^{2}] + \frac{ε (t)}{2} | u_{t} |_{2}^{2} + ε^{2} (t) ‖ u_{t} ‖_{0}^{2} - \frac{ν ε^{'} (t)}{2} ‖ u ‖_{0}^{2} \\ (26) & ⩽ ε (t) ‖ u_{t} ‖_{0} {‖ η^{t} ‖}_{μ, 0} + \frac{L}{2} | g |_{2}^{2} + {\tilde{β}}_{1} L | Ω | . \end{aligned}$ At this point, choosing $\begin{matrix} (27) & ϵ = \frac{1}{2} min {1, \frac{1}{L}, \frac{1}{2 μ (0) L}, \frac{- 2 + \sqrt{4 + δ L}}{L}} . \end{matrix}$

Moreover, for fixed $ϵ \in (0, 1)$ later, we define a functional $\begin{matrix} B_{ϵ} (t) = E (t) + ϵ ⟨ ε (t) F (u), 1 ⟩ + \frac{ϵ ν ε (t)}{2} ‖ u ‖_{0}^{2} + ϵ N_{0} (t) . \end{matrix}$ Then we can obtain that there exist constants $α_{0} : = α_{0} (ϵ) = min {\frac{1}{2} - ϵ ϱ_{0}, ϵ {\tilde{α}}_{1}} > 0$ and ${\tilde{α}}_{0} : = {\tilde{α}}_{0} (ϵ) = max {\frac{1}{2} + \frac{ϵ ν}{2} + ϵ ϱ_{0}, ϵ {\tilde{α}}_{2} L} > 0$ , such that $\begin{matrix} (28) & B_{ϵ} (t) ⩾ α_{0} (| u |_{2}^{2} + ε (t) ‖ u ‖_{0}^{2} + {‖ η^{t} ‖}_{μ, 0}^{2} + ε (t) | u |_{p}^{p}) - ϵ {\tilde{β}}_{1} L | Ω | \end{matrix}$ and $\begin{matrix} (29) & B_{ϵ} (t) ⩽ {\tilde{α}}_{0} (| u |_{2}^{2} + ε (t) ‖ u ‖_{0}^{2} + {‖ η^{t} ‖}_{μ, 0}^{2} + ε (t) | u |_{p}^{p}) + ϵ {\tilde{β}}_{2} L | Ω |, \end{matrix}$ hold respectively, where (21) and (10) are used.

Together with (20), (23) and (26), we obtain $\begin{aligned} \frac{d}{d t} B_{ϵ} + (\frac{1}{4} - \frac{ϵ μ (0) L}{2}) \int_{0}^{\infty} (- μ^{'} (s)) {‖ η^{t} (s) ‖}_{0}^{2} d s + \frac{δ}{4} {‖ η^{t} ‖}_{μ, 0}^{2} + \frac{ϵ ε (t)}{2} | u_{t} |_{2}^{2} \\ + ϵ ε^{2} (t) ‖ u_{t} ‖_{0}^{2} + [ν - ε^{'} (t) - \frac{ν ε^{'} (t)}{2} + \frac{ϵ ε (t)}{2}] ‖ u ‖_{0}^{2} + \frac{α_{1}}{2} | u |_{p}^{p} \\ (30) & ⩽ R_{1} + \frac{ϵ L}{2} | g |_{2}^{2} + ϵ {\tilde{β}}_{1} L | Ω | + 2 ϵ ε {‖ u_{t} (t) ‖}_{0} {‖ η^{t} ‖}_{μ, 0} - ϵ ε^{'} (t) ‖ u ‖_{0} {‖ η^{t} ‖}_{μ, 0} . \end{aligned}$ In addition, we have $\begin{aligned} (31) & 2 ϵ ε (t) ‖ u_{t} ‖_{0} {‖ η^{t} ‖}_{μ, 0} ⩽ \frac{ϵ ε^{2} (t)}{2} ‖ u_{t} ‖_{0}^{2} + 2 ϵ {‖ η^{t} ‖}_{μ, 0}^{2}, \end{aligned}$ and $\begin{aligned} (32) & - ϵ ε^{'} {‖ u (t) ‖}_{0} {‖ η^{t} ‖}_{μ, 0} ⩽ \frac{- ε^{'} (t)}{2} {‖ u (t) ‖}_{0}^{2} + \frac{ϵ^{2} L}{2} {‖ η^{t} ‖}_{μ, 0}^{2} . \end{aligned}$

Thus, combining with (30)–(32), we get $\begin{aligned} \frac{d}{d t} B_{ϵ} + (\frac{1}{4} - \frac{ϵ μ (0) L}{2}) \int_{0}^{\infty} (- μ^{'} (s)) {‖ η^{t} (s) ‖}_{0}^{2} d s + (\frac{δ}{4} - 2 ϵ - \frac{ϵ^{2} L}{2}) {‖ η^{t} ‖}_{μ, 0}^{2} \\ (33) & + \frac{1}{2} [2 ν - ε^{'} (t) - ν ε^{'} (t) + ϵ ε (t)] ‖ u ‖_{0}^{2} + \frac{α_{1}}{2} | u |_{p}^{p} ⩽ R_{1} + \frac{ϵ L}{2} | g |_{2}^{2} + ϵ {\tilde{β}}_{1} L | Ω |, \end{aligned}$ where we require $\begin{matrix} \frac{1}{4} - \frac{ϵ μ (0) L}{2} ⩾ 0, ϱ_{1} : = \frac{δ}{4} - 2 ϵ - \frac{ϵ^{2} L}{2} > 0 . \end{matrix}$ Since $| u |_{p}^{p} ⩾ \frac{p}{2} - \frac{p - 2}{2} | Ω |$ , then we can get by choosing appropriate $ϵ = min {1, \frac{1}{2 μ (0) L}, \frac{\sqrt{4 + \frac{δ L}{2}} - 2}{L}}$ $\begin{aligned} (34) & \frac{d}{d t} B_{ϵ} + ϱ_{1} {‖ η^{t} ‖}_{μ, 0}^{2} + \frac{ϵ ε (t)}{2} ‖ u ‖_{0}^{2} + \frac{p α_{1}}{8} | u |_{2}^{2} + \frac{α_{1} ε (t)}{4 L} | u |_{p}^{p} ⩽ R_{2}, \end{aligned}$ where $R_{2} = R_{1} + \frac{ϵ L}{2} | g |_{2}^{2} + (ϵ {\tilde{β}}_{1} L + \frac{p - 2}{2}) | Ω |$ .

According to (29), let $ϱ_{2} = min {ϱ_{1}, \frac{ϵ}{2}, \frac{p α_{1}}{8}, \frac{α_{1}}{4 L}} > 0$ , such that $\begin{matrix} (35) & \frac{d}{d t} B_{ϵ} + ϱ_{2} B_{ϵ} ⩽ R_{2} . \end{matrix}$ Applying Gronwall Lemma, it yields $\begin{matrix} B_{ϵ} (t) ⩽ B_{ϵ} (τ) e^{- ϱ_{2} (t - τ)} + \frac{R_{2}}{ϱ_{2}} . \end{matrix}$ In light of (28) and (29), we get $\begin{aligned} (36) & | u |_{2}^{2} + ε (t) ‖ u ‖_{0}^{2} + {‖ η^{t} ‖}_{μ, 0}^{2} + ε (t) | u |_{p}^{p} & ⩽ \frac{Q (R)}{α_{0}} e^{- ϱ_{2} (t - τ)} + \frac{R_{2} + ϵ ϱ_{2} {\tilde{β}}_{1} L | Ω |}{ϱ_{2} α_{0}}, \end{aligned}$ where $Q (R) = {\tilde{α}}_{0} (\frac{2 L (R_{1} + R)}{γ_{0}} + R) + ϵ {\tilde{β}}_{2} L | Ω |$ .

From (36), we know that there exists $t_{0} : = t_{0} (R) = \frac{1}{ϱ_{2}} ln \frac{Q (R) ϱ_{2}}{R_{2} + ϵ ϱ_{2} {\tilde{β}}_{1} L | Ω |} (> 0)$ , such that $\begin{matrix} | u |_{2}^{2} + ε (t) ‖ u ‖_{0}^{2} + {‖ η^{t} ‖}_{μ, 0}^{2} + ε (t) | u |_{p}^{p} ⩽ ρ_{0} \end{matrix}$ holds true for all $t - τ ⩾ t_{0}$ , where $ρ_{0} = 2 \frac{R_{2} + ϵ ϱ_{2} {\tilde{β}}_{1} L | Ω |}{ϱ_{2} α_{0}}$ . This proof is completed. □

3.2. The existence of pullback attractors

In subsection, we will prove the existence of pullback attractors in ${M_{t}}_{t \in R}$ through the process defined by (19). In order to get Theorem 3.12, firstly, we give the following lemmas.

Corollary 3.5.
Under the assumptions of Lemma 3.4 , there exists $K_{0} : = K_{0} (R)$ such that $\begin{matrix} (37) & {| u (t) |}_{2}^{2} + ε (t) {‖ u (t) ‖}_{0}^{2} + {‖ η^{t} ‖}_{μ, 0}^{2} + ε (t) {| u (t) |}_{p}^{p} ⩽ K_{0} \end{matrix}$ holds for all $t ⩾ τ$ .
Proof.
From (36), let $K_{0} = \frac{Q (R)}{α_{0}} + \frac{R_{2} + ϵ ϱ_{2} {\tilde{β}}_{1} L | Ω |}{ϱ_{2} α_{0}}$ , then it’s easy to get that the above conclusion is true. □
Corollary 3.6.
Under the assumptions of Lemma 3.4 , there exists $K_{1} : = K_{1} (ρ_{0}) > 0$ , such that the following estimate $\begin{array}{r} \int_{t}^{t + 1} ({‖ η^{s} ‖}_{μ, 0}^{2} + {| u (s) |}_{p}^{p}) d s ⩽ K_{1} \end{array}$ holds for any $τ ⩽ t - t_{0}$ .
Proof.
According to (23), we obtain $\begin{matrix} (38) & \begin{array}{r} \frac{d}{d t} (| u |_{2}^{2} + ε (t) ‖ u ‖_{0}^{2} + {‖ η^{t} ‖}_{μ, 0}^{2}) + ϱ_{3} ({‖ η^{t} ‖}_{μ, 0}^{2} + | u |_{p}^{p}) ⩽ R_{1}, \end{array} \end{matrix}$ where $ϱ_{3} = min {δ, α_{1}}$ .

Integrating (38) about t from t to $t + 1$ and using Lemma 3.4, we can get for any $t ⩾ t_{0} + τ$ $\begin{matrix} (39) & \int_{t}^{t + 1} [{‖ η^{s} ‖}_{μ, 0}^{2} + {| u (s) |}_{p}^{p}] d r ⩽ K_{1}, \end{matrix}$ where $K_{1} = \frac{1}{γ_{2}} (R_{1} + ρ_{0})$ . This proof is complete. □
Lemma 3.7.
Under the assumptions of Lemma 3.4 , there exists a constant $K_{2} : = K_{2} (R) > 0$ , such that $\begin{array}{r} \int_{t}^{t + 1} (ε (s) {| u_{t} (s) |}_{2}^{2} + ε^{2} (s) {‖ u_{t} (s) ‖}_{0}^{2}) d s ⩽ K_{2}, \forall t ⩾ τ . \end{array}$
Proof.
By using Young inequality, from (26), we can get $\begin{matrix} (40) & \frac{d}{d t} [⟨ 2 ε (t) F (u), 1 ⟩ + ν ε (t) ‖ u ‖_{0}^{2}] + ε (t) | u_{t} |_{2}^{2} + ε^{2} (t) ‖ u_{t} ‖_{0}^{2} ⩽ {‖ η^{t} ‖}_{μ, 0}^{2} + L | g |_{2}^{2} + 2 \tilde{β_{1}} L | Ω | . \end{matrix}$ Integrating (40) about t on $[t, t + 1]$ and combining with (5) and (10), we have $\begin{aligned} \int_{t}^{t + 1} (ε (s) {| u_{t} (s) |}_{2}^{2} + ε^{2} (s) {‖ u_{t} (s) ‖}_{0}^{2}) d s \\ ⩽ \int_{t}^{t + 1} {‖ η^{s} ‖}_{μ, 0}^{2} d s + ν ε (t) {‖ u (t) ‖}_{0}^{2} + 2 {\tilde{α}}_{2} ε (t) {| u (t) |}_{p}^{p} + L | g |_{2}^{2} + 2 L | Ω | (2 {\tilde{β}}_{1} + {\tilde{β}}_{2}) . \end{aligned}$ Thanks to Corollary 3.5, for any $t ⩾ τ$ , it yields $\begin{matrix} (41) & \int_{t}^{t + 1} (ε (s) {| u_{t} (s) |}_{2}^{2} + ε^{2} (s) {‖ u_{t} (s) ‖}_{0}^{2}) d s ⩽ K_{2}, \end{matrix}$ where $K_{2} = (ν + 1 + 2 {\tilde{α}}_{2} L) K_{0} + 2 L | Ω | (2 {\tilde{β}}_{1} + {\tilde{β}}_{2})$ . □

In the following, the asymptotic regularity of solutions will be obtained by using a operator decomposition method.

With regards to this, we decompose the solution $z (t) = U^{ν} (t, τ) z_{τ} = (u (t), η^{t})$ of equation (14) with initial data $z_{τ} = (u_{τ}, η^{τ}) \in M_{τ}$ into sum as follows: $\begin{matrix} (42) & U^{ν} (t, τ) z_{τ} = U_{1}^{ν} (t, τ) z_{τ} + U_{2}^{ν} (t, τ) z_{τ}, \end{matrix}$ where $z_{1} = U_{1}^{ν} (t, τ) z_{τ} = (v (t), ζ^{t})$ and $z_{2} = U_{2}^{ν} (t, τ) z_{τ} = (w (t), θ^{t})$ are the solutions to the following systems respectively: $\begin{matrix} (43) & \{\begin{array}{l} v_{t} - ε (t) Δ v_{t} - ν Δ v - \int_{0}^{\infty} μ (s) Δ ζ^{t} (s) d s + f (u) - f (w) + \tilde{μ} v = 0, \\ ζ_{t}^{t} = v - ζ_{s}^{t}, \end{array} \end{matrix}$ with initial-boundary conditions $\begin{array}{r} (44) & \{\begin{array}{l} v (x, t) |_{\partial Ω} = 0, ζ^{t} (x, s) |_{\partial Ω \times R^{+}} = 0, t ⩾ τ, \\ v (x, τ) = u_{τ} (x), ζ^{τ} (x, s) = \int_{0}^{s} u_{τ} (x, τ - r) d r, (x, s) \in Ω \times R^{+}, \end{array} \end{array}$ where $\tilde{μ} > l$ (from (9)) is a constant, and $\begin{matrix} (45) & \{\begin{array}{l} ω_{t} - ε (t) Δ ω_{t} - ν Δ ω - \int_{0}^{\infty} μ (s) Δ θ^{t} (s) d s + f (ω) - \tilde{μ} v = g, \\ θ_{t}^{t} = ω - θ_{s}^{t} \end{array} \end{matrix}$ with initial-boundary conditions $\begin{array}{r} (46) & \{\begin{array}{l} ω (x, t) |_{\partial Ω} = 0, θ^{t} (x, s) |_{\partial Ω \times R^{+}} = 0, t ⩾ τ, \\ ω (x, τ) = 0, θ^{τ} (x, s) = 0, (x, s) \in Ω \times R^{+} . \end{array} \end{array}$
Lemma 3.8.
Let f satisfy ( 8 )–( 9 ), $ε (t)$ satisfy ( 4 )–( 5 ), the kernel function μ satisfy $(H_{1})$ and $g \in L^{2} (Ω)$ . Assume that $U_{2}^{ν} (t, τ) z_{τ} = (ω (t), ζ^{t})$ is the solutions of the equation ( 45 ) with initial-boundary condition ( 46 ). Then there exist constants $K_{i} > 0 (i = 1, 2)$ , such that $\begin{matrix} {| ω (t) |}_{2}^{2} + ε (t) {‖ ω (t) ‖}_{0}^{2} + {‖ θ^{t} ‖}_{μ, 0}^{2} + ε (t) {| ω (t) |}_{p}^{p} ⩽ K_{1}, \end{matrix}$ and $\begin{matrix} \int_{t}^{t + 1} (ε (s) {| ω_{t} (s) |}_{2}^{2} + ε^{2} (s) {‖ ω_{t} (s) ‖}_{0}^{2}) d s ⩽ K_{2}, \end{matrix}$ hold for any $t ⩾ τ$ .
Proof.
This proof is exactly similar to that of $Lemma$ 3.4 and Lemma 3.7, and hence is omitted. □
Lemma 3.9.
Suppose that the assumptions of Lemma 3.8 hold. Then for any given $R > 0$ , whenever $‖ z_{τ} ‖_{M_{τ}} < R$ , then the relationship $\begin{array}{r} lim_{\begin{array}{c} τ \to - \infty \end{array}} ({| v (t) |}_{2}^{2} + ε (t) {‖ v (t) ‖}_{0}^{2} + {‖ ζ^{t} ‖}_{μ, 0}^{2}) = 0, \forall t ⩾ τ \end{array}$ holds true for any $ν \in [0, \infty)$ .
Proof.
Multiplying the first equation of (43) by v and then integrating it over Ω. we have $\begin{aligned} (47) & \frac{d}{d t} E_{1} (t) + 2 (\tilde{μ} - l) | v |_{2}^{2} + (2 ν - ε^{'} (t)) ‖ v ‖_{0}^{2} + \frac{1}{2} \int_{0}^{\infty} - μ^{'} (s) {‖ ζ^{s} ‖}_{0}^{2} d s + \frac{δ}{2} {‖ ζ^{t} ‖}_{μ, 0}^{2} ⩽ 0, \end{aligned}$ where $E (t) = | v (t) |_{2}^{2} + ε (t) ‖ v (t) ‖_{0}^{2} + ‖ ζ^{t} ‖_{μ, 0}^{2}$ .

By (47), we can get that there exists constant $K_{3} : = K_{3} (R) > 0$ such that $\begin{matrix} (48) & \int_{τ}^{t} {‖ ζ^{s} ‖}_{μ, 0}^{2} d s ⩽ K_{3}, \end{matrix}$ where $K_{3} = \frac{4 R}{δ}$

From Lemmas 3.7 and 3.8, we can obtain $\begin{aligned} \int_{t}^{t + 1} ε^{2} (s) {‖ v_{t} (s) ‖}_{0}^{2} d s & = \int_{t}^{t + 1} ε^{2} (s) {‖ u_{t} (s) - ω_{t} (s) ‖}_{0}^{2} d s \\ ⩽ 2 (\int_{t}^{t + 1} ε^{2} (s) {‖ u_{t} (s) ‖}_{0}^{2} d s + \int_{t}^{t + 1} ε^{2} (s) {‖ ω_{t} (s) ‖}_{0}^{2} d s) \\ (49) & ⩽ 2 (K_{2} + K_{2}) \end{aligned}$ for any $t ⩾ τ \in R$ .

As said in Remark 3.3, the equation (43) lacks also dissipation, but we need the uniform decay estimate of $U_{1}^{ν} (t, τ) z_{τ} = (v (t), ζ^{t})$ , thus, by using the same way with Lemma 3.2, we define the functional as follows: $\begin{matrix} N_{1} (t) = - {⟨ ε (t) v (t), ζ^{t} ⟩}_{V} . \end{matrix}$ Then the following differential inequality $\begin{aligned} \frac{d}{d t} N_{1} (t) + \frac{ε (t)}{2} {‖ v (t) ‖}_{0}^{2} \\ (50) & ⩽ ε (t) {‖ v_{t} (t) ‖}_{0} {‖ ζ^{t} ‖}_{μ, 0} - ε^{'} (t) {‖ v (t) ‖}_{0} {‖ ζ^{t} ‖}_{μ, 0} - \frac{μ (0) ε (t)}{2} \int_{0}^{\infty} μ^{'} (s) {‖ ζ^{t} (s) ‖}_{0}^{2} d s, \end{aligned}$ and the control $\begin{array}{r} | N_{1} (t) | ⩽ ϱ_{0} E_{1} (t), \end{array}$ are all true.

Furthermore, we can define the energy-like functional as follows: $\begin{matrix} (51) & F_{γ} (t, τ) = E_{1} (t) + γ N_{1} (t) \end{matrix}$ where $γ = min {1, \sqrt{\frac{δ}{L}}, \frac{1}{μ (0) L}}$ .

Then it’s easy to get $\begin{matrix} (52) & F_{γ} (t, τ) ⩽ (1 + γ) (| v |_{2}^{2} + ε (t) ‖ v ‖_{0}^{2} + {‖ ξ^{t} ‖}_{μ, 0}^{2}), \end{matrix}$ and $\begin{matrix} (53) & F_{γ} (t, τ) ⩾ (1 - γ) (| v |_{2}^{2} + ε (t) ‖ v ‖_{0}^{2} + {‖ ξ^{t} ‖}_{μ, 0}^{2}) . \end{matrix}$ Therefore, combining with (47), (50) and (51), we have $\begin{aligned} \frac{d}{d t} F_{γ} + 2 (\tilde{μ} - l) | v |_{2}^{2} + (2 ν - ε^{'} (t) + \frac{γ ε (t)}{2}) ‖ v ‖_{0}^{2} + \frac{δ}{2} {‖ ζ^{t} ‖}_{μ, 0}^{2} \\ (54) & + (\frac{1}{2} - \frac{γ L μ (0)}{2}) \int_{0}^{\infty} (- μ^{'} (s)) {‖ ζ^{t} (s) ‖}_{0}^{2} d s ⩽ γ ε (t) ‖ v_{t} ‖_{0} {‖ ζ^{t} ‖}_{μ, 0} - γ ε^{'} (t) ‖ v ‖_{0} {‖ ζ^{t} ‖}_{μ, 0} . \end{aligned}$ Similar to (32), it follows that $\begin{aligned} (55) & - γ ε^{'} (t) ‖ v ‖_{0} {‖ η^{t} ‖}_{μ, 0} ⩽ \frac{- ε^{'} (t)}{2} ‖ v ‖_{0}^{2} + \frac{γ^{2} L}{2} {‖ η^{t} ‖}_{μ, 0}^{2} . \end{aligned}$ Together with (54) and (55), we get $\begin{aligned} (56) & \frac{d}{d t} F_{γ} + \frac{γ}{2} ε (t) ‖ v ‖_{0}^{2} + (\frac{δ}{2} - \frac{γ^{2} L}{2}) {‖ ξ^{t} ‖}_{μ, 0}^{2} + 2 (\tilde{μ} - l) | v |_{2}^{2} ⩽ γ ε (t) ‖ v_{t} ‖_{0} {‖ ξ^{t} ‖}_{μ, 0} . \end{aligned}$ According to (52) and (56), let $ϱ_{4} = \frac{1}{1 + γ} min {\frac{γ}{2}, \frac{δ}{2} - \frac{γ^{2} L}{2}, 2 (\tilde{μ} - l)}$ , then $\begin{aligned} (57) & \frac{d}{d t} F_{γ} + ϱ_{4} F_{γ} ⩽ γ L ‖ v_{t} ‖_{0} {‖ ξ^{t} ‖}_{μ, 0} . \end{aligned}$ Integrating (57) about t on $[τ, t]$ , and combining with (52) and (53), we have $\begin{aligned} {| v (t) |}_{2}^{2} + ε (t) {‖ v (t) ‖}_{0}^{2} + {‖ ζ^{t} ‖}_{μ, 0}^{2} + ϱ_{4} \int_{τ}^{t} ({| v (r) |}_{2}^{2} + ε (r) {‖ v (r) ‖}_{0}^{2} + {‖ ζ^{r} ‖}_{μ, 0}^{2}) d r \\ ⩽ \frac{γ}{1 - γ} \int_{τ}^{t} ε (r) {‖ v_{t} (r) ‖}_{0} {‖ ζ^{r} ‖}_{μ, 0} d r + \frac{1 + γ}{1 - γ} R \\ (58) & ⩽ \frac{γ}{1 - γ} {(\int_{τ}^{t} ε^{2} (r) {‖ v_{t} (r) ‖}_{0}^{2} d r)}^{\frac{1}{2}} {(\int_{τ}^{t} {‖ ζ^{r} ‖}_{μ, 0}^{2} d r)}^{\frac{1}{2}} + \frac{1 + γ}{1 - γ} R . \end{aligned}$ However, from (47), it’s easy to know $\begin{matrix} \frac{d}{d t} ({| v (t) |}_{2}^{2} + ε (t) {‖ v (t) ‖}_{0}^{2} + {‖ ζ^{t} ‖}_{μ, 0}^{2}) ⩽ 0, \end{matrix}$ so $‖ U_{1} (t, τ) z_{τ} ‖_{M_{t}}^{2} : = | v (t) |_{2}^{2} + ε (t) ‖ v (t) ‖_{0}^{2} + ‖ ζ^{t} ‖_{μ, 0}^{2}$ is monotonically decreasing on t. Thus, we get $\begin{aligned} (59) & \int_{τ}^{t} ({| v (r) |}_{2}^{2} + ε (r) {‖ v (r) ‖}_{0}^{2} + {‖ ζ^{r} ‖}_{μ, 0}^{2}) d r ⩾ (t - τ) ({| v (t) |}_{2}^{2} + ε (t) {‖ v (t) ‖}_{0}^{2} + {‖ ζ^{t} ‖}_{μ, 0}^{2}) . \end{aligned}$ Combining with (48), (49), (58) and (59), we obtain $\begin{array}{r} (1 + ϱ_{4} (t - τ)) ({| v (t) |}_{2}^{2} + ε (t) {‖ v (t) ‖}_{0}^{2} + {‖ ζ^{t} ‖}_{μ, 0}^{2}) ⩽ \frac{γ}{1 - γ} {[2 K_{3} (K_{2} + K_{2})]}^{\frac{1}{2}} + \frac{1 + γ}{1 - γ} R . \end{array}$ Furthermore, we have $\begin{aligned} (60) & \begin{array}{r} {| v (t) |}_{2}^{2} + ε (t) {‖ v (t) ‖}_{0}^{2} + {‖ ζ^{t} ‖}_{μ, 0}^{2} ⩽ \frac{γ {[2 K_{3} (K_{2} + K_{2}) (t - τ)]}^{\frac{1}{2}}}{(1 - γ) (1 + ϱ_{4} (t - τ))} + \frac{(1 + γ) R}{(1 - γ) (1 + ϱ_{4} (t - τ))} \end{array} \end{aligned}$ Then for any fixed $t \in R_{τ}$ , $\begin{matrix} lim_{τ \to - \infty} {‖ U_{1} (t, τ) z_{τ} ‖}_{M_{t}}^{2} = 0 \end{matrix}$ This proof is completed. □
Lemma 3.10.
Suppose that the assumptions of Lemma 3.8 hold. Then there exists a positive constant $K_{3}$ , such that the solution $z_{2} (t) = (ω (t), θ^{t})$ of ( 45 )–( 46 ) satisfies $\begin{array}{r} {‖ ω (t) ‖}_{0}^{2} + ε (t) {‖ ω (t) ‖}_{1}^{2} + {‖ θ^{t} ‖}_{μ, 1}^{2} ⩽ K_{3}, \forall t ⩾ τ . \end{array}$
Proof.
Multiplying the first equation of (45) by $- 2 ε (t) Δ ω$ and then integrating it over Ω, we get by using the Hölder inequality and the Young inequality $\begin{aligned} \frac{d}{d t} [ε (t) ‖ ω ‖_{0}^{2} + ε^{2} (t) ‖ ω ‖_{1}^{2} + ε (t) {‖ θ^{t} ‖}_{μ, 1}^{2}] - ε^{'} (t) ‖ ω ‖_{0}^{2} - 2 ε (t) ε^{'} (t) ‖ ω ‖_{1}^{2} \\ + 2 ν ε (t) ‖ ω ‖_{1}^{2} - ε^{'} (t) {‖ θ^{t} ‖}_{μ, 1}^{2} + δ ε (t) {‖ θ^{t} ‖}_{μ, 1}^{2} + 2 (\tilde{μ} - l) ε (t) ‖ ω ‖_{0}^{2} \\ ⩽ ε (t) ‖ ω ‖_{0}^{2} + {\tilde{μ}}^{2} ε (t) ‖ u ‖_{0}^{2} + ε^{2} (t) ‖ ω ‖_{1}^{2} + | g |_{2}^{2} . \end{aligned}$ Combining with (5), $ε^{'} (t) ⩽ 0$ , $ε (t) > 0$ , $\tilde{μ} > l$ , and Corollary 3.5, the above formula becomes $\begin{aligned} \frac{d}{d t} [ε (t) ‖ ω ‖_{0}^{2} + ε^{2} (t) ‖ ω ‖_{1}^{2} + ε (t) {‖ θ^{t} ‖}_{μ, 1}^{2}] \\ ⩽ [ε (t) ‖ ω ‖_{0}^{2} + ε^{2} (t) ‖ ω ‖_{1}^{2} + ε (t) {‖ θ^{t} ‖}_{μ, 1}^{2}] + {\tilde{μ}}^{2} L K_{0} + | g |_{2}^{2} . \end{aligned}$ Using Gronwall lemma, we get $\begin{matrix} ε (t) ‖ ω ‖_{0}^{2} + ε^{2} (t) ‖ ω ‖_{1}^{2} + ε (t) {‖ θ^{t} ‖}_{μ, 1}^{2} ⩽ e^{(t - τ)} ({\tilde{μ}}^{2} L K_{0} + | g |_{2}^{2}) . \end{matrix}$ Moreover, since $ε (t)$ is monotonically decreasing, it follows that $ε (s) ⩾ ε (t) > 0$ for any $s \in [τ, t]$ , then $\begin{matrix} {‖ ω (s) ‖}_{0}^{2} + ε (s) {‖ ω (s) ‖}_{1}^{2} + {‖ θ^{s} ‖}_{μ, 1}^{2} ⩽ \frac{e^{(t - τ)}}{ε (t)} ({\tilde{μ}}^{2} L K_{0} + | g |_{2}^{2}) . \end{matrix}$ Let $K_{3} = \frac{e^{(t - τ)}}{ε (t)} ({\tilde{μ}}^{2} L K_{0} + | g |_{2}^{2})$ , then the result holds. This proof is completed. □

Although Lemma 3.9 and Lemma 3.10 hold, we still can’t guarantee the asymptotic compactness of the family of processes ${U^{ν} (t, τ)}_{t ⩾ τ}$ corresponding to system (14)–(15). This is that because the embedding $L_{μ}^{2} (R^{+}, D (A)) ↪ L_{μ}^{2} (R^{+}, H_{0}^{1} (Ω))$ is noncompact. In the following, we shall prove that the family of processes ${U^{ν} (t, τ)}_{t ⩾ τ}$ is an $M_{t}$ -contractive process in $M_{1}$ . Lemma 3.11.
The family of processes ${U^{ν} (t, τ)}$ corresponding to problem ( 14 )–( 15 ) is pullback asymptotical compact $M_{t}$ -contractive process in $M_{1}$ .
Proof.
Let $z_{i} (t) = (u_{i} (t), η_{i}^{t}) = U^{ν} (t, τ) z_{τ}^{i} (i = 1, 2)$ be the solutions of equation (14) with the perturbation parameters $ε (t)$ , ν and initial data $z_{τ}^{i} \in B_{τ}^{0} \in {\tilde{B}}^{0} (i = 1, 2)$ ( ${\tilde{B}}^{0}$ is from (37)).

By (42), it yields $\begin{matrix} z_{i} (t) = U^{ν} (t, τ) z_{τ}^{i} = U_{1}^{ν} (t, τ) z_{τ}^{i} + U_{2}^{ν} (t, τ) z_{τ}^{i} = (v_{i} (t), ζ_{i}^{t}) + (ω_{i} (t), θ_{i}^{t}) . \end{matrix}$ Thus, we have $\begin{aligned} {‖ U^{ν} (t, τ) z_{τ}^{1} - U^{ν} (t, τ) z_{τ}^{2} ‖}_{M_{t}}^{2} \\ ⩽ 2 ({| v_{1} (t) - v_{2} (t) |}_{2}^{2} + ε (t) {‖ v_{1} (t) - v_{2} (t) ‖}_{0}^{2} + {‖ ζ_{1}^{t} - ζ_{2}^{t} ‖}_{μ, 0}^{2}) \\ (61) & + 2 ({| ω_{1} (t) - ω_{2} (t) |}_{2}^{2} + ε (t) {‖ ω_{1} (t) - ω_{2} (t) ‖}_{0}^{2} + {‖ θ_{1}^{t} - θ_{2}^{t} ‖}_{μ, 0}^{2}) . \end{aligned}$ In addition, by Lemma 3.9, it’s easy to get $\begin{aligned} lim_{τ \to - \infty} ({| v_{1} (t) - v_{2} (t) |}_{2}^{2} + ε (t) {‖ v_{1} (t) - v_{2} (t) ‖}_{0}^{2} + {‖ ζ_{1}^{t} - ζ_{2}^{t} ‖}_{μ, 0}^{2}) \\ ⩽ 2 lim_{τ \to - \infty} [({| v_{1} (t) |}_{2}^{2} + ε (t) {‖ v_{1} (t) ‖}_{0}^{2} + {‖ ζ_{1}^{t} ‖}_{μ, 0}^{2}) + (v_{2} (t) |_{2}^{2} + ε (t) {‖ v_{2} (t) ‖}_{0}^{2} + {‖ ζ_{2}^{t} ‖}_{μ, 0}^{2})] = 0 . \end{aligned}$ Then for any $ε > 0$ and fixed t, there exists $T : = T (ε)$ such that $\begin{matrix} (62) & 2 ({| v_{1} (t) - v_{2} (t) |}_{2}^{2} + ε (t) {‖ v_{1} (t) - v_{2} (t) ‖}_{0}^{2} + {‖ ζ_{1}^{t} - ζ_{2}^{t} ‖}_{μ, 0}^{2}) < ε, \end{matrix}$ holds for $T ⩽ t$ .

Let $(ϖ (t), ς^{t}) = (ω_{1} (t) - ω_{2} (t), θ_{1}^{t} - θ_{2}^{t})$ , then $(ϖ (t), ς^{t})$ fulfills the following problem $\begin{array}{r} (63) & \{\begin{array}{l} ϖ_{t} - ε (t) Δ ϖ_{t} - ν Δ ϖ - \int_{0}^{\infty} μ (s) Δ ς^{t} (s) d s + f (ω_{1}) - f (ω_{2}) + \tilde{μ} ϖ = \tilde{μ} (u_{1} - u_{2}), \\ ς_{t}^{t} = - ς_{s}^{t} + ϖ \end{array} \end{array}$ with the initial-boundary value conditions $\begin{array}{r} \{\begin{array}{l} ϖ (x, t) |_{\partial Ω} = 0, ς^{t} (x, s) |_{\partial Ω \times R^{+}} = 0, t ⩾ τ, \\ ϖ (x, τ) = 0, ς^{τ} (x, s) = 0, (x, s) \in Ω \times R^{+} . \end{array} \end{array}$ For any $ν \in [0, \infty)$ , using $2 ϖ$ to act on the first equation of (63) in $L^{2} (Ω)$ , it yields $\begin{aligned} \frac{d}{d t} ({| ω_{1} (t) - ω_{2} (t) |}_{2}^{2} + ε (t) {‖ ω_{1} (t) - ω_{2} (t) ‖}_{0}^{2} + {‖ ζ_{1}^{t} - ζ_{2}^{t} ‖}_{μ, 0}^{2}) \\ (64) & ⩽ ϱ_{5} ({| u_{1} (t) - u_{2} (t) |}_{2}^{2}), \end{aligned}$ where $ϱ_{5} = \frac{2 \tilde{μ}}{\tilde{μ} - l}$ .

Then integrating (64) about t on $[T, t]$ , we obtain $\begin{aligned} (65) & {| ω_{1} (t) - ω_{2} (t) |}_{2}^{2} + ε (t) {‖ ω_{1} (t) - ω_{2} (t) ‖}_{0}^{2} + {‖ ζ_{1}^{t} - ζ_{2}^{t} ‖}_{μ, 0}^{2} ⩽ ϱ_{5} \int_{T}^{t} {| u_{1} (s) - u_{2} (s) |}_{2}^{2} d s . \end{aligned}$ Setting $\begin{aligned} (66) & ϕ_{T}^{t} (z_{1}, z_{2}) = ϱ_{5} \int_{T}^{t} {| u_{1} (s) - u_{2} (s) |}_{2}^{2} d s . \end{aligned}$ Combining with Corollary 3.5, Lemma 3.7 and Lemma 2.16, for any $ν \in [0, \infty)$ fixed, then the sequence ${u_{n} (s)}_{n = 1}^{\infty}$ is relatively compact in $L^{2} (T, t; L^{2} (Ω))$ . That is to say, for any sequences ${z_{n} (T) = (u_{n} (T), η_{n}^{T})} \subset B_{T}^{0} \in {\tilde{B}}^{0}$ , ${z_{n} (t) = (u_{n} (t), η_{n}^{t})}$ is the solution of equation (14) with the initial value ${z_{n} (T) = (u_{n} (T), η_{n}^{T})}$ . Then there exists a subsequence ${z_{n_{k}} (t)}_{k = 0}^{\infty}$ of ${z_{n} (t)}_{k = 0}^{\infty}$ satisfying: $\begin{matrix} lim_{k \to \infty} lim_{l \to \infty} ϕ_{T}^{t} (z_{n_{k}}, z_{n_{l}}) = ϱ_{5} lim_{k \to \infty} lim_{l \to \infty} \int_{T}^{t} {| u_{n_{k}} (s) - u_{n_{l}} (s) |}_{2}^{2} d s = 0 . \end{matrix}$ It is follows that $ϕ_{T}^{t} \in E (B_{T})$ . Substituting (65) and (62) into (61), one gets $\begin{matrix} {‖ U^{ν} (t, T) x - U^{ν} (t, T) y ‖}_{M_{t}}^{2} ⩽ ε + ϕ_{T}^{t} (x, y) . \end{matrix}$ Owing to Definitions 2.13 and 2.14, it can be seen that $φ_{T}^{t}$ is contractive function in $B_{T}^{0}$ . Therefore, it’s easy to obtain that the process $U^{ν} (t, τ)$ is $M_{t}$ -contractive process on $B_{T}^{0} \in {\tilde{B}}^{0}$ . This proof is finished. □
Theorem 3.12.
Under the hypotheses of Lemma 3.4 . The process ${U^{ν} (t, τ)}_{t ⩾ τ}$ defined by ( 19 ) possesses a pullback attractor $A^{ν}$ in ${M_{t}}_{t \in R}$ for any $ν \in [0, \infty)$ , and $A^{ν} = {A^{ν} (t)}_{t \in R}$ is non-empty, compact, invariant in ${M_{t}}_{t \in R}$ and pullback attracts every bounded set of $M_{t}$ with respect to $M_{t}$ -norm.
Proof.
In virtue of Theorem 3.4 and Lemma 3.11, we can obtain the existence of time-dependent global attractors $A^{ν}$ for the process $U (t, τ)$ defined by (19) in time-dependent spaces ${M_{t}}_{t \in R}$ . □
Remark 3.13.
In the light of Lemmas 3.9 and 3.10, the asymptotic regularity of the solutions to the equation (1) can be obtained.

4. Upper semi-continuity (w.r.t. ν)

In this section, we will claim the upper semi-continuity of the family of pullback attractors ${A^{ν}}_{ν > 0}$ with respect to $ν \in (0, + \infty)$ .

Lemma 4.1.
Let ( 4 )–( 5 ) and $(H_{1})$ – $(H_{2})$ hold. Then, for any $τ \in R$ , $z_{τ} \in M_{τ}$ , the following conclusion $\begin{matrix} {‖ U^{ν} (s, τ) z_{τ} - U^{0} (s, τ) z_{τ} ‖}_{M_{s}} ⩽ \sqrt{Q} ν \end{matrix}$ holds for all $t ⩾ s ⩾ τ$ and $ν \in (0, + \infty)$ .
Proof.
Let $z^{1} (t) = (u^{1} (t), η_{1}^{t})$ , $z^{2} (t) = (u^{2} (t), η_{2}^{t})$ be two solutions of (1) with parameter $ν \in (0, + \infty)$ and initial value $z_{τ} \in M_{τ}$ respectively, i.e., $\begin{matrix} z^{1} (t) = U^{ν} (t, τ) z_{τ} and z^{2} (t) = U^{0} (t, τ) z_{τ} . \end{matrix}$ Setting $z (t) = (ϖ (t), ξ^{t}) = z^{1} (t) - z^{2} (t)$ , then $z (t) = (ϖ (t), ξ^{t})$ satisfies the following equation $\begin{array}{r} (67) & \{\begin{array}{l} ϖ_{t} - ε (t) Δ ϖ_{t} - \int_{0}^{\infty} μ (s) Δ ξ^{t} (s) d s + f (u^{1}) - f (u^{2}) = ν Δ u^{1}, \\ ξ_{t}^{t} = - ξ_{s}^{t} + ϖ . \end{array} \end{array}$ The problem is supplemented with the boundary condition $\begin{matrix} (68) & z (x, t) |_{\partial Ω} = (0, 0), for all t ⩾ τ, \end{matrix}$ and initial condition $\begin{matrix} (69) & z (x, t) |_{t = τ} = 0 . \end{matrix}$ Multiplying (69) by $ε (t) ϖ$ and integrating it over Ω and arranging (7) and monotone decreasing property $ε (t)$ , we have $\begin{aligned} \frac{d}{d t} (ε (t) | ϖ |_{2}^{2} + ε^{2} (t) ‖ ϖ ‖_{0}^{2} + ε (t) {‖ ξ^{t} ‖}_{μ, 0}^{2}) + δ ε (t) {‖ ξ^{t} ‖}_{μ, 0}^{2} \\ (70) & ⩽ 2 l ε (t) | ϖ |_{2}^{2} + 2 ν ε (t) {‖ u^{1} ‖}_{0} ‖ ϖ ‖_{0} . \end{aligned}$ Then $\begin{aligned} \frac{d}{d t} (ε (t) | ϖ |_{2}^{2} + ε^{2} (t) ‖ ϖ ‖_{0}^{2} + ε (t) {‖ ξ^{t} ‖}_{μ, 0}^{2}) \\ (71) & ⩽ χ_{1} (ε (t) | ϖ |_{2}^{2} + ε^{2} (t) ‖ ϖ ‖_{0}^{2} + ε (t) {‖ ξ^{t} ‖}_{μ, 0}^{2}) + ν^{2} {‖ u^{1} ‖}_{0}^{2}, \end{aligned}$ where $χ_{1} = max {2 l, 1}$ . Combining with Lemma 3.7 and Gronwall inequality, it follows that $\begin{matrix} ε (t) {| ϖ (t) |}_{2}^{2} + ε^{2} (t) {‖ ϖ (t) ‖}_{0}^{2} + ε (t) {‖ ξ^{t} ‖}_{μ, 0}^{2} ⩽ R_{3} e^{χ_{1} (t - τ)} ν^{2} \end{matrix}$ holds true for any $t ⩾ τ$ . Using the hypothesis about $ε (t)$ , we have that $\forall s \in [τ, t]$ , $ε (s) ⩾ ε (t) > 0$ , then $\begin{matrix} {| ϖ (s) |}_{2}^{2} + ε (s) {‖ ϖ (s) ‖}_{0}^{2} + {‖ ξ^{s} ‖}_{μ, 0}^{2} ⩽ \frac{R_{3}}{ε (t)} e^{χ_{1} (t - τ)} ν^{2} \end{matrix}$ i.e., $\begin{matrix} {‖ U^{ν} (s, τ) z_{τ} - U^{0} (s, τ) z_{τ} ‖}_{M_{s}}^{2} ⩽ Q ν^{2}, \end{matrix}$ holds for any $t ⩾ s ⩾ τ \in R$ , where $Q = \frac{R_{3}}{ε (t)} e^{χ_{1} (t - τ)}$ . □

In the following, we will give the another crucial Theorem of this paper, this result can be obtained by Theorem 2.1 in [15], but here we directly prove it based on above conclusions (Theorem 3.12 and Lemma 4.1).
Theorem 4.2 (Upper semicontinuity).

Suppose that $Ω \subset R^{n} (n ⩾ 3)$ is a bounded domain with smooth boundary $\partial Ω$ , the kernel function $μ (s)$ , the nonlinearity f and $ε (t)$ satisfy $(H_{1})$ , $(H_{2})$ and ( 4 )–( 5 ) respectively, and $g \in L^{2} (Ω)$ . Let ${A^{ν}}_{ν ⩾ 0} = {{A^{ν} (t)}_{t \in R}}_{ν ⩾ 0}$ be the family of pullback attractors given by Theorem 3.12 , then it follows that $\begin{matrix} (72) & lim_{ν \to 0} {dist}_{M_{t}} (A^{ν} (t), A^{0} (t)) = 0, \forall t \in R, \end{matrix}$ where ${dist}_{M_{t}}$ denotes the standard Hausdorff semidistance in $M_{t}$ (see ( 17 )).

Proof.

For any given $t \in R$ and $ϵ > 0$ , because $A^{0} (t)$ attracts $B (t) \subset M_{t}$ (see Theorem 3.4), then it’s to know that there exists $T > 0$ , such that $\begin{matrix} {dist}_{M_{t}} (U^{0} (t, t - T) B (t - T), A^{0} (t)) < \frac{ϵ}{2}, \end{matrix}$ this is $\begin{aligned} (73) & U^{0} (t, t - T) B (t - T) \subset O_{t}^{ϵ / 2} (A^{0} (t)) . \end{aligned}$ By virtue of the invariance of $A^{ν} (t)$ , we have $\begin{aligned} (74) & A^{ν} (t) = U^{ν} (t, t - T) A^{ν} (t - T) \subset U^{ν} (t, t - T) B (t - T) . \end{aligned}$ Furthermore, from Lemma 4.1, we know $\begin{matrix} sup_{z_{τ} \in B (t)} {‖ U^{ν} (t, τ) z_{τ} - U^{0} (t, τ) z_{τ} ‖}_{M_{t}} ⩽ \sqrt{Q} | ν | . \end{matrix}$ Therefore, $\begin{matrix} lim_{ν \to 0} sup_{z_{t - T} \in B (t - T)} {‖ U^{ν} (t, t - T) z_{t - T} - U^{0} (t, t - T) z_{t - T} ‖}_{M_{t}} = 0 . \end{matrix}$ Then for aforementioned ϵ, there exists $δ = δ (ϵ) > 0$ , such that when $dist (ν, 0) < δ$ , $\begin{aligned} (75) & U^{ν} (t, t - T) B (t - T) \subset O_{t}^{ϵ / 2} (U^{0} (t, t - T) B (t - T)) . \end{aligned}$ By virtue of (73)–(75), we obtain that $\begin{matrix} A^{ν} (t) \subset O_{t}^{ϵ / 2} (U^{0} (t, t - T) B (t - T)) \subset O_{t}^{ϵ} (A^{0} (t)) \end{matrix}$ This shows that when $dist (ν, 0) < δ$ , there is $\begin{matrix} {dist}_{M_{t}} (A^{ν} (t), A^{0} (t)) < ϵ, \end{matrix}$ so (72) holds true by the arbitrariness of ϵ. This proof is completed. □

Footnotes

Acknowledgement

The authors would like to thank the referees for their many helpful comments and suggestions. The research is financially supported by General Project of Education Department of Hunan Province (Nos. 21C0660), and National Natural Science Foundation of China (Nos. 11101053, 71471020).

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Long-time behavior of nonclassical diffusion equations with memory on time-dependent spaces

Abstract

Keywords

1. Introduction

3. The pullback attractors in time-dependent M t

Footnotes

Acknowledgement

References

3. The pullback attractors in time-dependent $M_{t}$