1. Introduction
It is well-known that the study of global, pullback and uniform attractors is of great significance for characterizing the long-time behavior of the solutions of nonlinear evolutionary partial differential equations (see [Reference Chepyzhov and Vishik7, Reference Chipot and Lovat8, Reference Lions and Magenes17, Reference Robinson27]). Therefore, in recent decades, as an important class of the nonlinear partial differential equations, the autonomous and non-autonomous diffusion equations have been extensively studied (see [Reference Anguiano, Kloeden and Lorenz2, Reference Caballero, Marín-Rubio and Valero3, Reference Sun, Wang and Zhong28, Reference Sun and Yang29]). However, there are still relatively few results on the existence of time-dependent pullback attractors in the Sobolev space 
${H^{2}}(\Omega ) \cap H_0^{1}(\Omega )$. To this end, this paper is devoted to studying the existence and regularity of the time-dependent pullback attractors of the non-autonomous diffusion equations with nonlocal diffusion in the time-dependent space 
$\mathcal {H}_{t}(\Omega )$.
Let 
${\Omega }$ be a bounded domain in 
$\mathbb {R}^{n}$ with smooth boundary 
$\partial \Omega.$ We consider the long-time behavior of the solutions for the following non-autonomous nonclassical diffusion equations:
\begin{equation} \left\{\begin{array}{@{}ll} u_{t}-\varepsilon(t) \Delta u_{t}-a(l(u)) \Delta u=f(u)+h(t) & \text{in}\ \Omega \times(\tau, \infty), \\ u=0 & \text{on}\ \partial \Omega\times(\tau, \infty), \\ u(x, \tau)=u_{\tau}, & x \in \Omega, \end{array}\right.\end{equation}
where 
$\tau \in \mathbb {R}$ is the initial time and 
$\varepsilon (t) \in C^{1}(\mathbb {R})$ is a decreasing bounded function with respect to the parameter 
$t$ satisfying
\begin{equation} \lim _{t \rightarrow+\infty} \varepsilon(t)=1, \end{equation}
and there exists a constant 
$L>0$ such that
\begin{equation} \sup _{t \in \mathbb{R}}\left(|\varepsilon(t)|+\left|\varepsilon^{\prime}(t)\right|\right) \leq L. \end{equation} For the nonlocal functional 
$a(l(u))$, we assume that 
$l(u)$ is a linear functional acting on 
$u$ that satisfies 
$l(u)=(u,l)$, whose definition of 
$(\cdot, \cdot )$ is below, and 
$a \in C(\mathbb {R}; \mathbb {R}_{+})$ is a locally Lipschitz continuous function satisfying
\begin{equation} \frac{1}{2} < m \leqslant a(s) \leqslant M, \quad \forall s \in \mathbb{R}, \end{equation}
where 
$m$ and 
$M$ are constant. In addition, suppose the nonlinear term 
$f \in C^{1}(\mathbb {R})$ and satisfies the following assumptions:
\begin{gather} \limsup _{|s| \rightarrow \infty} \frac{f(s)}{s}<\lambda_{1}, \end{gather}
\begin{gather} f^{\prime}(s) \leq \eta, \quad \forall s \in \mathbb{R}, \end{gather}
\begin{gather} |f(s)| \leqslant C\left(1+|s|^{p}\right), \quad \forall s \in \mathbb{R}, \end{gather}
where 
$\lambda _{1}>0$ is the first eigenvalue of 
$-\Delta$ in 
$\Omega$ with the homogeneous Dirichlet boundary conditions, 
$\eta$ and 
$C$ are arbitrarily positive constants.
Throughout this paper, the inner product of 
$L^{2}(\Omega )$ is represented by 
$(\cdot, \cdot )$, and the corresponding norm is denoted by 
$\|\cdot \|_{2}.$ For simplicity, 
$\|\cdot \|_{2}$ is written as 
$\|\cdot \|$. The norm of 
$H^{-1}(\Omega )$ is denoted as 
$\|\cdot \|_{-1}$, the norm of 
$H_{0}^{1}(\Omega )$ is denoted as 
$\|\cdot \|_{1}$, and the dual product between them will be represented by 
$\langle \cdot, \cdot \rangle$. From [Reference Evans15], the chain of dense and continuous embeddings 
$H_{0}^{1}(\Omega ) \subset L^{2}(\Omega ) \subset H^{-1}(\Omega )$ holds. In particular, 
$H_{0}^{1}(\Omega ) \subset L^{2}(\Omega )$ is compact.
When hypothesis (1.7) holds, the Sobolev embedding theorem (see [Reference Adams and Fournier1]) shows that when 
$n=1,2$ and 
$n \geqslant 3$, the index 
$p$ satisfies 
$p>1$ and 
$1< p<\frac {n}{n-2}$, respectively, hence it can be concluded that there is an embedding 
$H_{0}^{1}(\Omega ) \subset L^{2p}(\Omega )$, and then by using the Poincaré inequality, we can deduce that
\begin{equation} \left|f\left(u_{n}\right)\right|^{2}_{2} \leq C+C \int_{\Omega}\left|u_{n}\right|^{2 p} dx \leq C+C\left\|u_{n}\right\|^{2 p}. \end{equation}
It is easy to check that if the function 
$u_{n}$ is bounded in 
$L^{2p}(\Omega )$, then 
$f(u_{n})$ is bounded in 
$L^{\infty }(\tau, t ; L^{2}(\Omega ))$. Furthermore, let the external force term 
$h \in L_{l o c}^{2}(\mathbb {R} ; H^{-1}(\Omega ))$.
Let us recall some works of problem (1.1) in the literature, which are based on the classic general diffusion equation 
${u_t} - \Delta u = f(u).$ The asymptotic behavior of nonlocal problem 
$u_{t}-a(l(u(t))) A u=f$ was studied when there was only one equilibrium point in [Reference Chipot and Lovat9]. Later, this result was expanded to investigate the convergence of the solution towards a steady state in [Reference Chipot and Molinet10]. In [Reference Chipot, Valente and Vergara-Caffarelli11], the well-posedness of the solutions to
\[ u_{t}-a\left(\int_{\Omega}|\nabla u|^{2}\,\textrm{d}x\right) \Delta u=f \]
was obtained by using the energy method. In order to make the solutions not only exist in finite-time interval, hypothesis (1.4) was introduced and the asymptotic behavior of the solutions was studied in [Reference Chipot and Zheng12]. The existence of pullback attractors in 
$L^{2}(\Omega )$ for
\[ \frac{\textrm{d}u}{\textrm{d}t}-a(l(u)) \Delta u=f(u)+h(t) \]
was obtained when 
$f$ is a sublinear function in [Reference Caraballo, Herrera-Cobos and Marín-Rubio4]. Besides, some researchers assumed that 
$f$ satisfies
\[ -\kappa-\alpha_{1}|s|^{p} \leq f(s) s \leq \kappa-\alpha_{2}|s|^{p}, \quad \forall s \in \mathbb{R}, \]and got similar results (see [Reference Caraballo, Herrera-Cobos and Marín-Rubio5, Reference Caraballo, Herrera-Cobos and Marín-Rubio6]).
In addition, there are several results in 
$\mathcal {H}_{t}(\Omega ).$ Note that the definition of time-dependent space 
$\mathcal {H}_{t}(\Omega )$ can be checked in § 2. The existence and regularity of the time-dependent global attractors of problem
\[ u_{t}-\varepsilon(t) \Delta u_{t}-\Delta u+\lambda u=f(u)+g(x) \]are established by the decomposition method in [Reference Ma, Wang and Xu19]. Besides, the method of contraction function was used to prove the existence of the time-dependent global attractors in [Reference Zhu, Xie and Zhou34] of problem
\[ u_{t}-\varepsilon(t) \Delta u_{t}-{\Delta u}+f(u)=g(x). \]The Lebesgue-dominated convergence theorem was applied in [Reference Wang and Ma30] to verify the pullback asymptotic compactness of the global attractors of problem
\[ u_{t}-\varepsilon(t) \Delta u_{t}-\triangle u+\lambda u+f(u)=g. \]Furthermore, time-dependent attractors have also been extensively studied in [Reference Conti, Pata and Temam14, Reference Ma, Wang and Liu18, Reference Meng and Liu20, Reference Pata and Conti21, Reference Plinio, Duan and Temam23] and other papers.
Since problem (1.1) contains 
$a(\cdot )$, 
$\varepsilon (t)$, 
$f(u)$ and 
$h(t)$, which results in some difficulties to study the long-time behavior of solutions. To this end, our main solutions are as follows:
(1) When using the condition C, the contraction function or the decomposition method to prove the asymptotic compactness of pullback attractors, we are supposed to select suitable test functions and inequalities to transform (1.1)
$_{1}$ into a formula to ensure it satisfies the Gronwall lemma. But because $a=a(l(u))$ is a compound function, these methods do not work. To overcome this prominent technical difficulty, we use the diagonal method in functional analysis to obtain the upper and lower bounds of the process of problem (1.1), which is very challenging.(2) The time-dependent function
$\varepsilon (t)$ complicates calculations of energy estimation. Some common techniques, such as multiplying by (1.1)$_{1}$ with $u$ or $u_{t}$ as a test function, does not offer any meaningful results. In order to make the Gronwall inequality work, we bring
into a priori estimate, then obtain (3.1). Although this seems to complicate the energy equation, it actually helps to discuss the existence of solutions in\[ \varepsilon(t) \frac{\textrm{d}}{\textrm{d}t}\|\nabla u\|^{2}=\frac{\textrm{d}}{\textrm{d}t}\left(\varepsilon(t)\|\nabla u\|^{2}\right)-\varepsilon^{\prime}(t)\|\nabla u\|^{2} \]$\mathcal {H}_{t}(\Omega ).$(3) It is worth mentioning that the nonlinear term
$f(u)$ and the external force term $h(t)$ makeproblem (1.1) be studied in a more general functional framework. To obtain the dissipative properties of the process, we assume that $f$ satisfies (1.5)–(1.7), which is weaker than the conditions in [Reference Peng, Shang and Zheng22].
The structure of our paper is organized as follows. In § 2, some function spaces, abstract definitions and functions to be used later are introduced. As is known, when studying the pullback attractors of an equation, it is often necessary to attain the existence and uniqueness of the solution at first, which will be obtained by the standard Faedo-Galerkin approximations in § 3. The most important parts § 4 and § 5 are arranged in the last two sections. We mention here the energy method, the diagonal method, the decomposition method and multiple inequalities are used to overcome the difficulties in proving the existence and regularity caused by the nonlocal function and nonlinear term.
2. Preliminaries
In this section, we shall introduce the definitions of some spaces and functions involved in the paper, and some abstract concepts related to the time-dependent pullback attractors theory.
For any 
$t$, let 
$X_{t}$ be a family of normed spaces, where the sphere with radius 
$R$ is denoted as
\[ {\bar B_{{X_t}}}(R) = \left\{ {u \in {X_t}:\left\| u \right\|_{{X_t}}^{2} \le R} \right\}. \]
Besides, for any 
$t \in \mathbb {R}$, the time-dependent space 
$\mathcal {H}_{t}(\Omega )$ is endowed with the norms
\[ \|u\|_{\mathcal{H}_{t}}^{2}=\|u\|_{2}^{2}+\varepsilon(t)\|\nabla u\|_{2}^{2}, \]
and the space 
${\mathcal {H}}_t^{1}(\Omega )$, more regular than 
$\mathcal {H}_{t}(\Omega )$, is endowed with the norms
\[ \|u\|_{\mathcal{H}_{t}^{1}}^{2}=\|\nabla u\|_{2}^{2}+\varepsilon(t)\|\Delta u\|_{2}^{2}. \]
We define the Hausdorff semidistance of two nonempty sets 
$A,B \subset \mathcal {H}_{t}(\Omega )$ by
\[ dist_{\mathcal{H}_{t}}(A,B)=\sup _{x \in A} \inf _{y \in B}\|x-y\|_{\mathcal{H}_{t}} . \]Lemma 2.1 Aubin-Lions Compactness Lemma [Reference Lions16]
Let 
$X_{0}$, 
$X$ and 
$X_{1}$ be Banach spaces, and satisfy that 
$X \subset X_{0} \subset X_{1}$ are dense and continuous embeddings and 
$X \subset X_{0}$ is a compact embedding. Assuming that 
$p \ge 1$, 
$1 \le q \le + \infty$ and 
$T>0$ is given. Let
\[ \overline W=\left\{u \in L^{p}([0, T]; X), \frac{\textrm{d}u}{\textrm{d} t} \in L^{q}([0, T]; X)\right\}, \]then
(i) if
$p < + \infty$, $\overline W \subset {L^{p}}([0,T];{X_0})$ is compact;(ii) if
$p = + \infty$, $\overline W \subset {C}([0,T];{X_0})$ is compact.
Remark 2.1 Let 
$p = q = 2$, 
$X_{0}$, 
$X$ and 
$X_{1}$ are Hilbert spaces, 
$X \subset X_{0} \subset X_{1}$ are dense and continuous embeddings, if 
$X_{0}$ is the interpolation space between 
$X$ and 
$X_{1}$ and the coefficient is 
$\frac {1}{2}$, then 
$W \subset {C}([0,T];{X_0})$ is a continuous embedding.
Definition 2.2 [Reference Ma, Wang and Xu19, Reference Zhu, Xie and Zhou34]
Let 
$\{\mathcal {H}_{t}\}_{t \in \mathbb {R}}$ be a family of time-dependent normed spaces. A process or a two-parameter semigroup on 
$\mathcal {H}_{t}$ is a family 
$\{U(t, \tau ) \mid t, \tau \in \mathbb {R}, t \geqslant \tau \}$ of continuous mapping 
$U(t, \tau ): \mathcal {H}_{\tau } \rightarrow \mathcal {H}_{t}$ satisfies that 
$U(\tau, \tau )u=u$ for any 
$u \in \mathcal {H}_{\tau }$ and 
$U(t, s) U(s, \tau )=U(t, \tau )$ for all 
$t \geq s \geqslant \tau$.
Definition 2.3 [Reference Caraballo, Herrera-Cobos and Marín-Rubio6, Reference Peng, Shang and Zheng22]
For any 
$\delta >0$, let 
$\mathcal {D}_{\delta, \mathcal {H}_{t}}$ be a nonempty class of all families of parameterized sets 
$\widehat {D}_{\delta }=\{D_{\delta }(t): t \in \mathbb {R}\} \subset \Gamma (\mathcal {H}_{t})$ such that
\[ \lim _{\tau \rightarrow-\infty}\left(e^{\delta \tau} \sup _{u \in D_{\delta}(\tau)}\|u\|_{\mathcal{H}_{t}}^{2}\right)=0, \]
where 
$\Gamma (\mathcal {H}_{t})$ denotes the family of all nonempty subsets of 
$\mathcal {H}_{t}(\Omega )$.
Definition 2.4 [Reference Peng, Shang and Zheng22, Reference Zhu and Sun33]
The process 
$\{ U(t,\tau )\} _{t \ge \tau }$ is said to be pullback 
$\mathcal {D}_{\delta, \mathcal {H}_{t}}$-asymptotically compact if for any 
$t \in \mathbb {R}$, any 
${\widehat D_\delta } \in {{\mathcal {D}}_{\delta,{{\mathcal {H}}_t}}}$, any sequence 
$\tau _{n} \to -\infty$, and any sequence 
${x_n} \in {D_\delta }( {{\tau _n}} ) \subset \mathcal {H}_{t}(\Omega )$, the sequence 
$\{U (t, \tau ) x_{n}\}_{n=1}^{\infty }$ is relatively compact in 
$\mathcal {H}_{t}(\Omega )$.
Definition 2.5 [Reference Peng, Shang and Zheng22, Reference Zhu and Sun33]
It is said that 
${\widehat D_{0} } \in {{\mathcal {D}}_{\delta,{{\mathcal {H}}_t}}}$ is pullback 
$\mathcal {D}_{\delta, \mathcal {H}_{t}}$-absorbing for the process 
${\{ U(t,\tau )\} _{t \ge \tau }}$ if for any 
$t \in \mathbb {R}$ and 
${\widehat D_\delta } \in {{\mathcal {D}}_{\delta,{{\mathcal {H}}_t}}}$, there exists a 
${\tau _0} = {\tau _0}(t,{\widehat D_\delta }) < t$ such that 
$U(t, \tau ) D_{\delta }(\tau ) \subset D_{0}(t),$ for all 
$\tau \leqslant \tau _{0}(t, \widehat {D}_{\delta })$.
Definition 2.6 [Reference Peng, Shang and Zheng22, Reference Zhu and Sun33]
A family 
$\widehat {\mathcal {A}}_{t}=\{\mathcal {A}(t): t \in \mathbb {R}\} \subset \Gamma (\mathcal {H}_{t}(\Omega ))$ is said to be a time-dependent pullback 
$\mathcal {D}_{\delta, \mathcal {H}_{t}}$-attractor for the process 
${\{ U(t,\tau )\} _{t \ge \tau }}$ in 
$\mathcal {H}_{t}(\Omega )$ if
(i)
$\mathcal {A}(t)$ is compact in $\mathcal {H}_{t}(\Omega )$ for any $t \in \mathbb {R}$;(ii)
$\widehat {\mathcal {A}}_{t}$ is pullback $\mathcal {D}_{\delta, \mathcal {H}_{t}}$-attracting in $\mathcal {H}_{t}(\Omega )$, i.e.,
for all\[ \lim _{\tau \rightarrow-\infty} \operatorname{dist}_{\mathcal{H}_{t}}\left(U(t, \tau) D_{\delta}(\tau), \mathcal{A}(t)\right)=0, \]${\widehat D_\delta } \in {{\mathcal {D}}_{\delta,{{\mathcal {H}}_t}}}$ and $t \in \mathbb {R}$;(iii)
$\widehat {\mathcal {A}}_{t}$ is invariant, i.e., $U(t, \tau )\mathcal {A}(\tau )={\mathcal {A}(t)}$, for any $-\infty <\tau \leq t<+\infty$.
3. Existence and uniqueness of solutions
In order to obtain the existence of the time-dependent pullback attractors of problem (1.1), we need to establish the existence and uniqueness of solutions. First, we give the definition of the weak solution.
Definition 3.1 A weak solution to problem (1.1) is a function 
$u \in C([\tau, t], \mathcal {H}_{t}(\Omega ))$ for any 
$t \ge \tau$, with 
$u(\tau )=u_{\tau }$, and such that for all 
$\varphi \in H_{0}^{1}(\Omega )$, it holds that
\begin{align} & \frac{\textrm{d}}{\textrm{d} t}[(u(t), \varphi)+\varepsilon(t)(\nabla u(t), \nabla \varphi)]+\left(2 a(l(u))-\varepsilon^{\prime}(t)\right)(\nabla u(t), \nabla \varphi)\nonumber\\ &\quad = 2(f(u(t)), \varphi)+2\left\langle {h(t),\varphi } \right\rangle. \end{align}Remark 3.1 The equation (3.1) is supposed to be understood in the sense of the generalized function space 
$\mathcal {D}^{\prime }(\tau,+\infty )$.
Remark 3.2 If 
$u(x,t)$ is a weak solution of problem (1.1), then it satisfies the following energy equality:
\begin{align} &\|u(t)\|^{2}+\varepsilon(t)\|\nabla u(t)\|^{2}+\int_{s}^{t}\left(2 a(l(u))-\varepsilon^{\prime}(r)\right)\|\nabla u(r)\|^{2}\,\textrm{d} r\nonumber\\ &\quad=\|u(s)\|^{2}+\varepsilon(s)\|\nabla u(s)\|^{2}+2 \int_{s}^{t}(f(u(r)), u(r))\,\textrm{d}r+2 \int_{s}^{t}(h(r), u(r))\,\textrm{d}r. \end{align}The following theorems, theorems 3.2–3.3 will clarify the existence and uniqueness of the solution to problem (1.1). In addition, the former theorem is proved by the classic Faedo-Galerkin method, and the proof also involves the energy estimate method.
Theorem 3.2 Assume that 
$a(\cdot )$ is a local Lipschitz continuous function and satisfies (1.4), 
$f \in C^{1}(\mathbb {R})$ and satisfies (1.5)–(1.7), 
$h \in L_{l o c}^{2}(\mathbb {R}; H^{-1})$, 
$l(\cdot )$ is given, and the initial value 
${u_\tau } \in {{\mathcal {H}}_t}(\Omega )$, then for any 
$\tau \in \mathbb {R}$ and 
$t \ge \tau$, there exists a weak solution to problem (1.1), which satisfies 
$u \in C([\tau, t] ;\mathcal {H}_{t}(\Omega ))$ and 
$u_{t} \in L^{2}(\tau, t;\mathcal {H}_{t}(\Omega )).$ Moreover, the solution 
$u$ in 
$\mathcal {H}_{t}(\Omega )$ depends continuously on the initial values.
Proof. Using the spectral theory, we conclude that there exists a sequence 
$\{ {{\omega _j}} \}_{j = 1}^{\infty }$ of eigenfunctions of 
$-\Delta$ in 
$H_{0}^{1}(\Omega )$, which is a Hilbert basis of 
$L^{2}(\Omega ).$ Fix an integer 
$j$, then 
$\{\omega _{1}, \omega _{2}, \cdots, \omega _{j}\}$ are 
$j$ linearly independent functions in 
$H_{0}^{1}(\Omega )$, which can be expanded into a 
$j$-dimensional linear subspace, denoted as 
$W_{j}(\Omega )=\operatorname {span}\{\omega _{1}, \omega _{2}, \cdots, \omega _{j}\}$. The Faedo-Galerkin method needs to find the approximate sequence 
$u_{k}(t, \tau ; u_{\tau })$ 
$=\sum\limits_{j=1}^{k} r_{k, j}(t) \omega _{j}(x)$, so that for any 
$k \geqslant n$, the following approximate system holds:
\begin{equation} \left\{{\begin{array}{@{}l} {\dfrac{\textrm{d}}{{\textrm{d}t}}[({u_k}(t),{\omega _j}) + \varepsilon (t)(\nabla {u_k}(t),\nabla {\omega _j})] + \left({2a(l({u_k})) - {\varepsilon ^{\prime} }(t)} \right)(\nabla {u_k}(t),\nabla {\omega _j})}\\ = 2(f({u_k}(t)),{\omega _j}) + 2\left\langle {h(t),{\omega _j}} \right\rangle,\quad\forall {u_k} \in {W_j}(\Omega ),\\ {u_{k,\tau }}(x) = {u_\tau }(x). \end{array}} \right. \end{equation} Step 1: (A priori estimate) In order to ensure that for any 
$t \in [\tau,+\infty )$, there is a weak solution 
$u_{k}$ of the system (3.3), therefore a priori estimate needs to be established. Taking 
${\gamma _{k,j}}(t)$ as the test function of the above approximate system, and then summing 
$j$ from 
$1$ to 
$k$, we obtain
\begin{align} &\frac{\textrm{d}}{\textrm{d}t}(\left\|u_{k}(t)\right\|^{2}+\varepsilon(t)\left\|\nabla u_{k}(t)\right\|^{2})+\left(2 m-\varepsilon^{\prime}(t)\left\|\nabla u_{k}(t)\right\|^{2}\right)\notag\\ &\quad \leqslant 2\left(f\left(u_{k}\right), u_{k}\right)+2\left(h(t), u_{k}\right).\end{align}
From (1.5) and the Poincaré inequality, it follows that there is a constant 
$w \in (0,m{\lambda _1})$ such that
\begin{equation} \left(f\left(u_{k}\right), u_{k}\right) \leqslant\left(\lambda_{1} u_{k}, u_{k}\right) \leqslant \frac{w}{\lambda_{1}}\left\|\nabla u_{k}\right\|^{2} + C. \end{equation}By the Hölder, Young and Poincaré inequalities, we have
\begin{equation} \left(h(t), u_{k}\right) \leqslant\|h(t)\|_{{-}1}\left\|u_{k}\right\|_{1} \leqslant \varepsilon\left\|\nabla u_{k}\right\|^{2}+c(\varepsilon)\|h(t)\|_{{-}1}^{2}, \end{equation}
where 
$\varepsilon =m-\frac {w}{\lambda _{1}}$, 
$c(\varepsilon )=(2 \varepsilon )^{-1} 2^{-1}=\frac {\lambda _{1}}{4(m \lambda _{1}-w)}.$ Substituting (3.5) and (3.6) into (3.4), we can derive
\begin{equation} \frac{\textrm{d}}{\textrm{d} t}(\left\|u_{k}(t)\right\|^{2}+\varepsilon(t)\left\|\nabla u_{k}(t)\right\|^{2})-\varepsilon^{\prime}(t)\left\|\nabla u_{k}(t)\right\|^{2}\leqslant\frac{\lambda_{1}}{2\left(m \lambda_{1}-w\right)}\|h(t)\|_{{-}1}^{2}. \end{equation}
Integrating (3.7) with respect to t in 
$[\tau, t]$, we deduce that
\begin{align*} &\left\|u_{k}(t)\right\|^{2}+\varepsilon(t)\left\|\nabla u_{k}(t)\right\|^{2}-\int_{\tau}^{t} \varepsilon^{\prime}(s)\left\|\nabla u_{k}(s)\right\|^{2}\,\textrm{d} s \\ &\quad\leqslant\left\|u_{k}(\tau)\right\|^{2}+\varepsilon(\tau)\left\|\nabla u_{k}(\tau)\right\|^{2}+\frac{\lambda_{1}}{2\left(m \lambda_{1}-w\right)} \int_{\tau}^{t}\|h(s)\|_{{-}1}^{2}\,\textrm{d}s. \end{align*}
From 
$\varepsilon ^{\prime }(t)<0$, we conclude that
\begin{align} \left\|u_{k}(t)\right\|^{2}+\varepsilon(t)\left\|\nabla u_{k}(t)\right\|^{2} \leqslant\left\|u_{k}(\tau)\right\|^{2}+\varepsilon(\tau)\left\|\nabla u_{k}(\tau)\right\|^{2}+C \int_{\tau}^{t}\|h(s)\|_{{-}1}^{2}\,\textrm{d} s. \end{align} It can be seen from (3.8) that for any 
$t \ge \tau$, 
$\{u_{k}\}_{k\geqslant n}$ is bounded in 
$L^{\infty }(\tau, t ; H_{t}(\Omega )) \cap L^{2}(\tau, t ; H_{0}^{1}(\Omega )) \cap L^{p}(\tau, t ; L^{p}(\Omega ))$, hence 
$\{-a(l(u_{n}) \Delta u_{n}\}$ is bounded in 
$L^{2}(\tau, t; H^{-1}(\Omega ))$. From (1.5), the Hölder inequality and Sobolev embedding theorem, we have
\begin{equation} \int_{\tau}^{t} \int_{\Omega}\left|f\left(u_{k}(s)\right)\right|^{q}\,\textrm{d}x\,\textrm{d}s \leqslant C \int_{\tau}^{t}\left\|u_{k}(s)\right\|_{L^{p}(\Omega)}^{p}\,\textrm{d}s+C, \end{equation}
where 
$\frac {1}{p}+\frac {1}{q}=1$. Therefore, for any 
$t \geqslant \tau$, we have
\begin{equation} \left\{f\left(u_{k}\right)\right\}_{k\geqslant n}\ is\ bounded \ in \ L^{q}\left(\tau, t ; L^{q}(\Omega)\right). \end{equation} Step 2: (Uniform estimate for the time derivatives) Multiplying the approximate system (3.3) by 
$\gamma _{k, j}^{\prime }(t)$ and summing from 1 to 
$k$, we obtain that
\begin{align} &\left\|\left(u_{k}(t)\right)_{t}\right\|^{2}+\varepsilon(t)\left\|\nabla\left(u_{k}(t)\right)_{t}\right\|^{2}+a\left(l\left(u_{k}\right)\right) \frac{\textrm{d}}{\textrm{d}t}\left\|\nabla u_{k}(t)\right\| \notag\\ &\quad =2\left(f\left(u_{k}\right)+h(t),\left(u_{k}(t)\right)_{t}\right).\end{align}
Integrating (3.11) in 
$[\tau, t]$, and then using (1.4), and the boundedness of function 
$f$ and 
$h$, the following equation can be obtained through the similar estimates in step 1:
\begin{align} m\left\|\nabla u_{k}(t)\right\|^{2}+\int_{\tau}^{t}\left\|\left(u_{k}(s)\right)_{s}\right\|^{2}+\varepsilon(s)\left\|\nabla\left(u_{k}(s)\right)_{s}\right\|^{2}\,\textrm{d}s \leqslant m\left\|\nabla u_{k}(\tau)\right\|^{2}+C. \end{align}
Then it is easy to get that 
$\{(u_{k})_{t}\}_{k \geqslant n}$ is bounded in 
$L^{\infty }(\tau, \tau ; \mathcal {H} _{t}(\Omega ))$.
Step 3: (Existence of solutions) From the boundedness of functions 
$\{u_{k}\}_{k\geqslant n}$, 
$\{f(u_{k})\}_{k \geqslant n}$, 
$\{(u_{k})_{t}\}_{k \geqslant n}$ and lemma 2.1, we can deduce that for any 
$t \geqslant \tau$, there exist functions 
$u \in L^{\infty }(\tau, t ; H_{t}(\Omega )) \cap L^{2}(\tau, t ; H_{0}^{1}(\Omega ))$ 
$\cap L^{p}(\tau, t ; L^{p}(\Omega ))$, 
$u_{t} \in L^{\infty }(\tau, \tau ; \mathcal {H} _{t}(\Omega ))$, 
$\xi \in L^{2}(\tau, t; H_{0}^{1}(\Omega ))$ and 
$\chi \in L^{q}(\tau, t; L^{q}(\Omega ))$ such that
\begin{gather} u_{k} \rightharpoonup u \quad \text{weakly-star in}\ L^{\infty}\left(\tau, t ;\mathcal{H}_{t}(\Omega)\right); \end{gather}
\begin{gather} a\left(l(u_{k})\right) u_{k} \rightharpoonup \xi\quad \text{weakly in}\ L^{2}\left(\tau, t ; H_{0}^{1}(\Omega)\right); \end{gather}
\begin{gather} f\left(u_{k}\right) \rightharpoonup \chi \quad \text{weakly in}\ L^{q}\left(\tau, t ; L^{q}(\Omega)\right); \end{gather}
\begin{gather} \left.\left(u_{k}\right)_{t} \rightharpoonup u_{t} \quad \text{weakly in}\ L^{2}(\tau, t; \mathcal{H}_{t}(\Omega)\right); \end{gather}
\begin{gather} \left.u_{k} \rightarrow u \quad \text{strongly in}\ L^{2}\left(\tau, t ; L^{2} (\Omega\right)\right); \end{gather}
\begin{gather} u_{k} \rightarrow u \quad a. e.\ (x, t) \in \Omega \times[\tau,+\infty). \end{gather}
From lemma 2.1, it is easy to prove that 
$a(l(u)) u=\xi$ and 
$f(u)=\chi$. Then we can conclude that
\[ f\left(u_{k}\right) \rightarrow f(u) \quad \text{a. e. in}\ \Omega \times[\tau,+\infty). \]
When 
$W_{j}(\Omega )$ is dense in 
$H_{0}^{1}(\Omega )$, then from the convergence obtained above, we can see that 
$u$ is a weak solution of problem (1.1), while when 
$W_{j}(\Omega )$ is not dense in 
$H_{0}^{1}(\Omega )$, this more general case will be demonstrated below. Let 
$u_{l}$ be a weak solution of problem (1.1). Now estimating the energy of 
$u_{k}$ and 
$u_{l}$ respectively, and then using (1.6) and the Poincaré, Hölder inequalities, and the Sobolev embedding theorem, we can obtain
\begin{align} &\frac{\textrm{d}}{\textrm{d} t}\left[\left\|u_{k}-u_{l}\right\|^{2}+\varepsilon(t)\left\|u_{k}-u_{l}\right\|_{1}^{2}\right]+(2 a l(u))-\varepsilon^{\prime}(t)\left\|u_{k}-u_{l}\right\|_{1}^{2}\nonumber\\ &\quad\leq 2\left|\left(f\left(u_{k}\right)-f\left(u_{l}\right), u_{k}-u_{l}\right)\right|\nonumber\\ &\quad=2\left|\left(f^{\prime}\left(\theta u_{k}+(1-\theta) u_{l}\right)\left(u_{k}-u_{l}\right), u_{k}-u_{l}\right)\right|\nonumber\\ &\quad\leq C\left\|u_{k}-u_{l}\right\|^{2}\nonumber\\ &\quad\leq C\left\|u_{k}-u_{l}\right\|_{1}^{2}. \end{align} Moreover, from 
$\varepsilon ^{\prime }(t) \leq 0$ and noting that 
$a (\cdot )$ is a positive bounded function, we can obtain that
\begin{equation} \frac{\textrm{d}}{\textrm{d} t}[\|u_{k}-u_{l}\|^{2}+\varepsilon(t)\|u_{k}-u_{l}\|_{1}^{2}] \leq C\|u_{k}-u_{l}\|_{1}^{2}. \end{equation}Applying the generalized Gronwall lemma (see [Reference Qin24–Reference Qin26]) to (3.20), we can conclude
\begin{equation} \left\|u_{k}-u_{l}\right\|_{\mathcal{H}_{t}} \leqslant\,\textrm{e}^{C(t-\tau)}\left\|u_{k}-u_{l}\right\|_{\mathcal{H}_{t}}^{2}, \end{equation}
which implies that 
$\{ {u_k}\} \to \{ {u_l}\}$ in 
$u \in C([\tau, t] ;\mathcal {H}_{t}(\Omega ))$, for any 
$t \geqslant \tau$.
Obviously, 
$u$ is a weak solution of problem (1.1).
Theorem 3.3 Under the assumptions of theorem 3.2, if the weak solution of problem (1.1) exists, then it is a unique solution.
Proof. Assuming that the solutions corresponding to the initial values 
$u_{\tau,1}$ and 
$u_{\tau,2}$ are 
$u_{1}$ and 
$u_{2}$, respectively, they satisfy the following equations, respectively:
\begin{equation} \left\{\begin{array}{@{}ll} (u_{1})_{t}-\varepsilon(t) \Delta (u_{1})_{t}-a(l(u_{1})) \Delta u_{1}=f(u_{1})+h(t) & \text{in}\ \Omega \times(\tau, \infty), \\ u_{1}=0 & \text{on}\ \partial \Omega\times(\tau, \infty), \\ u_{1}(x, \tau)=u_{\tau,1}(x), & x \in \Omega, \end{array}\right.\end{equation}and
\begin{equation} \left\{\begin{array}{@{}ll} (u_{2})_{t}-\varepsilon(t) \Delta (u_{2})_{t}-a(l(u_{2})) \Delta u_{2}=f(u_{2})+h(t) & \text{in}\ \Omega \times(\tau, \infty), \\ u_{2}=0 & \text{on}\ \partial \Omega\times(\tau, \infty), \\ u_{2}(x, \tau)=u_{\tau,2}(x), & x \in \Omega. \end{array}\right.\end{equation} Subtracting (3.22) from (3.23), and setting 
$u=u_{1}-u_{2}$, then 
$u$ satisfies
\begin{equation} \left\{\begin{array}{@{}ll} u_{t}-\varepsilon(t) \Delta u_{t}-a\left(l\left(u_{1}\right)\right) \Delta u_{1}+a\left(l\left(u_{2}\right)\right) \Delta u_{2}&\\ \quad =f\left(u_{1}\right)-f\left(u_{2}\right) & \text{in}\ \Omega \times(\tau, \infty) \\ u(x, t)=0 & \text{on}\ \partial \Omega \times(\tau, \infty) \\ u(x, \tau)=u_{\tau, 1}-u_{\tau, 2}, & \, x \in \Omega. \end{array}\right.\end{equation} Taking 
$u=u_{1}-u_{2}$ as a test function of (3.24), then we can obtain
\begin{align} &\frac{\textrm{d}}{\textrm{d} t}\left[\|u\|^{2}+\varepsilon(t)\|\nabla u\|^{2}\right]+\left(2 a\left(l\left(u_{1}\right)\right)-\varepsilon^{\prime}(t)\right)\|\nabla u\|^{2}\nonumber\\ &\quad=2\left[a\left(l\left(u_{2}\right)\right)-a\left(l\left(u_{1}\right)\right)\right]\left(\nabla u_{2}, \nabla u\right)+2\left(f\left(u_{1}\right)-f\left(u_{2}\right), u\right). \end{align} Since 
$l \in L^{2}(\Omega )$ is a bounded linear functional and by the H
$\ddot {\mathrm {o}}$lder inequality, it is easy to get
\begin{equation} l\left(u_{1}\right)-l\left(u_{2}\right)=l\left(u_{1}-u_{2}\right)=\left(l, u_{1}-u_{2}\right) \leq\|l\|_{2}\|u\|_{2}. \end{equation} Besides, since 
$a(\cdot )$ is a locally Lipschitz continuous function, when 
$l({u_1}),l({u_2}) \in [0,{C_1}]$, we have
\begin{equation} \left|a\left(l\left(u_{2}\right)\right)-a\left(l\left(u_{1}\right)\right)\right| \leq \operatorname{Lip}\left(C_{1}\right)\|l\|_{2}\|u\|_{2}, \end{equation}
where 
$Lip({C_1})$ is the Lipschitz constant of 
$a(\cdot )$. By (3.26) and (3.27) and the Young inequality, we obtain
\begin{align} \left|\left(a\left(l\left(u_{2}\right)\right)-a\left(l\left(u_{1}\right)\right)\right)\left(\nabla u_{2}, \nabla u\right)\right| &\leq d\|\nabla u\|^{2}+\frac{1}{4 d}\left(\operatorname{Lip}\left(C_{1}\right)\right)^{2} \notag\\ &\quad{}\times\|l\|^{2}\|u\|^{2}\left\|\nabla u_{2}\right\|^{2}. \end{align}From the proof of theorem 3.2, we deduce that
\begin{equation} \left|\left(f\left(u_{1}\right)-f\left(u_{2}\right), u\right)\right| \leqslant c\|\nabla u\|^{2}. \end{equation}In addition, substituting (3.28)–(3.29) into (3.25), then we can obtain the following inequality
\begin{equation} \frac{\textrm{d}}{\textrm{d} t}\left[\|u\|^{2}+\varepsilon(t)\|\nabla u\|^{2}\right] \leq C\left(\|u\|^{2}+\varepsilon(t)\|\nabla u\|^{2}\right). \end{equation}Applying the Gronwall lemma to (3.30), we conclude
\begin{equation} \|u(t)\|^{2}+\varepsilon(t)\|\nabla u(t)\|^{2} \leq \textrm{e}^{C(t-\tau)}\left(\left\|u_{\tau}\right\|^{2}+\varepsilon(\tau)\left\|\nabla u_{\tau}\right\|^{2}\right). \end{equation}Consequently, the uniqueness of the solution follows readily.
4. The time-dependent pullback attractors
In this section, we will verify the existence of the time-dependent pullback attractors in 
$\mathcal {H}_{t}$. In order to prove the existence of time-dependent pullback attractors for the process 
$\{U(t,\tau )\} _{t \ge \tau }$, we need to check that the process 
$U$ is pullback 
$D_{s,\mathcal {H}_{t}}$-asymptotically compact, thus we first need to give the following lemma.
Lemma 4.1 Under the assumptions of theorem 3.2, then for any 
$t \ge \tau$, the solution of problem (1.1) satisfies
\begin{align} \|u(t)\|^{2}+\varepsilon(t)\|\nabla u(t)\|^{2} &\leq \textrm{e}^{-\delta(t-\tau)}\left\|u_{\tau}\right\|_{\mathcal{H}_{t}}^{2}+\frac{C}{\delta}\notag\\ &\quad +\frac{\textrm{e}^{-\delta t}}{2\left(m-w \lambda _1^{{-}1}\right)-\delta} \int_{\tau}^{t}\,\textrm{e}^{\delta s}\|h(s)\|_{{-}1}^{2}\,\textrm{d}s, \end{align}
where 
$0<\delta <\min \{2(m-w \lambda _{1}^{-1}), \frac {-\varepsilon ^{\prime }(t) \lambda _{1}}{1+\lambda _{1} \varepsilon (t)-\lambda _{1}}\}$.
Proof. From the energy equality (3.2) of problem (1.1), it can be seen that the weak solution 
$u$ satisfies
\begin{align} &\frac{\textrm{d}}{\textrm{d}t}\left[\|u(t)\|^{2}+\varepsilon(t)\|\nabla u(t)\|^{2}\right]+\left(2 a(l(u(t)))-\varepsilon^{\prime}(t)\right)\|\nabla u(t)\|^{2}\nonumber\\ &\quad=2(f(u(t)), u(t))+2(h(t), u(t)). \end{align} Using the Young, Cauchy-Schwartz and Poincaré inequalities, from (1.2), (4.2) and 
$w \in (0,m{\lambda _1})$, we can derive
\begin{align} &\|u(t)\|^{2}+\varepsilon(t)\|\nabla u(t)\|^{2}+\delta\left(\|u(t)\|^{2}+\varepsilon(t)\|\nabla u(t)\|^{2}\right)\notag\\ &\quad \leq C+\frac{1}{2\left(m-w \lambda _1^{{-}1}\right)-\delta}\|h(s)\|_{{-}1}^{2}. \end{align}Then by the Gronwall lemma, (4.1) follows directly.
Remark 4.1 The choice of 
$\delta$ in lemma 4.1 is not unique. For example, if for any 
$t \ge \tau$, 
$0<\delta <\min \{\frac {-\varepsilon ^{\prime }(t)}{\varepsilon (t)}, 2(m-w \lambda _{1}^{-1}) \lambda _{1}\}$ is selected, the solution u of problem (1.1) can be obtained through similar calculations, which satisfies
\begin{align} \|u(t)\|^{2}+\varepsilon(t)\|\nabla u(t)\|^{2} &\leq \textrm{e}^{-\delta(t-\tau)}\left\|u_{\tau}\right\|_{\mathcal{H} t}^{2}+\frac{C}{\delta}\notag\\ &\quad +\frac{\textrm{e}^{-\delta t}}{2\left(m-w \lambda^{{-}1}\right)-\delta \lambda_{1}^{{-}1}} \int_{\tau}^{t}\,\textrm{e}^{\delta s}\|h(s)\|_{{-}1}^{2}\,\textrm{d}s. \end{align}As a consequence, the existence of the time-dependent pullback attractors can be obtained by both (4.3) and (4.4).
Lemma 4.2 Under the assumptions of theorem 3.2, moreover if 
$h$ satisfies that
\begin{equation} \int_{-\infty}^{t}\,\textrm{e}^{\delta s}\|h(s)\|_{{-}1}^{2} d_{s}<{+}\infty, \end{equation}
for some 
$0<\delta <\min \{2(m-w \lambda _{1}^{-1}), \frac {-\varepsilon ^{\prime }(t) \lambda _{1}}{1+\lambda _{1} \varepsilon (t)-\lambda _{1}}\}.$ Then the family 
${{\widehat D}_\delta } = \{{D_\delta }(t): t \in \mathbb {R}\}$ with 
${D_\delta }(t) = {{\bar B}_{{\mathcal {H}_t}}}(0,\rho (t))$, the closed ball in 
$\mathcal {H}_t(\Omega )$ of centre zero and radius 
$\rho (t)$, where
\begin{equation} \rho^{2}(t)=1+\frac{C}{\delta}+\frac{\textrm{e}^{-\delta t}}{2\left(m-w \lambda^{{-}1}\right)-\delta} \int_{\tau}^{t}\,\textrm{e}^{\delta s}\|h(s)\|_{{-}1}^{2}\,\textrm{d} s \end{equation}
is pullback 
$\mathcal {D}_{\delta, \mathcal {H}_{t}}$-absorbing for the process 
${\{ U(t,\tau )\} _{t \ge \tau }}$ of the solution of the equation (1.1). Moreover, we have 
$\widehat D_\delta \in \mathcal {D}_{\delta, \mathcal {H}_{t}}.$
Proof. It is clear that 
$\widehat D_{\delta }$ is pullback 
$\mathcal {D}_{\delta, \mathcal {H}_{t}}$-absorbing for the process 
$\{ U(t,\tau )\} _{t \ge \tau }$ is an immediate consequence of lemma 4.1. Thanks to (4.5), for any 
$t \ge \tau$, we have 
$\textrm {e}^{\delta t}{\rho ^{2}}(t) \to 0$, as 
$t \to \infty$. Then, 
$\widehat D_\delta$ belongs to 
$\mathcal {D}_{\delta, \mathcal {H}_{t}}$.
In order to prove the existence of time-dependent pullback attractors for the process 
$\{ U(t,\tau )\} _{t \ge \tau }$, we need to check that the process 
$U$ is pullback 
$D_{s,\mathcal {H}_{t}}$-asymptotically compact, which is stated in the next lemma.
Lemma 4.3 Under the assumptions of theorem 3.2, the process 
$\{ {U(t,\tau )\} _{t \ge \tau }}$ is pullback 
$D_{s,\mathcal {H}_{t}}$-asymptotically compact in 
$\mathcal {H}_{t}(\Omega ).$
Proof. To prove the lemma, obviously we ought to estimate that for any 
${u_{{\tau _n}}} \in {D_\delta }({\tau _n})$, 
${\tau _n} \in (- \infty,t)$, 
$n \in \mathbb {N}^{+}$ and 
$t \in \mathbb {R}$, the sequence 
${\{ U(t,{\tau _n}){u_{{\tau _n}}}} \}_{n = 1}^{\infty }$ is relatively compact in 
$\mathcal {H}_{t}(\Omega ).$
To simplify the notation, let 
$U(t,{\tau _n}){u_{{\tau _n}}} = {u_n}(t)$. In the following, we will use the diagonal method to get the compactness of the process 
$\{ U(t,{\tau _n}){u_{{\tau _n}}}\}_{n = 1}^{\infty }$.
From lemma 4.3, we can conclude that for any 
$r \ge 0$, there is 
${\tau _r}({{\widehat D}_\delta },t)\le t - r$ such that 
$U(t - r){{\widehat D}_\delta }(\tau ) \subset {D_\delta }(t - r)$ for any 
$\tau \le {\tau _r}({{\widehat D}_\delta },t)$. Besides, we can also obtain that 
$\widehat D_\delta (t)$ is bounded in 
$\mathcal {H}_{t}(\Omega ).$
According to the diagonal parameter method, there is a subsequence 
$\{ {u_{{\tau _m}}}\} \subset \{ {u_{{\tau _n}}}\}$ for any 
$r \geqslant 0$ such that
\begin{equation} U\left(t-r, \tau_{m}\right) u_{\tau_{m}} \rightharpoonup u_{k} \quad in\ \mathcal{H}_{t}(\Omega), \end{equation}
where 
${u_k} \in {D_\delta }(t - r)$.
From theorems 3.2–3.3, (1.4) and (1.7), we can conclude that for a fixed interval 
$[t - r,t]$, the sequence 
$\{ {u_n}\}$ is bounded in 
$L^{\infty }(t-r, t; \mathcal {H}_{t}(\Omega ))$, 
$\{-a(l( u)) \Delta u_{n}\}$ is bounded in 
${L^{2}}(t - r,t;{H^{- 1}(\Omega )})$, and 
$f({u_n})$ is bounded in 
${L^{q}}(t - r,t;{L^{q}}(\Omega )).$ Therefore, there is a subsequence 
$\widetilde {u}_{n}$ of 
$\{u_{n}\}_{n=1}^{\infty }$ that belongs to 
$L^{\infty }(t-r, t; \mathcal {H}_{t}(\Omega ))$ and satisfies
\begin{equation} \begin{aligned} & u_{m} \rightharpoonup \tilde{u}_{n} \quad \text{weakly-star in}\ L^{\infty}\left(t-r, t ; \mathcal{H}_{t}(\Omega)\right); \\ & u_{m} \rightharpoonup \tilde{u}_{n} \quad \text{weakly in}\ L^{2}\left(t-r, t ; \mathcal{H}_{t}(\Omega)\right); \\ & u_{m} \rightarrow \tilde{u}_{n} \quad \text{ strongly in}\ L^{2}\left(t-r, t ; L^{2}(\Omega)\right) ; \\ & f\left(u_{m}\right) \rightharpoonup f\left(\tilde{u}_{n}\right) \quad \text{weakly in}\ L^{q}\left(t-r, t ; L^{q}(\Omega)\right). \end{aligned} \end{equation}Furthermore, from (4.7) and (4.8), we can conclude that
\begin{gather} \tilde{u}_{n}=U(t, t-r) u_{r}, \end{gather}
\begin{gather} U(t, t-r) U\left(t-r, \tau_{m}\right) u_{\tau_{m}} \rightharpoonup U(t, t-r) u_{r} \quad \text{in}\ L^{2}\left(t-r, t ; \mathcal{H}_{t}(\Omega)\right), \end{gather}and
\begin{equation} U(t, t-r) U\left(t-r, \tau_{m}\right) u_{\tau_{m}} \rightarrow U(t, t-r) u_{r} \quad \text{in}\ L^{2}\left(t-r, t ; L^{2}(\Omega)\right). \end{equation}Thanks to (4.8), we have
\begin{equation} \left\|\tilde{u}_{n}\right\| \leq \lim _{m \rightarrow \infty} i n f\left\|U\left(t, \tau_{m}\right) u_{\tau_{m}}\right\|. \end{equation}In order to prove the lemma, we only need to check that
\begin{equation} \lim _{m \rightarrow \infty} sup \left\|U\left(t, \tau_{m}\right) u_{\tau_{m}}\right\| \leqslant\left\|{{\tilde u}_n}\right\|. \end{equation} Multiplying both sides of the energy equation (3.2) by 
$\textrm {e}^{\delta t}$, and then integrating it on 
$[t - r,t]$, we have
\begin{align} \|u(t)\|^{2}+\varepsilon(t)\|\nabla u(t) \|^{2} &=\textrm{e}^{-\delta r}\left(\|u(t-r)\|^{2}+\varepsilon(t)\|\nabla u(t-r)\|^{2}\right)\nonumber\\ &\quad+\delta \int_{t-r}^{t}\,\textrm{e}^{\delta(s-t)}\|u(s)\|^{2}\,\textrm{d}s \nonumber\\ &\quad+\delta \int_{t-r}^{t}\,\textrm{e}^{\delta(s-t)} \varepsilon(s)\|\nabla u(s)\|^{2}\,\textrm{d}s \nonumber\\ &\quad-\int_{t-r}^{t}\,\textrm{e}^{\delta(s-t)}\left(2 a(l(u(s)))-\varepsilon^{\prime}(s)\right)\|\nabla u(s)\|^{2}\,\textrm{d}s\nonumber\\ &\quad+2 \int_{t-r}^{t}\,\textrm{e}^{\delta(s-t)}(f(u(s)), u(s)\,\textrm{d}s\nonumber\\ &\quad+2 \int_{t-r}^{t}\,\textrm{e}^{\delta(s-t)}(h(s), u(s))\,\textrm{d}s. \end{align} Furthermore, from (4.14) and definition 2.2, for any solution 
$U(t,{\tau _m}){u_{{\tau _m}}} \le t - r$ with 
${\tau _m} \le t - r$, it is easy to check that
\begin{align} &\left\|U\left(t, \tau_{m}\right) u_{\tau_{m}}\right\|^{2}+\varepsilon(t)\left\|\nabla U\left(t, \tau_{m}\right) u_{\tau_{m}}\right\|^{2}\nonumber\\ &\quad=\textrm{e}^{-\delta t}\left(\left\|U\left(t-r, \tau_{m}\right) u_{\tau_{m}}\right\|^{2}+\varepsilon(t)\left\|\nabla U\left(t-r, \tau_{m}\right) u_{\tau_{m}}\right\|^{2}\right)\nonumber\\ &\qquad+\delta \int_{t-r}^{t}\,\textrm{e}^{\delta(s-t)}\left\|U(s, t-r) U\left(t-r, \tau_{m}\right) u_{\tau_{m}}\right\|^{2}\,\textrm{d}s\nonumber\\ &\qquad+\delta \int_{t-r}^{t}\,\textrm{e}^{\delta(s-t)} \varepsilon(s)\left\|\nabla U(s, t-r) U\left(t-r, \tau_{m}\right) u_{\tau_{m}}\right\|^{2}\,\textrm{d}s\nonumber\\ &\qquad-\int_{t-r}^{t}\,\textrm{e}^{\delta(s-t)}\left(2 a\left(l\left(U\left(s, \tau_{m}\right) u_{\tau_{m}}\right)\right)-\varepsilon^{\prime}(s)\right)\left\|\nabla U\left(s, \tau_{m}\right) u_{\tau_{m}}\right\|^{2}\,\textrm{d}s\nonumber\\ &\qquad+2 \int_{t-r}^{t}\,\textrm{e}^{\delta(s-t)}\left(f\left(U\left(s, \tau_{m}\right) u_{\tau_{m}}\right), U\left(s, \tau_{m}\right) u_{\tau_{m}}\right)\,\textrm{d}s\nonumber\\ &\qquad+ 2 \int_{t-r}^{t}\,\textrm{e}^{\delta(s-t)}(h(s), U(s, t-r) U(t-r, \tau_{m}) u_{\tau_{m}})\,\textrm{d} s. \end{align}By (4.11), we have
\begin{align} &\lim_{m \rightarrow \infty} \delta \int_{t-r}^{t}\,\textrm{e}^{\delta(s-t)}\left\|U(s, t-r) U\left(t-r, \tau_{m}\right) u_{\tau_{m}}\right\|^{2}\,\textrm{d}s\notag\\ &\quad =\delta \int_{t-r}^{t}\,\textrm{e}^{\delta(s-t)}\left\|U(s, t-r) u_{r}\right\|^{2}\,\textrm{d}s, \end{align}and
\begin{align} &\lim _{m \rightarrow \infty} 2 \int_{t-r}^{t}\,\textrm{e}^{\delta(s-t)}\left(f\left(U\left(s, \tau_{m}\right) u_{\tau_{m}}\right), U\left(s, \tau_{m}\right) u_{\tau_{m}}\right)\,\textrm{d}s\nonumber\\ &\quad=2 \lim _{m \rightarrow \infty} \int_{t-r}^{t}\,\textrm{e}^{\delta(s-t)}(f\left(U(s, t-r) U\left(t-r, \tau_{m}\right) u_{\tau_{m}}\right),\notag\\ &\qquad \times U(s, t-r) U\left(t-r, \tau_{m}\right) u_{\tau_{m}})\,\textrm{d}s\nonumber\\ &\quad=2 \int_{t-r}^{t}\,\textrm{e}^{\delta(s-t)}(f(U(s, t-r) u_{r}), U(s, t-r) u_{r})\,\textrm{d}s \end{align}According to (1.4) and (4.10), we obtain that
\begin{align} &\int_{t-r}^{t}\,\textrm{e}^{\delta(s-t)}\left(2 a\left(l\left(U\left(s, \tau_{m}\right) u_{\tau_{m}}\right)\right)-\varepsilon^{\prime}(s)-\delta \varepsilon(s)\right)\left\|\nabla U\left(s, \tau_{m}\right) u_{\tau_{m}}\right\|^{2}\,\textrm{d}s\nonumber\\ &\quad\leq \lim _{m \rightarrow \infty} i n f \int_{t-r}^{t}\,\textrm{e}^{\delta(s-t)}\left(2 a\left(l\left(U\left(s, \tau_{m}\right) u_{\tau_{m}}\right)\right)-\varepsilon^{\prime}(s)-\delta \varepsilon(s)\right)\notag\\ &\qquad \times \left\|\nabla U(s, t-r) U\left(t-r, \tau_{m}\right) u_{\tau_{m}}\right\|^{2}\,\textrm{d}s\nonumber\\ &\quad=\int_{t-r}^{t}\,\textrm{e}^{\delta(s-t)}\left(2 a\left(l\left(U(s, t-r) u_{r}\right)\right)-\varepsilon^{\prime}(s)-\delta \varepsilon(s)\right)\left\|\nabla U(s, t-r) u_{r}\right\|^{2}\,\textrm{d}s. \end{align} By 
$\textrm {e}^{\delta (s-t)} h(s) \in L^{2}(t-r, t ; H^{-1}(\Omega ))$ and (4.10), we have
\begin{align} &\lim _{m \rightarrow \infty}2 \int_{t-r}^{t}\,\textrm{e}^{\delta(s-t)}\left(h(s), U(s, t-r) U\left(t-r, \tau_{m}\right) u_{\tau_{m}}\right) \textrm{d}s\nonumber\\ &\quad=2\int_{t-r}^{t}\,\textrm{e}^{\delta(s-t)}\left(h(s), U(s, t-r) u_{r}\right)\,\textrm{d}s. \end{align}Using lemmas 4.1 and 4.2, we deduce that
\begin{equation} \textrm{e}^{-\delta r}\left(\left\|U\left(t-r, \tau_{m}\right) u_{\tau_{m}}\right\|^{2}+\varepsilon(t)\left\|\nabla U\left(t-r, \tau_{m}\right) u_{\tau_{m}}\right\|^{2}\right) \leqslant \textrm{e}^{-\delta r} \rho^{2}(t-r) \end{equation}Taking (4.16)–(4.20) into (4.15), we have
\begin{align} \| & \tilde{u}_{n}\|^{2}+\varepsilon(t) \lim _{m \rightarrow \infty} sup \| \nabla U\left(t, \tau_{m}\right) u_{\tau_{m}} \|^{2}\nonumber\\ &\leq\, \textrm{e}^{-\delta r} \rho^{2}(t-r)+\delta \int_{t-r}^{t}\,\textrm{e}^{\delta(s-t)}\left\|U(s, t-r) u_{r}\right\|^{2}\,\textrm{d}s\nonumber\\ &\quad-\int_{t-r}^{t}\,\textrm{e}^{\delta(s-t)}\left(2 a\left(l\left(U(s, t-r) u_{r}\right)\right)-\varepsilon^{\prime}(s)-\delta \varepsilon(s)\right)\left\|\nabla U(s, t-r) u_{r}\right\|^{2}\,\textrm{d}s\nonumber\\ &\quad+2 \int_{t-r}^{t}\,\textrm{e}^{\delta(s-t)}(f(U(s, t-r) u_{r}, U(s, t-r) u_{r})\,\textrm{d}s\nonumber\\ &\quad+2 \int_{t-r}^{t}\,\textrm{e}^{\delta(s-t)}\left(h(s), U(s, t-r) u_{r}\right)\,\textrm{d}s. \end{align}In addition, substituting (4.9) into the equation (1.1) and performing a calculation similar to (4.14), we can deduce that
\begin{align} \left\|\tilde{u}_{n}\right\|^{2}+\varepsilon(t)\left\|\nabla \tilde{u}_{n}\right\|^{2} &\leq \textrm{e}^{-\delta r}\left(\left\|u_{r}\right\|^{2}+\varepsilon(t)\left\|\nabla u_{r}\right\|^{2}\right)\notag\\ &\quad +\delta \int_{t-r}^{t}\,\textrm{e}^{\delta(s-t)}\left\|U(s, t-r) u_{r}\right\|^{2}\,\textrm{d}s\nonumber\\ &\quad-\int_{t-r}^{t}\,\textrm{e}^{\delta(s-t)}\left(2 a\left(l\left(U(s, t-r) u_{r}\right)\right)-\varepsilon^{\prime}(s)-\delta \varepsilon(s)\right)\notag\\ &\quad \times \left\|\nabla U(s, t-r) u_{r}\right\|^{2}\,\textrm{d}s\nonumber\\ &\quad+2 \int_{t-r}^{t}\,\textrm{e}^{\delta(s-t)}\left(f\left(U(s, t-r) u_{r}\right), U(s, t-r) u_{r}\right)\,\textrm{d}s\nonumber\\ &\quad+2 \int_{t-r}^{t}\,\textrm{e}^{\delta(s-t)}\left(h(s), U(s, t-r) u_{r}\right)\,\textrm{d}s. \end{align}Furthermore, by (4.21), (4.22) and the Poincaré inequality, we have
\[ \lim _{m \rightarrow \infty} s u p\left\|U\left(t, \tau_{m}\right) u_{\tau_{m}}\right\|^{2} \leq \textrm{e}^{-\delta r} \lambda_{1}^{{-}1} \rho^{2}(t-r)+\left\|\tilde{u}_{n}\right\|^{2}. \]
Obviously, 
$\lim \limits _{r \rightarrow \infty }$ 
$\textrm {e}^{-\delta r} \lambda _{1}^{-1} \rho ^{2}(t-r)=0$ can be obtained from lemma 4.2, therefore (4.13) holds. From (4.12) and (4.13), we have
\[ \lim _{m \rightarrow \infty} i n f\left\|U\left(t, \tau_{m}\right) u_{\tau_{m}}\right\|=\lim _{m \rightarrow \infty} sup \left\|U\left(t, \tau_{m}\right) u_{\tau_{m}}\right\|=\left\|{{\tilde u}_n}\right\|, \]
which implies the process 
$U$ converges to 
$\tilde {u}_{n}$, thus we can obtain that process 
$U$ is relatively compact. Hence, the proof is complete.
Theorem 4.4 Under the assumptions of theorem 3.2, then from the lemmas in this section, we can conclude that the process of problem (1.1) exists time-dependent pullback attractors 
$\{ {\widehat {\mathcal {A}}}_{t}\}_{t \in \mathbb {R}}$, which satisfy nonempty, compact, invariant and pullback attracting.
5. Regularity of the attractors
In this section, we shall establish the regularity of time-dependent pullback attractors for non-autonomous system (1.1). The methods used in the proof process can also be seen in [Reference Conti and Pata13, Reference Wang, Zhu and Li31, Reference Zelik32].
Theorem 5.1 Under the assumptions of theorem 3.2, then 
$\{\widehat {\mathcal {A}}_{t}\}_{t \in \mathbb {R}}$ is bounded in 
$\mathcal {H}_{t}^{1}(\Omega ).$
Proof. Since 
$L^{2}(\Omega ) \subset H^{-1}(\Omega )$ is dense, for any 
$h$, there exists a function 
${h^{\xi }} \in {L^{2}}(\Omega )$ such that
\begin{equation} \|h-h^{\xi}\|<\xi, \end{equation}
where 
$\xi \ge 0$ is a constant.
Fix 
$\tau \in \mathbb {R}$, suppose 
$u_{\tau } \in \{\widehat {\mathcal {A}}_{t}\} _{t \in \mathbb {R}}$, decompose 
$U(t,\tau ){u_\tau } = u(t)$ into 
$U(t, \tau ) u_{\tau }=U_{1}(t, \tau ) u_{\tau }+U_{2}(t, \tau ) u_{\tau },$ where 
${U_1}(t,\tau ){u_\tau } = v(t)$ and 
${U_2}(t,\tau ){u_\tau } = g(t)$ satisfy the following two equations respectively,
\begin{equation} \left\{\begin{array}{@{}ll} v_{t}-\varepsilon(t) \Delta v_{t}-a\left(l(u)\right) \Delta v=h(t)-h^{\xi}(t) & \text{in}\ \Omega \times(\tau, \infty), \\ v=0 & \text{on}\ \partial \Omega \times(\tau, \infty), \\ v(x, \tau)=u_{\tau}(x), & x \in \Omega, \end{array}\right. \end{equation}and
\begin{equation} \left\{\begin{array}{@{}ll} g_{t}-\varepsilon(t) \Delta g_{t}-a\left(l(u)\right) \Delta g=f(u)+h^{\xi}(t) & \text{in}\ \Omega \times(\tau, \infty), \\ g=0 & \text{on}\ \partial \Omega \times(\tau, \infty), \\ g(x, \tau)=0, & x \in \Omega. \end{array}\right. \end{equation} Multiplying (5.2)
$_{1}$ by 
$- \Delta v$ and integrating it in 
$\Omega$, we have
\[ \frac{\textrm{d}}{\textrm{d}t}\left(\|\nabla v\|^{2}+\varepsilon(t)\|\Delta v\|^{2}\right)+\left(2 a(l(u))-\varepsilon^{\prime}(t)\right)\|\Delta v\|^{2}=2\left(h(t)-h^{\xi}(t),-\Delta v\right). \]By (1.4), (5.1), the Cauchy and Poincaré inequalities, we deduce that
\begin{equation} \frac{\textrm{d}}{\textrm{d}t} E_{1}(t)+\delta_{1} E_{1}(t) \leqslant \xi^{2}, \end{equation}
where 
$\left.E_{1}(t)=\|\nabla v\|^{2}+\varepsilon (t)\right \|\Delta v\|^{2}$ and 
$0<\delta _{1} \leqslant \frac {2 m-\varepsilon ^{\prime }(t)-1}{\lambda _{1}^{-1}+\varepsilon (t)}$.
Using the Gronwall lemma, we obtain
\begin{equation} \|v(t)\|_{\mathcal{H}_{t}^{1}}^{2}=\left\|U_{1}(t, \tau) u_{\tau}\right\|_{\mathcal{H}_{t}^{1}}^{2} \leqslant\,\textrm{e}^{-\delta_{1}(t-\tau)}\left\|u_{\tau}\right\|_{\mathcal{H}_{t}^{1}}^{2}+\frac{\xi^{2}}{\delta_{1}}. \end{equation} Then, multiplying (5.3)
$_{1}$ by 
$- \Delta g$ and integrating it in 
$\Omega$, we obtain
\begin{align} &\frac{\textrm{d}}{\textrm{d}t}\left(\|\nabla g\|^{2}+\varepsilon(t)\|\Delta g\|^{2}\right)+\left(2 a(l(u))-\varepsilon^{\prime}(t)\right)\|\Delta g\|^{2}\notag\\ &\quad =2(f(u),-\Delta g)+2\left(h^{\xi}(t),-\Delta g\right). \end{align}Besides, from (1.5) and the Young inequality, we have
\begin{equation} 2(f(u),-\Delta g) \leqslant 2 \lambda_{1}^{2}\|u\|^{2}+\frac{1}{2}\|A g\|^{2}, \end{equation}and
\begin{equation} 2(h^{ \xi}(t),-\Delta g) \leqslant 2\|h^{\xi}(t)\|^{2}+\frac{1}{2}\|A g\|^{2}, \end{equation}
where 
$A=-\Delta.$
In addition, taking (5.7), (5.8) into (5.6), and from (1.4), we have
\begin{equation} \frac{\textrm{d}}{\textrm{d}t} E_{2}(t)+\delta_{1} E_{2}(t) \leqslant 2 \lambda_{1}^{2}\|u\|^{2}+2\|h^{\xi}(t)\|^{2}, \end{equation}
where 
$\left.E_{2}(t)=\|\nabla g\|^{2}+\varepsilon (t)\right \|\Delta g\|^{2}$.
It follows from (4.1) that
\begin{equation} \frac{\textrm{d}}{\textrm{d} t} E_{2}(t)+\delta_{1} E_{2}(t) \leq 2 \lambda_{1}^{2} R_{1}+2\|h^{\xi}(t)\|^{2}, \end{equation}
where 
$R_{1}=\textrm {e}^{-\delta (t-\tau )}\left \|u_{\tau }\right \|_{\mathcal {H}_{t}}^{2}+\frac {C}{\delta }+\frac {\textrm {e}^{-\delta t}}{2(m-w \lambda _{1}^{-1})-\delta } \int _{\tau }^{t}\,\textrm {e}^{\delta s}\|h(s)\|_{-1}^{2}\,\textrm {d}s.$
By the Gronwall lemma, we conclude that
\begin{equation} \|g(t)\|_{\mathcal{H}_{t}^{1}}^{2}=\left\|U_{2}(t, \tau) u_{\tau}\right\|_{\mathcal{H}_{t}^{1}}^{2} \leqslant R_{2}, \end{equation}
where 
$R_{2}=\textrm {e}^{-\delta _{1} t} \int _{\tau }^{t}\,\textrm {e}^{\delta _{1} s}(2 \lambda _{1}^{2} R_{1}+2\left \|h^{\xi }(s)\right \|_{-1}^{2})\,\textrm {d}s.$
Thanks to (5.5) and (5.11), for any 
$t \in \mathbb {R}$, we can deduce that
\begin{equation} \operatorname{dist}\left(\mathcal{A}_{t}, {\bar B}_{\mathcal{H}_{t}^{1}}\left(R_{2}\right)\right)=\operatorname{dist}\left(U(t, \tau) \mathcal{A}_{\tau}, {\bar B}_{\mathcal{H}_{t}^{1}}\left(R_{2}\right)\right) \leq C\,\textrm{e}^{-\delta_{2}(t-\tau)} \rightarrow 0, \quad \tau \rightarrow-\infty, \end{equation}
where 
$\delta _{2} > 0$.
As a result, we have 
${{\mathcal {A}}_t} \subseteq {\bar B_{{\mathcal {H}}_t^{1}}}$, which represents the time-dependent pullback attractor 
$\{\widehat {\mathcal {A}}_{t}\}_{t \in \mathbb {R}}$ is bounded in 
$\mathcal {H}_{t}^{1}(\Omega ).$
Acknowledgments
This paper was partially supported by the NNSF of China with contract number 12171082 and by the fundamental funds for the central universities with contract number 2232021G-13.
The authors have no conflict of interest.
