2. Notations and preliminaries

In this section, we review some notations about function spaces and preliminary results.

As in [Reference Dafermos12], a new variable which reflects the history of (1.1) is introduced, that is

\begin{equation*}\eta^t(x,s)=\eta (x,t,s)= \int_0^su(x,t-r)dr, \ s \ge 0,\end{equation*}

then we can check that

\begin{equation*}\partial_t \eta^t(x,s)= u(x,t ) -\partial_s \eta^t(x,s) , \ s \ge 0.\end{equation*}

Since $\mu_t(s)=- k_t^{\prime}(s)$ , problem (1.1) can be transformed into the following system

(2.1) \begin{equation}\begin{aligned}\begin{cases}\partial_t u -\partial_t \Delta u - \Delta u - \int_0^\infty \mu_t(s) \Delta \eta^t(s)ds+ f(u) = g(x), &x\in \Omega, t > \tau,\\\partial_t \eta^t (x,s) = - \partial_s \eta^t (x,s)+ u(x,t), &x\in \Omega, t > \tau, s\geq 0,\\u(x,t) = 0, &x\in \partial \Omega, t > \tau,\\\eta^t(x,s)=0, &(x,s )\in \partial \Omega \times \mathbb R^+, t > \tau,\\u(x, \tau ) = u_{\tau }(x),& x\in \Omega,\\\eta^\tau (x,s)=\eta_\tau (x,s)\,{:\!=}\, \int_\tau^s g_0 (x,r)dr,& (x,s )\in \Omega \times \mathbb R^+.\end{cases} \end{aligned}\end{equation}

Now, let

\begin{equation*} z(t)= (u(t), \eta^t), \text{ and } z_\tau = (u_\tau , \eta_\tau ). \end{equation*}

Unless otherwise specified, it is understood that we consider spaces of functions which are defined on the domain $\Omega$ . Let $\left\langle{\cdot,\cdot}\right\rangle$ and $\|\cdot\|$ denote the $L^2(\Omega)$ -inner product and $L^2(\Omega)$ -norm, respectively.

Let $L^2_{\mu_t}(\mathbb{R}^+,L^2(\Omega))$ be the Hilbert space of functions $\varphi\colon\mathbb{R}^+\rightarrow L^2(\Omega)$ endowed with the inner product

\begin{equation*}\left\langle{\varphi_1,\varphi_2}\right\rangle_{\mu_t }=\int_{0}^{\infty}\mu_t(s)\left\langle{\varphi_1(s),\varphi_2(s)}\right\rangle ds,\end{equation*}

and let $\|\varphi\|_{\mu_t}$ be the corresponding norm. In a similar manner, we introduce the inner products $\left\langle{\cdot,\cdot}\right\rangle_{1,\mu_t}, \left\langle{\cdot,\cdot}\right\rangle_{2,\mu_t}$ on $L^2_{\mu_t}(\mathbb{R}^+, H_0^1(\Omega))$ and $L^2_{\mu_t}(\mathbb{R}^+, H^2(\Omega) \cap H_0^1(\Omega) )$ by

\begin{equation*}\left\langle{\cdot,\cdot}\right\rangle_{1,\mu_t}=\left\langle{\nabla\cdot,\nabla\cdot}\right\rangle_{\mu_t}\;\; ,\;\; \left\langle{\cdot,\cdot}\right\rangle_{2,\mu_t}=\left\langle{\Delta\cdot,\Delta\cdot}\right\rangle_{\mu_t},\end{equation*}

and the corresponding norms are denoted by $\|\cdot\|_{1,\mu_t}, \|\cdot\|_{2,\mu_t}$ .

We now introduce the following Hilbert spaces

\begin{equation*}\begin{split}\mathcal{V}_t&=H_0^1(\Omega)\times L^2_{\mu_t}(\mathbb{R}^+, H_0^1(\Omega)),\\[3pt] \mathcal{W}_t&=\left (H^2(\Omega) \cap H_0^1(\Omega)\right)\times L^2_{\mu_t}(\mathbb{R}^+, H^2(\Omega) \cap H_0^1(\Omega)), \end{split} \end{equation*}

which are, respectively, endowed with the inner products

\begin{align*}\left\langle{w_1,w_2}\right\rangle_{\mathcal V_t}&=\left\langle{\nabla \psi_1, \nabla \psi_2}\right\rangle +\left\langle{\varphi_1,\varphi_2}\right\rangle_{1,\mu_t},\\[3pt] \left\langle{w_1,w_2}\right\rangle_{\mathcal W_t}&=\left\langle{\Delta \psi_1, \Delta \psi_2}\right\rangle +\left\langle{\varphi_1,\varphi_2}\right\rangle_{2,\mu_t},\end{align*}

where $w_i=(\psi_i, \varphi_i)\in \mathcal V_t, \mathcal W_t, \,\, i=1,2$ .

The norms induced on $\mathcal V_t, \mathcal W_t$ , are

\begin{align*}\|(\psi,\varphi)\|_{\mathcal V_t}^2&=\|\nabla \psi\|^2 +\int_{0}^{\infty}\mu_t(s)\|\nabla\varphi(s)\|^2ds,\\\|(\psi,\varphi)\|_{\mathcal W_t}^2&= \|\Delta\psi\|^2 +\int_{0}^{\infty}\mu_t(s)\|\Delta\varphi(s)\|^2ds.\end{align*}

The following results will be used to prove the existence of time-dependent global attractors.

For $t \in R$ , let $X_t$ be a family of normed spaces, the two-parameter family of operators

\begin{equation*}U(t,\tau)\,{:}\, X_\tau \to X_t, \quad t\ge \tau,\end{equation*}

is called a process on time-dependent spaces (see [Reference Conti, Pata and Temam10, Reference Di Plinio, Duane and Temam13]), characterized by the two properties:

1. (i) $U(\tau,\tau)$ is the identity map on $X_\tau$ for every $\tau$ ;

2. (ii) $U(t,\tau)U(\tau,s)=U(t,s)$ for every $t\ge \tau\ge s$ .

As introduced in [Reference Conti, Pata and Temam10], we consider the following definitions and theorem.

Definition 2.1. A time-dependent absorbing set for the process $U(t, \tau)$ is a uniformly bounded family $\mathcal{B} = \{B_t\}_{t\in \mathbb{R}}$ with the following property: for every $R \ge 0$ there exists $\theta_e=\theta_e(R) \ge 0$ such that

\begin{equation*}\tau \le t-\theta_e \to U(t,\tau)\mathbb{B}_\tau(R) \subset B_t.\end{equation*}

Definition 2.2. Let $\mathbb K = \{ \mathcal K=\{K_t\}_{t \in \mathbb R}\,{:}\, K_t \subset X_t \text{ compact, }\mathcal K \text{ pullback attracting} \}$ . The family $\mathcal A= \{A_t\}_{t\in \mathbb R} \in \mathbb K$ is said to be a time-dependent global attractors if $ \mathcal A$ is the smallest element of $\mathbb K$ such that

\begin{equation*} A_t \subset K_t, \quad \forall t \in \mathbb R, \end{equation*}

for any element $\mathcal K=\{K_t\}_{t \in \mathbb R} \in \mathbb K$ .

We know that the minimal element of $\mathbb K$ exists (and it is unique) if and only if $\mathbb K$ is not empty.

Theorem 2.1. If $U(t,\tau)$ is asymptotic compact, that is,

\begin{equation*}\mathbb K = \{ \mathcal K=\{K_t\}_{t \in \mathbb R}\,{:}\, K_t \subset X_t {\ compact,\ }\mathcal K {\ pullback\ attracting} \}, \mathbb K \ne \varnothing \end{equation*}

then $\{ U(t,\tau)\}$ has a unique time-dependent global attractors $\mathcal A= \{A_t\}_{t\in \mathbb R}$ and

\begin{equation*} A_t = \bigcap_{s \le t} \overline{ \bigcup_{\tau \le s}U(t,\tau)B_{\tau}}.\end{equation*}

Moreover, if $U(t,\tau)$ is a continuous (or norm-to-weak continuous) map for all $t \ge \tau$ , then $\mathcal A$ is invariant.

3. Existence and uniqueness of weak solutions

Definition 3.1. A function $z=(u, \eta^t)$ is called a weak solution of problem (2.1) on the interval $(\tau, T)$ with the initial datum $z(\tau)=z_\tau \in \mathcal {V_\tau}$ if

\begin{equation*}\begin{aligned}&u\in C([\tau ,T];\ H^{1}_0 (\Omega) ), f(u) \in L^1(Q_T),\\[3pt] & \partial_tu \in L^2(\tau ,T;H_0^1(\Omega) ),\eta^t \in C([\tau ,T];L^2_{\mu_t} (\mathbb R^+, H^{1}_0(\Omega) )),\end{aligned}\end{equation*}

and

\begin{equation*}\begin{aligned}&\langle\partial_t u ,\varphi\rangle +\langle\partial_t \nabla u ,\nabla \varphi \rangle +\langle \nabla u ,\nabla \varphi \rangle +\langle \eta^t, \varphi \rangle_{1,\mu_t } +\langle f(u),\varphi \rangle_{L^1, L^\infty} =\langle g,\varphi \rangle_{H^{-1}, H^1_0}, \\[3pt] &\langle \partial_t \eta^t + \partial_s \eta^t ,\xi \rangle_{1,\mu_t } = \langle u ,\xi \rangle_{1,\mu_t } ,\end{aligned}\end{equation*}

for all test functions $\varphi \in W = H_0^1(\Omega)\cap L^\infty (\Omega), \xi \in L^2_{\mu_t}(\mathbb R^+, H_0^1(\Omega))$ , and for a.e. $t\in [\tau , T]$ .

We are now ready to state the existence and uniqueness result for problem (2.1).

Theorem 3.1. Assume that hypotheses (H1)–(H3) hold. Then for any $ z_\tau =(u_\tau ,\eta_\tau )\in \mathcal{V}_\tau$ and $T>\tau$ given, problem (2.1) has a unique weak solution $z=(u,\eta^t)$ on the interval $(\tau ,T)$ satisfying

\begin{equation*}u \in C([\tau,T]; H_0^1(\Omega)), \quad \eta^t \in L^2_{\mu_t} (\mathbb R^+, H_0^1(\Omega)).\end{equation*}

Moreover, the weak solutions depend continuously on the initial data.

Proof. We use the Faedo–Galerkin method. As argued in [Reference Anh, Thanh and Toan5], because of the separability of $H_0^1(\Omega)$ , one can choose a sequence $\{\omega_j\}_{j=1}^{\infty}$ which forms a smooth orthonormal basis in both $L^2(\Omega)$ and $H_0^1(\Omega)$ spaces. For instance, one can take a complete set of normalized eigenfunctions for $-\Delta$ in $H_0^1(\Omega)$ , such that $-\Delta\omega_j=\nu_j\omega_j$ , where $\nu_j$ is the eigenvalue corresponding to $\omega_j$ . Next, we want to choose an orthonormal basis $\{\zeta_j\}_{j=1}^{\infty}$ of $L^2_{\mu_t}(\mathbb{R}^+, H_0^1(\Omega))$ which also belong to $\mathcal{D}(\mathbb{R}^+, H_0^1(\Omega))$ , where $\mathcal{D}(I, X)$ is the space of infinitely differentiable X-valued functions with compact support in $I\subset\mathbb{R}$ . For this purpose, we select vectors of the form $l_k\omega_j$ ( $k,j=1,\ldots,\infty$ ), where $\{l_j\}_{j=1}^{\infty}$ is an orthonormal basis in both $L_{\mu_t}^2(\mathbb{R}^+)$ and $\mathcal{D}(\mathbb{R}^+)$ spaces.

1. (i) Existence. Given an integer n, denote by $P_n$ and $Q_n$ the projections on the subspaces

   \begin{equation*}\mathrm{span}(\omega_1,\ldots,\omega_n)\in H_0^1(\Omega)\quad\text{ and }\quad\mathrm{span}(\zeta_1,\ldots,\zeta_n)\in L_{\mu_t}^2(\mathbb{R}^+,H_0^1(\Omega)),\end{equation*}
   respectively. We look for a function $z_n=(u_n,\eta^t_n)$ of the form
   \begin{equation*}u_n(t)=\sum_{j=1}^{n}a_j(t)\omega_j\quad\text{ and }\quad\eta^t_{n} (s)=\sum_{j=1}^{n}b_j(t)\zeta_j(s)\end{equation*}
   satisfying
   (3.1) \begin{equation}\begin{split}&\left\langle{(\partial_t u_n - \Delta \partial_t u_n , \partial_t \eta^t_n ),(\omega_k,\zeta_j)}\right\rangle_{\mathcal V_t}\\ &\qquad = \left\langle{( \Delta u_n + \int_0^\infty \mu_t(s) \Delta \eta^t_n(s)ds +g - f(u_n) , u_n - \partial_s \eta^t_n),(\omega_k,\zeta_j)}\right\rangle_{\mathcal V_t}\\&u_n(\tau)= P_n u_\tau \to u_\tau = \sum_{j=1}^{\infty}\alpha_j \omega_j \quad \text{ in } H_0^1(\Omega), \text{ as } n \to \infty,\\&\eta^t_n(\tau) = Q_n\eta_\tau \to \eta_\tau =\sum_{j=1}^{\infty}\beta_j\zeta_j(s) \quad\text{ in } L_{\mu_t}^2(\mathbb{R}^+,H_0^1(\Omega)), \text{ as } n \to \infty,\end{split}\end{equation}
   for a.e. $\tau \leq t\leq T$ , for every $k,j=0,\ldots,n$ , where $\omega_0$ and $\zeta_0$ are the zero vectors in the respective spaces. Taking $(\omega_k,\zeta_0)$ and $(\omega_0,\zeta_k)$ in (3.1), and applying the divergence theorem to the term
   \begin{equation*}\left\langle{\int_{0}^{\infty}\mu_t(s)\Delta\eta^t_n(s)ds,\omega_k}\right\rangle,\end{equation*}
   we get a system of Ordinary Differential Equation (ODE) in the variables $a_k(t)$ and $b_k(t)$ of the form
   (3.2) \begin{equation}\begin{split}&\frac{d}{dt}\bigg( (1+\nu_k) a_k \bigg) =-\nu_k a_k - \sum_{j=1}^{n}b_j\left\langle{\zeta_j,\omega_k}\right\rangle_{1,\mu_t}+\left\langle{g,\omega_k}\right\rangle - \left\langle{f(u_n),\omega_k}\right\rangle,\\[3pt] &\frac{d}{dt}b_k=\sum_{j=1}^{n}a_j\left\langle{\omega_j,\zeta_k}\right\rangle_{1,\mu_t}- \sum_{j=1}^{n}b_j\left\langle{\zeta^{\prime}_j,\zeta_k}\right\rangle_{1,\mu_t},\end{split}\end{equation}
   subject to the initial conditions
   (3.3) \begin{equation}\begin{split}&a_k(\tau )=\left\langle{u_\tau ,\omega_k}\right\rangle_{H_0^1(\Omega)},\\[3pt] &b_k(\tau )=\left\langle{\eta_\tau ,\zeta_k}\right\rangle_{1,\mu_\tau}.\end{split}\end{equation}
   According to standard existence theory for ODEs, there exists a continuous solution $(a_k, b_k)$ of (3.2)–(3.3) on some interval $(\tau ,T_n)$ for each n. The a priori estimates below imply that in fact $T_n=+\infty$ .

   Multiplying the first equation of (3.2) by $a_k$ and the second by $b_k$ , then summing over k and adding the results, we get

   (3.4) \begin{equation}\begin{aligned}\frac{1}{2}\frac{d}{dt}\|z_n\|^2_{\mathcal{V}_t}= - \|\nabla u_n\|^2-&\left\langle{\partial_s\eta^t_n,\eta^t_n}\right\rangle_{1,\mu_t} + \left\langle{g,u_n}\right\rangle_{H^{-1}, H^1_0}\\&-\left\langle f(u_n),u_n\right\rangle + \int_{0}^{\infty}\partial_t \mu_t(s)\|\nabla\eta^t_n(s)\|^2ds. \end{aligned}\end{equation}
   Using (1.3) and the Cauchy inequality, we have
   (3.5) \begin{equation} \left\langle{g,u_n}\right\rangle_{H^{-1}, H^1_0}-\left\langle f(u_n),u_n\right\rangle \leq \varepsilon \| \nabla u_n\|^2+\frac{1}{4\varepsilon}\|g\|^2_{H^{-1}(\Omega)}+ \beta \|u_n\|^{2}+C_0|\Omega|,\end{equation}
   where $\varepsilon >0$ will be chosen later. From (3.4) and (3.5) we have
   \begin{equation*}\begin{aligned}&\frac{d}{dt}\|z_n\|_{\mathcal{V}_t}^2+2\left\langle{\partial_s\eta^t_n,\eta^t_n}\right\rangle_{1,\mu_t} - 2\int_{0}^{\infty}\partial_t \mu_t(s)\|\nabla\eta^t_n(s)\|^2ds+ 2\!\left(1- \frac{\beta }{\lambda_1}-\varepsilon \right)\|\nabla u_n\|^{2}\\\leq & \, \, \frac{1}{2\varepsilon}\|g\|^2_{H^{-1}(\Omega)} + 2C_0|\Omega| .\end{aligned}\end{equation*}
   Integrating by parts and using (M2), we get
   (3.6) \begin{equation}\begin{aligned}&2\left\langle{\partial_s\eta^t_n,\eta^t_n}\right\rangle_{1,\mu_t} - 2\int_{0}^{\infty}\partial_t \mu_t(s)\|\nabla\eta^t_n(s)\|^2ds \\=& \,\,-2\int_{0}^{\infty}(\partial_s \mu_t(s) +\partial_t\mu_t(s) ) \|\nabla\eta^t_n(s)\|^2ds\geq 0.\end{aligned}\end{equation}
   Thus,
   \begin{equation*}\frac{d}{dt}\|z_n\|_{\mathcal{V}_t}^2 + 2(1- \frac{\beta }{\lambda_1}-\varepsilon )\|\nabla u_n\|^{2}\leq C( \|g\|^2_{H^{-1}(\Omega)} +1).\end{equation*}
   Choosing $\varepsilon >0$ small enough so that $1- \dfrac{\beta }{\lambda_1}-\varepsilon >0$ and integrating on $(\tau ,t)$ , $t\in(\tau ,T)$ , lead to the following estimate
   \begin{equation*}\|z_n(t)\|_{\mathcal{V}_t}^2+2\!\left(1- \frac{\beta }{\lambda_1}-\varepsilon \right)\int_{\tau }^{t}\|\nabla u_n(r)\|^{2} dr \leq \|z_n(\tau) \|_{\mathcal{V}_\tau}^2+ CT( \|g\|^2_{H^{-1}(\Omega)} +1).\end{equation*}
   Hence, in particular, we see that
   (3.7) \begin{equation}\begin{aligned}&\{u_n\} \quad\text{ is bounded in }\quad L^{\infty}( \tau ,T;\ H_0^1(\Omega)),\\[3pt]&\{\eta^t_n\} \quad\text{ is bounded in }\quad L^{\infty}( \tau ,T;\ L^2_{\mu_t}(\mathbb{R}^+,H_0^1(\Omega))).\end{aligned}\end{equation}
   Therefore, by the Banach–Alaoglu theorem, there exists a function $z=(u,\eta^t)$ such that
   (3.8) \begin{align}&u_n\rightharpoonup u\quad\text{ weakly star in }\quad L^{\infty}(\tau ,T;\ H_0^1(\Omega)),\nonumber\\[3pt]&\eta^t_n\rightharpoonup \eta^t\quad\text{ weakly star in }\quad L^{\infty}(\tau ,T;\ L_{\mu_t}^{2}(\mathbb{R}^+,H_0^1(\Omega))),\end{align}
   and
   (3.9) \begin{align}&\Delta u_n\rightharpoonup \Delta u\quad\text{ weakly in }\quad L^{2}( \tau ,T;\ H^{-1}(\Omega)),\nonumber\\[3pt]& \Delta \eta^t_n\rightharpoonup \Delta \eta^t\quad\text{ weakly in }\quad L^{2}( \tau ,T;\ L_{\mu_t}^{2}(\mathbb{R}^+,H^{-1}(\Omega))),\end{align}
   up to a subsequence. Now, we estimate $\partial_t z_n$ . From (3.4) and (3.6), we get
   (3.10) \begin{equation}\frac{d}{dt}\|z_n\|_{\mathcal{V}_t}^2 + \|\nabla u_n\|^{2}+ 2\int_\Omega f(u_n)u_ndx \leq \|g\|^2_{H^{-1}(\Omega)}.\end{equation}
   Integrating (3.10) from $\tau $ to T, we obtain
   \begin{equation*} \|z_n(T) \|_{\mathcal{V}_T}^2 + \int_\tau ^T \|\nabla u_n(t)\|^{2}dt+ 2\int_{Q_T} f(u_n)u_ndxdt \leq \|z_\tau \|_{\mathcal{V}_n(\tau)}^2 + T\|g\|^2_{H^{-1}(\Omega)}.\end{equation*}
   In particular,
   (3.11) \begin{equation}\int_{Q_T}f(u_n)u_ndxdt \leq C.\end{equation}
   Multiplying the first equation of (3.2) by $a_k+ \varepsilon a^{\prime}_k$ and the second by $b_k$ , then summing over k and adding the results, we get
   (3.12) \begin{equation}\begin{aligned}\frac{1}{2}\frac{d}{dt}&\left( \|u_n\|^2+(1+\varepsilon)\|\nabla u_n\|^2 + \int_0^\infty \mu_t(s)\|\eta_n^t\|^2ds+ 2\varepsilon \langle F(u_n),1\rangle \right) + \|\nabla u_n\|^2 \\[3pt] &+\varepsilon \left(\|\partial_t u_n\|^2+\|\nabla \partial_t u_n\|^2\right) +\langle f(u_n),u_n \rangle - \int_{0}^{\infty}(\partial_s \mu_t(s) +\partial_t\mu_t(s) ) \|\nabla\eta^t_n(s)\|^2ds\\[3pt] &= - \int_{0}^{\infty} \mu_t(s)\langle \nabla\eta^t_n(s), \nabla \partial_tu_n \rangle ds+ \left\langle{g,u_n + \varepsilon \partial_t u_n}\right\rangle_{H^{-1}, H^1_0}. \end{aligned}\end{equation}
   Using Young inequality and $\kappa(t)= \int_{0}^{\infty} \mu_t(s)ds$ , we have
   \begin{equation*} \begin{aligned} - \int_{0}^{\infty} \mu_t(s)\langle \nabla\eta^t_n(s), \nabla \partial_tu_n \rangle ds& \leq \int_{0}^{\infty} \mu_t(s) \| \nabla\eta^t_n(s)\| \| \nabla \partial_tu_n \| ds\\[3pt]& \leq \sqrt{\varepsilon} \delta \kappa(t) \int_{0}^{\infty} \mu_t(s) \| \nabla\eta^t_n(s)\|^2 ds + \frac{\varepsilon \sqrt{\varepsilon} }{\delta} \| \nabla \partial_tu_n \|^2,\end{aligned}\end{equation*}
   and
   (3.13) \begin{equation} \left\langle{g,u_n + \varepsilon \partial_t u_n}\right\rangle_{H^{-1}, H^1_0} \leq \frac{1}{2} \| \nabla u_n \|^2+ \frac{\varepsilon}{2} \|\partial_t \nabla u_n \|^2+ C(\varepsilon) \|g\|^2_{H^{-1}(\Omega)}.\end{equation}
   Combining (3.12)–(3.13), and owing to (M2) and (1.4), we get
   (3.14) \begin{equation}\begin{aligned} \frac{d}{dt}&\left( \|u_n\|^2+(1+\varepsilon)\|\nabla u_n\|^2 + \int_0^\infty \mu_t(s)\|\eta_n^t\|^2ds+ 2\varepsilon \langle F(u_n),1\rangle \right) + \|\nabla u_n\|^2 \\[3pt]&+2\left(\varepsilon - \frac{\varepsilon \sqrt{\varepsilon} }{\delta} \right) \left( \|\partial_t u_n\|^2+\|\nabla \partial_t u_n\|^2 \right) +2\langle f(u_n),u_n \rangle\\[3pt]&+ 2(1-\sqrt{\varepsilon}) \delta \kappa(t) \int_{0}^{\infty} \mu_t(s) \|\nabla\eta^t_n(s)\|^2ds \leq C(\varepsilon) \|g\|^2_{H^{-1}(\Omega)}. \end{aligned}\end{equation}
   Integrating (3.14) from $\tau $ to t and using (1.4), (3.11) and (3.7), we can deduce that
   \begin{align*}\{\partial_t u_n\} \quad\text{ is bounded in }\quad L^{2}( \tau ,T;\ H_0^1(\Omega)) ,\end{align*}
   so, up to a subsequence,
   (3.15) \begin{align} \begin{split}&\partial_t u_n \rightharpoonup \partial_tu \quad \text{ weakly in }\quad L^{2}( \tau ,T;\ H_0^1(\Omega)),\\[3pt]& \partial_t \Delta u_n \rightharpoonup \partial_t\Delta u \quad \text{ weakly in }\quad L^{2}( \tau ,T;\ H^{-1}(\Omega)).\end{split}\end{align}
   We now prove that $\{f(u_n)\}$ is bounded in $L^1(Q_T)$ where $Q_T=\Omega \times (\tau, T)$ . Putting $h(s) = f(s)-f(0) + \gamma s$ , where $\gamma > \ell$ . Note that $h(s)s=(f(s)-f(0))s+\gamma s^2 =f^{\prime}(c) s^2+\gamma s^2\geq (\gamma -\ell)s^2 \ge 0$ for all $s\in \mathbb R$ , we have
   \begin{align*}\int\limits_{Q_T} \left| h(u_n) \right|dxdt &\le \int\limits_{Q_T \cap \{ \left| u_n \right| > 1\} } \left| h(u_n)u_n \right|dxdt + \int\limits_{Q_T \cap \{ \left| u_n\right| \le 1\}} \left| h(u_n)\right|dxdt\\[3pt] &\le \int\limits_{Q_T} h({u_n}){u_n}dxdt + \mathop {\sup }\limits_{\left| s \right| \le 1} \left| h(s) \right|\left| Q_T\right|\\[3pt]& \le \int\limits_{Q_T} f({u_n}){u_n}dxdt + |f(0)|\|u_n\|_{L^1(Q_T)}+\gamma \|u_n\|^2_{L^2(Q_T)}\\[3pt]&\;\;\;+ \mathop {\sup }\limits_{\left| s \right| \le 1} \left| h(s) \right|\left| Q_T\right|\\[3pt]&\le C, \end{align*}
   where we have used (3.7), (3.11) and the boundedness of $\{u_n\}$ in $L^\infty (\tau,T;\ H^1_0(\Omega ))$ . Hence it implies that $\{h(u_n)\}$ , and therefore $\{f(u_n)\}$ is bounded in $L^1(Q_T)$ .

   Using the Aubin–Lions lemma in [Reference Lions18], we can suppose that $u_n \to u$ strongly in $L^2(\tau ,T;\ L^2(\Omega))$ . Hence $u_n \to u$ a.e. in $Q_T$ , up to a subsequence. Besides, using the definition of h(s) and (3.10), (3.7), we have

   \begin{equation*}\int_{Q_T}h(u_n)u_ndxdt \leq C.\end{equation*}
   Therefore, by Lemma 6.1 in [Reference Geredeli14], we obtain that $h(u) \in L^{1}(Q_T ) $ and for all test functions $\varphi \in C_0^\infty ([\tau,T];\ H_0^1(\Omega) \cap L^\infty (\Omega)), $
   \begin{equation*}\int_{Q_T} h(u_n) \varphi dxdt \to \int_{Q_T} h(u) \varphi dxdt.\end{equation*}
   Hence, $f(u)\in L^1(Q_T)$ and
   (3.16) \begin{equation}\int_{Q_T} f(u_n) \varphi dxdt \to \int_{Q_T} f(u) \varphi dxdt, \text{ for all }\varphi \in C_0^\infty ([\tau,T];\ H_0^1(\Omega) \cap L^\infty (\Omega)).\end{equation}

   We are now ready to show that the limit $z=(u, \eta^t)$ is a weak solution of (2.1). Choose an arbitrary test function

   \begin{equation*} \phi=(\varphi, \xi) \in \mathcal D ( [\tau ,T], H_0^1(\Omega)\cap L^\infty(\Omega)) \times \mathcal D ( [\tau ,T], \mathcal D(\mathbb R^+, H_0^1(\Omega)) )\end{equation*}
   of the form
   \begin{equation*} \varphi(t)= \sum\limits_{j=1}^{m} a_j(t)\omega_j \quad \text{ and }\quad \xi(t)=\sum\limits_{j=1}^{m} b_j(t)\zeta_j,\end{equation*}
   where m is a fixed integer, $\{a_j\}_{j=1}^m$ and $\{b_j\}_{j=1}^m$ are given functions in $\mathcal D( (\tau ,T))$ . Then (3.1) holds with $(v(t), \xi(t))$ in place of $(\omega_k, \zeta_j)$ . Integrating the resulting equation over $(\tau , T)$ and passing to the limits, in view of (3.8), (3.9), (3.15) and (3.16), we get
   \begin{equation*}\begin{aligned}\int_\tau ^T \bigg[\langle \partial_tu , \varphi\rangle +\langle\partial_t \nabla u ,\nabla \varphi\rangle +&\langle \partial_t\eta^t , \xi \rangle_{1,\mu_t}\bigg] dt \\[3pt]=& -\int_\tau ^T \bigg[ \langle \nabla u, \nabla \varphi \rangle + \langle \eta^t, \varphi \rangle_{1,\mu_t}\bigg]dt \\[3pt]&-\int_\tau ^T\bigg[\int_\Omega f(u )\varphi dx- \langle g, \varphi \rangle_{H^{-1}, H^1_0}\bigg] dt \\[3pt]&+\int_\tau ^T\bigg[ - \langle \partial_s\eta^t, \xi \rangle_{1,\mu_t} + \langle u, \xi \rangle_{1,\mu_t}\bigg]dt.\end{aligned}\end{equation*}
   Using a density argument, we conclude that $z=(u,\eta^t)$ satisfies the equation in the weak sense. By standard arguments, we can check that z satisfies the initial condition $z(\tau ) =z_{\tau}$ . This implies that $z(\!\cdot\!)$ is a weak solution of problem (2.1).

2. (ii) Uniqueness and continuous dependence on the initial data. We assume that $z_1=(u_1, \eta^t_1)$ and $z_2=(u_2, \eta^t_2)$ are two solutions of (2.1) with initial data $z_{1\tau }$ and $z_{2\tau }$ , respectively. Denote $w = z_1 -z_2 = (u_3, \eta^t_3)$ , then

   (3.17) \begin{equation} \partial_t u_3 - \partial_t \Delta u_3 -\Delta u_3 - \int_0^{\infty} \mu_t(s) \Delta \eta^t_3(x,s)ds + (\hat{f}(u_1(t) -\hat{f}(u_2)) - \ell u_3 =0, \text{ for all } t >0,\end{equation}
   where $\hat{f}(s) = f(s) +\ell s$ . Here, because $u_3(t)$ does not belong to $W=H_0^1(\Omega)\cap L^\infty(\Omega)$ , we cannot choose $u_3(t)$ as a test function. Consequently, the proof will be more involved than that in [Reference Conti, Marchini and Pata9, Reference Wang, Yang and Zhong29, Reference Wang and Zhong30].

   We use some ideas in [Reference Geredeli and Khanmamedov15]. Let

   \begin{equation*}B_k(s) =\begin{cases}k & \text{ if } s>k,\\[3pt]s & \text{ if } |s| \leq k,\\[3pt]-k & \text{ if } s< -k.\end{cases}\end{equation*}
   Consider the corresponding Nemytskii mapping $\hat{B}_k\,{:}\, W \to W$ defined as follows
   \begin{equation*}\hat{B}_k(u_3)(x)=B_k(u_3(x)), \text{ for all } x \in \Omega.\end{equation*}
   By Theorem 4.7 in [Reference Krasnoselskii, Zabreiko, Pustylnik and Sobolevskii17] (see also Lemma 2.3 in [Reference Geredeli and Khanmamedov15]), we have that $\|\hat{B}_k(u_3) - u_3\|_W\to 0$ as $k \to \infty$ . Now multiplying (3.17) by $\hat{B}_k(u_3)$ , then integrating over $\Omega $ we get
   \begin{equation*} \begin{aligned} \frac{d}{dt}& \bigg( \int_\Omega u_3 \hat{B}_k(u_3)dx + \int_\Omega \nabla u_3 \nabla \hat{B}_k(u_3)dx - \frac{1}{2}\bigl(\|\hat{B}_k(u_3)\|^2 + \|\nabla\hat{B}_k(u_3)\|^2\bigr) \bigg)\\[3pt]& + \int_\Omega \nabla u_3 \nabla\hat{B}_k(u_3)dx + \int_0^\infty \mu_t(s) \int_\Omega \nabla \eta^t_3 \nabla \hat{B}_k(u_3)dxds \\[3pt]&+ \int_\Omega (\hat{f}(u_1) - \hat{f}(u_2)) \hat{B}_k(u_3)dx - \ell \int_\Omega u_3 \hat{B}_k(u_3) dx=0.\end{aligned}\end{equation*}
   Thus,
   (3.18) \begin{equation} \begin{aligned}\frac{d}{dt}& \bigg( \int_\Omega u_3 \hat{B}_k(u_3)dx + \int_\Omega \nabla u_3 \nabla \hat{B}_k(u_3)dx - \frac{1}{2}\bigl(\|\hat{B}_k(u_3)\|^2 + \|\nabla\hat{B}_k(u_3)\|^2\bigr) \bigg)\\[3pt]& + \int_{ \{x \in \Omega\,{:}\, |u_3(x,t)| \leq k \}} | \nabla u_3 |^2 dx + \int_0^\infty \mu_t(s) \int_{ \{x \in \Omega\,{:}\, |u_3(x,t)| \leq k \}}\nabla \eta^t_3 \nabla u_3 dxds \\[3pt] &+ \int_\Omega \hat{f}^{\prime} (\xi) u_3\hat{B}_k(u_3) dx = \ell \int_\Omega u_3 \hat{B}_k(u_3) dx.\end{aligned} \end{equation}
   Note that $\hat{f}^{\prime}(s) \geq 0$ and $sB_k(s) \geq 0$ for all $s \in \mathbb{R}$ , we have
   \begin{equation*}\int_\Omega \hat{f}^{\prime} (\xi) u_3\hat{B}_k(u_3) dx \geq 0.\end{equation*}
   Moreover,
   \begin{equation*}\int_{ \{x \in \Omega\,{:}\, |u_3(x,t)| \leq k \}} | \nabla u_3 |^2 dx \geq 0,\end{equation*}
   and
   \begin{align*} \int_0^\infty \mu_t(s) &\int_{ \{x \in \Omega\,{:}\, |u_3(x,t)| \leq k \}} \nabla \eta^t_3 \nabla u_3 dxds \\[3pt]= & \int_0^\infty \mu_t(s) \int_{ \{x \in \Omega\,{:}\, |u_3(x,t)| \leq k \}}\nabla \eta^t_3 \partial_t \nabla \eta^t_3 dxds \\[3pt]&\qquad + \int_0^\infty \mu_t(s) \int_{ \{x \in \Omega\,{:}\, |u_3(x,t)| \leq k \}}\nabla \eta^t_3 \partial_s\nabla \eta^t_3 dxds \\[3pt] = & \int_0^\infty \mu_t(s) \int_{ \{x \in \Omega\,{:}\, |u_3(x,t)| \leq k \}} \nabla \eta^t_3 \partial_t \nabla \eta^t_3 dxds\\[3pt]&\qquad - \frac{1}{2}\int_0^\infty \partial_s \mu_t(s) \int_{ \{x \in \Omega\,{:}\, |u_3(x,t)| \leq k \}}| \nabla \eta^t_3 |^2 dxds.\end{align*}
   From the above inequalities we deduce from (3.18) that
   \begin{equation*} \begin{aligned}\frac{d}{dt}& \bigg( \int_\Omega u_3 \hat{B}_k(u_3)dx + \int_\Omega \nabla u_3 \nabla \hat{B}_k(u_3)dx - \frac{1}{2}\bigl(\|\hat{B}_k(u_3)\|^2 + \|\nabla\hat{B}_k(u_3)\|^2\bigr) \bigg)\\[3pt]& + \frac{1}{2} \int_0^\infty \mu_t(s) \int_{ \{x \in \Omega\,{:}\, |u_3(x,t)| \leq k \}} \frac{d}{dt} | \nabla \eta^t_3|^2 dxds \\[3pt] & \leq \ell \int_\Omega u_3 \hat{B}_k(u_3) dx + \frac{1}{2}\int_0^\infty \partial_s \mu_t(s) \int_{ \{x \in \Omega\,{:}\, |u_3(x,t)| \leq k \}}| \nabla \eta^t_3 |^2 dxds.\end{aligned}\end{equation*}
   Integrating from $\tau$ to t, where $t \in (\tau,T)$ , then letting $ k \to \infty$ , we obtain
   \begin{align*}&\| u_3(t) \|^2 + \| \nabla u_3(t) \|^2 + \| \eta_3^t \|^2_{1, \mu_t } \\[3pt] \leq & \quad \| u_3(\tau) \|^2 + \| \nabla u_3(\tau) \|^2+ \| \eta_3^\tau \|^2_{1, \mu_\tau} + 2 \ell \int_\tau^t \| u_3 (s) \|^2 ds \\[3pt]& \; + \int_\tau^t \biggl( \int_0^\infty ( \partial_s \mu_r(s) + \partial_r \mu_r(s))\| \nabla \eta^t_3 \|^2 ds \biggr)dr \\[3pt] \leq & \quad \| u_3(\tau) \|^2 + \| \nabla u_3(\tau) \|^2 + \| \eta_3^\tau \|^2_{1, \mu_\tau} + 2 \ell \int_\tau^t ( \| u_3 (s) \|^2 + \| \nabla u_3 (s) \|^2 + \| \eta_3^s \|^2_{1, \mu_s}) ds.\end{align*}
   By the Gronwall inequality of integral form, we get
   \begin{equation*}\| w(t) \|^2_{\mathcal V_t } \leq \| w(\tau) \|^2_{\mathcal V _\tau } e^{2 \ell (t - \tau)} \leq \| w(\tau) \|^2_{\mathcal V_\tau } e^{2 \ell (T-\tau)}, \text { for all } t \in [\tau,T].\end{equation*}
   Hence we get the continuous dependence on the initial data of the solutions, and in particular, the uniqueness when $ w(\tau) = 0$ .

4. Existence of a time-dependent global attractor

Theorem 3.1 allows us to define a process on time-dependent spaces $U(t, \tau): \mathcal V_\tau \to \mathcal V_t $ associated to problem (2.1) by the formula

\begin{equation*} U(t, \tau) z_\tau\,{:\!=}\, z(t), \end{equation*}

where $z(\!\cdot\!) $ is the unique global weak solution of (2.1) with the initial datum $z_\tau \in \mathcal V_\tau$ .

4.1. Existence of a time-dependent absorbing set

Lemma 4.1. Under assumptions (H1)–(H3), there exists a time-dependent absorbing set in $\mathcal V_t $ for the process $U(t, \tau)$ .

Proof. Multiplying the first equation of (2.1) by u(t) and integrating over $\Omega$ , we obtain

(4.1) \begin{equation}\begin{aligned}\frac{1}{2}\frac{d}{dt} \left(\|u\|^2+ \|\nabla u\|^2 \right) +\|\nabla u\|^2 +\int_\Omega f(u)u dx+ \int_0^\infty &\mu_t(s) \langle \nabla \eta^t(s), \nabla u \rangle ds \\& = \langle g, u \rangle_{H^{-1}(\Omega), H^1_0(\Omega)}. \end{aligned}\end{equation}

Using the hypothesis (1.3) and the Cauchy inequality, we have

(4.2) \begin{equation} \begin{aligned} \int_\Omega f(u)u dx &\geq -\beta\|u\|^2-C_0|\Omega|,\\ \langle g, u \rangle_{H^{-1}(\Omega), H^1_0(\Omega)}& \leq \varepsilon_0 \| \nabla u \|^2 + \dfrac{1}{4\varepsilon_0} \|g\|^2_{H^{-1}(\Omega)}. \end{aligned} \end{equation}

Recalling that $ u(t) =\partial_t\eta^t(s) + \partial_s \eta^t(s)$ , we have

(4.3) \begin{equation} \begin{aligned} &\int_0^\infty \mu_t(s) \langle \nabla \eta^t(s), \nabla u \rangle ds \\[2pt]=& \int_0^\infty \mu_t(s) \langle \nabla \eta^t(s), \nabla \partial_s \eta^t(s) \rangle ds +\int_0^\infty \mu_t(s) \langle \nabla \eta^t(s), \nabla \partial_t\eta^t(s) \rangle ds \\[2pt]=& -\dfrac{1}{2}\int_0^\infty \partial_s \mu_t(s) \|\eta^t(s)\|^2ds +\dfrac{1}{2}\dfrac{d}{dt} \|\eta^t(s)\|^2-\dfrac{1}{2}\int_0^\infty \partial_t \mu_t(s) \|\eta^t(s)\|^2ds \\[2pt] =& -\dfrac{1}{2}\int_0^\infty \left( \partial_s \mu_t(s)+\partial_t \mu_t(s)\right) \|\eta^t(s)\|^2ds +\dfrac{1}{2}\dfrac{d}{dt} \|\eta^t \|^2_{1,\mu_t}. \end{aligned} \end{equation}

From (4.3) and (M2), we get

(4.4) \begin{equation} \begin{aligned} \int_0^\infty \mu_t(s) \langle \nabla \eta^t(s), \nabla u \rangle ds \geq \dfrac{1}{2} \delta \kappa(t) \|\eta^t \|^2_{1,\mu_t} +\dfrac{1}{2}\dfrac{d}{dt} \|\eta^t \|^2_{1,\mu_t}. \end{aligned} \end{equation}

Combining (4.2), (4.4) and (4.1), we get

\begin{equation*}\begin{aligned}\frac{d}{dt} \left(\|u\|^2+ \|\nabla u\|^2+ \|\eta^t\|^2_{1,\mu_t} \right) + 2 \left(1- \dfrac {\beta}{\lambda_1}-\varepsilon_0 \right) & \| \nabla u \|^2 + \delta \kappa(t) \|\eta^t(s)\|^2_{1,\mu_t}\\[2pt]&\leq \dfrac{1}{2\varepsilon} \|g\|^2_{H^{-1}(\Omega)} +2C_0|\Omega|. \end{aligned}\end{equation*}

Using hypothesis (M3), we deduce $ \exists \delta_0 >0$ , so that $ \kappa(t) \ge \delta_0 >0 \,\, \forall t \in \mathbb{R}$ . Therefore

\begin{equation*}\begin{aligned}\frac{d}{dt} \left(\|u\|^2+ \|\nabla u\|^2+ \|\eta^t\|^2_{1,\mu_t} \right) + \gamma&\left(\|u\|^2+ \|\nabla u\|^2+ \|\eta^t\|^2_{1,\mu_t} \right) \\&\leq C \left( \|g\|^2_{H^{-1}(\Omega)} +1\right). \end{aligned}\end{equation*}

By the Gronwall inequality, we get

\begin{equation*}\begin{aligned} y(t) \leq e^{-\gamma (t - \tau)} y(\tau) + C \bigg( \|g\|^2_{H^{-1}(\Omega)} + 1 \bigg),\end{aligned}\end{equation*}

where

\begin{equation*}y(t)= \|u\|^2+ \|\nabla u\|^2 + \|\eta^t\|^2_{1,\mu_t}.\end{equation*}

Hence, there exists $\rho_{0}>0$ such that

(4.5) \begin{equation} \|z(t)\|^2_{\mathcal V_t} \leq \rho_{0},\end{equation}

for all $z_\tau \in B$ and $t\geq t_\tau=t_\tau(B)$ , where B is an arbitrary bounded subset of $\mathcal{V}_t$ . This completes the proof.

4.2 Asymptotic compactness

Recall that in this paper we only assume the external force $g \in H^{-1}(\Omega)$ . However, we know that for any $g\in H^{-1}(\Omega)$ and $\varepsilon >0$ given, there is a $g^\varepsilon \in L^2(\Omega)$ , which depends on g and $\varepsilon$ , such that

(4.6) \begin{equation} \|g-g^{\varepsilon}\|_{H^{-1}(\Omega)}< \varepsilon. \end{equation}

Now, in order to show that the process is asymptotically compact, we shall exhibit a pullback attracting family of compact sets. To this aim, the strategy classically consists in finding a suitable decomposition of the process in the sum of a decaying part and of a compact one.

4.2.1 Decomposition of the equation

Since $\mathcal{B}=\{\mathbb{B}_t(R)\}_{t \in\mathbb{R}}$ is a time-dependent absorbing set for $U(t, \tau)z_\tau$ , then for each initial data $z_\tau \in \mathbb{B}_t(R)$ , we decompose $U(t, \tau)z_\tau$ as follows

\begin{equation*} U(t,\tau) z_\tau =U_1(t,\tau) z_\tau + U_2(t,\tau) z_\tau,\end{equation*}

where $U_1(t,\tau) z_\tau=z_1(t)$ and $U_2(t,\tau) z_\tau=z_2(t)$ , that is, $z= (u,\eta^t)=z_1+z_2$ , the decomposition is of the following form

\begin{equation*} u=v^\varepsilon+w^\varepsilon, \quad \eta^t = \zeta^{t\varepsilon} + \xi^{t\varepsilon}, \end{equation*}

\begin{equation*} z_1 = (v^\varepsilon,\zeta^{t\varepsilon}), \quad z_2 = (w^\varepsilon,\xi^{t\varepsilon}),\end{equation*}

where $z_1(t)$ is the unique solution of the following problem

(4.7) \begin{equation}\begin{aligned}\begin{cases}&\partial_t v^\varepsilon -\partial_t \Delta v^\varepsilon - \Delta v^\varepsilon + f(u)-f(w^\varepsilon) - \int_0^\infty \mu_t(s)\Delta \zeta^{t\varepsilon}(s)ds + \lambda v^\varepsilon = g- g^\varepsilon, \; \lambda > \ell,\\ \\[-8pt]&\partial_t \zeta^{t\varepsilon} = - \partial_s \zeta^{t\varepsilon} +v^\varepsilon,\\ \\[-8pt] &v^\varepsilon(x,t)|_{\partial \Omega} = 0, \; v^\varepsilon(x,t)|_{t = \tau}=u_\tau(x),\\ \\[-8pt]&\zeta^{t\varepsilon}(x,s)|_{\partial \Omega} = 0, \; \zeta^\tau(x,s)=\zeta_\tau(x,s)\,{:\!=}\,\int\limits_\tau^sg_0(x, r)dr,\end{cases}\end{aligned} \end{equation}

and $z_2(t)$ is the unique solution of the following problem

(4.8) \begin{equation}\begin{aligned}\begin{cases} \partial_tw^\varepsilon - \partial_t\Delta w^\varepsilon - \Delta w^\varepsilon + f(w^\varepsilon) - \int_0^\infty \mu_t(s)\Delta \xi^{t\varepsilon}(s)ds - \lambda ( u - w^\varepsilon ) = g^\varepsilon, \; \lambda > \ell, \\ \\[-8pt] \partial_t \xi^{t\varepsilon} = - \partial_s \xi^{t\varepsilon} +w^\varepsilon,\\ \\[-8pt] w^\varepsilon(x,t)|_{\partial \Omega} = 0, \; w^\varepsilon(x,t)|_{t =\tau}=0,\\ \\[-8pt] \xi^{t\varepsilon}(x,s)|_{\partial \Omega} = 0, \; \xi^\tau(x,s)=\xi_\tau(x,s)=0.\end{cases}\end{aligned}\end{equation}

By using similar arguments as in the proof of Theorem 3.1, one can prove the existence and uniqueness of solutions to problems (4.7) and (4.8). Moreover, for problem (4.8), because the external force $g^{\varepsilon}\in L^2(\Omega)$ and the initial data are zero (so it belong to $\mathcal{W}_t\,{:\!=}\,\left (H^2(\Omega) \cap H_0^1(\Omega)\right)\times L^2_{\mu_t}(\mathbb{R}^+, H^2(\Omega) \cap H_0^1(\Omega))$ ), we can show that the solution $(w^\varepsilon, \xi^{t\varepsilon})$ is in fact a strong solution. In particular, we will have $w^{\varepsilon}\in C([0,T];\ H^2(\Omega)\cap H^1_0(\Omega))$ for any $T>0$ . This will be used in the proof of Lemma 4.3 below.

We begin with the decay estimate for solutions of (4.7).

Lemma 4.2. For any $\varepsilon >0$ , the solutions of equation (4.7) satisfy the following estimates: there is a constant $d_0$ which depends on $\lambda_1, \ell$ , such that for every $t \ge 0 $ ,

\begin{equation*} \|U_1(t,\tau) z_\tau \|^2_{\mathcal {V}_t} \leq Q(\|z_\tau \|_{\mathcal {V}_t})e^{-d_0 t } + \varepsilon.\end{equation*}

Proof. Multiplying the first equation of (4.7) by $v^\varepsilon$ we get

\begin{equation*} \begin{aligned} \frac{1}{2}\frac{d}{dt} \bigl(\|v^\varepsilon\|^2 + \|\nabla v^\varepsilon\|^2\bigr) + \lambda \|v^\varepsilon\|^2 + \|\nabla v^\varepsilon\|^2+ \int_0^\infty \mu_t(s) \int_\Omega \nabla \zeta^{t\varepsilon} \nabla v^\varepsilon dxds \\+ \langle f(u) - f(w^\varepsilon), v^\varepsilon \rangle = \langle g- g^\varepsilon , v^\varepsilon \rangle_{H^{-1}, H^1_0}.\end{aligned}\end{equation*}

Applying Cauchy inequality, we get

\begin{equation*}\begin{aligned}\langle g- g^\varepsilon , v^\varepsilon \rangle_{H^{-1}, H^1_0} \leq \frac{1}{2} \| \nabla v^\varepsilon \|^2 + \frac{1}{2} \|g- g^\varepsilon \|^2_{H^{-1}(\Omega)}.\end{aligned}\end{equation*}

Noting that $\partial_t\zeta^{t\varepsilon}=-\partial_s\zeta ^{t\varepsilon}+v^{\varepsilon}$ and reasoning exactly as in (4.3), (4.4), we obtain

(4.9) \begin{equation} \begin{aligned} \int_0^\infty \mu_t(s) \langle \nabla \zeta^{t\varepsilon}(s), \nabla v^\varepsilon \rangle ds \geq \dfrac{1}{2} \delta \kappa(t) \|\zeta^{t\varepsilon}(s)\|^2_{1,\mu_t} +\dfrac{1}{2}\dfrac{d}{dt} \|\zeta^{t\varepsilon}(s)\|^2_{1,\mu_t}. \end{aligned} \end{equation}

Therefore, because $f^{\prime}(\xi) \geq -\ell$ , we have

\begin{equation*} \begin{aligned} &\frac{d}{dt}\bigg( \|v^\varepsilon\|^2 + \|\nabla v^\varepsilon\|^2 +\|\zeta^{t\varepsilon}\|^2_{1,\mu_t}\bigg) + \| \nabla v^\varepsilon\|^2+2(\lambda - \ell) \|v^\varepsilon\|^2+ \delta \kappa(t) \|\zeta^{t\varepsilon}(s)\|^2_{1,\mu_t} \\ \leq &\,\, \|g- g^\varepsilon \|^2_{H^{-1}(\Omega)}.\end{aligned}\end{equation*}

Using hypothesis (M3), we deduce $ \exists \delta_0 >0$ , so that $ \kappa(t) \ge \delta_0 >0 \,\, \forall t \in \mathbb{R}$ .

Thus, similarly to the proof of Lemma 4.1, we obtain for some $d_0>0$ ,

\begin{equation*} \|U_1(t,\tau) z_\tau \|^2_{\mathcal {V}_t}\leq Q(\|z_\tau \|_{\mathcal {V}_t})e^{-d_0 t } + \frac{1}{C}\|g- g^\varepsilon \|^2_{H^{-1}(\Omega)}.\end{equation*}

Taking $\varepsilon^2 \leq C \varepsilon$ in (4.6) we have

\begin{align*} \|U_1(t,\tau) z_\tau \|^2_{\mathcal {V}_t}\leq Q(\|z_\tau \|_{\mathcal {V}_t})e^{-d_0 t } +\varepsilon.\\[-40pt] \end{align*}

About the solution $z_2(t)$ of (4.8), we have the following lemma.

Lemma 4.3. For any $\varepsilon >0$ , there is $M>0$ such that for any $z_\tau \in \mathcal W_t$ , there exists $T>0$ large enough, which depends on $\|g \|^2_{H^{-1}(\Omega)}$ , $\varepsilon$ , such that

(4.10) \begin{equation} \|U_2(t,\tau) z_\tau \|^2_{\mathcal {W}_t} \leq M, \text{ for all }t \geq T. \end{equation}

Proof. Multiplying the first equation of (4.8) by $- \Delta w^\varepsilon$ , then using (1.2), (M2) and the Cauchy inequality, we have

\begin{equation*} \begin{aligned} \dfrac{d}{dt}\bigg( \|\nabla w^\varepsilon\|^2 &+ \|\Delta w^\varepsilon\|^2 +\|\xi^{t\varepsilon}\|^2_{2,\mu_t}\bigg) + \| \Delta w^\varepsilon\|^2+ 2 (\lambda - \ell) \|\nabla w^\varepsilon\|^2\\& + \delta \kappa(t) \|\xi^{t\varepsilon}(s)\|^2_{1,\mu_t} \leq \dfrac{1}{2} \left(\|g^\varepsilon\|^2_{H^{-1}(\Omega)}+ \| u\|_{H_0^1(\Omega)}^2 \right)\leq C( \|g^\varepsilon\|^2_{H^{-1}(\Omega)}+ \rho_0)\end{aligned}\end{equation*}

when $t \ge t_0(B)$ . Note that we used the estimate (4.5) for this expression.

Hence, similarly to the proof of Lemma 4.1, we obtain a number $T>0$ large enough such that

\begin{equation*} \|U_2(t,\tau) z_\tau \|^2_{\mathcal {W}_t} \leq M , \text{ for all }t \ge T.\end{equation*}

The proof is complete.

Remark 4.1. From Lemma 4.3, we immediately have the following regularity result: $\mathcal A_t$ is bounded in $\mathcal W_t$ (with a bound independent of t).

Since $\mathcal{B}=\{B_t\}_{t\in \mathbb{R}}$ is a time-dependent absorbing set, collecting Lemmas 4.2 and 4.3 we infer that the family of $\mathcal W_t$ -ball $\mathcal K = \{K_t(r)\}_{t\in \mathbb{R}}$ is pullback attracting provided that $r > 0$ is sufficiently large, for

\begin{equation*}dist_{\mathcal V_t}(U (t,\tau)B_\tau, K_t(r)) \leq \sup_{z_\tau \in B_\tau} \|D (t,\tau)z_\tau\|_{\mathcal V_t} \leq Ce^{-\frac{\gamma }{2}(t-\tau)}. \end{equation*}

Unfortunately, there are not enough conditions to conclude the existence of the unique time-dependent global attractor. Indeed, although closed balls of $\mathcal{W}_t$ are uniformly bounded by the embedding constant of $ \mathcal{W}_t \hookrightarrow \mathcal V_t$ is independent of t, they fail to be compact in $\mathcal V_t$ due to the lack of compactness of the embedding $L^2_{\mu_t}(\mathbb{R}^+, H_0^1(\Omega)) \hookrightarrow L^2_{\mu_t}(\mathbb{R}^+, H^2(\Omega) \cap H_0^1(\Omega)) $ . The argument as in the proof of [Reference Wang and Zhong30, Theorem 3.13], where the same model is considered for a constant-in-time memory kernel, we get that the process $U(t, \tau)$ is asymptotically compact, which proves the existence of the unique time-dependent global attractor in below.

Theorem 4.4. Assume that (H1)–(H3) hold. Then the process $\{U(t,\tau )\}_{t \ge \tau }$ generated by problem (2.1) admits an invariant time-dependent global attractor $\mathcal A = \{ A_t \}_{t \in \mathbb R}$ .

5. Recovering nonclassical diffusion equation

In the last section, using the idea of Conti et al. in [Reference Conti, Danese and Pata7], we discuss the case when $k_t \to m\delta_0$ for some $m >0$ , that is,

(5.1) \begin{equation} \begin{aligned} \lim_{t\to \infty} \int_\varepsilon^\infty k_t(s) ds = \begin{cases} m \quad \text{ if } \varepsilon = 0,\\ 0 \quad \text{ if } \varepsilon > 0. \end{cases} \end{aligned} \end{equation}

Then, for long times our problem becomes the nonclassical diffusion equation (without memory kernel)

(5.2) \begin{equation} u_{t} - \Delta u_{t} - (1+m)\Delta u + f( u) = g.\end{equation}

Equation (5.2) generates a $\mathcal{C}_0$ -semigroup of solutions

\begin{equation*}S(t)\,{:}\, H_0^1(\Omega) \to H_0^1(\Omega),\end{equation*}

possessing the global attractor $\hat{A}$ in the classical sense. Besides, $\hat{A}$ is a bounded subset of $H^2(\Omega)\cap H_0^1(\Omega) $ , and coincides with the sections at (any) time $t_0 \in \mathbb{R}$ of the set of all complete bounded trajectories (CBT) of S(t) (see, [Reference Wang, Li and Zhong28]). Namely, for any fixed $t_0 \in \mathbb{R}$ ,

\begin{equation*} \hat{A} = \{\hat{z}(t_0)\,{:}\, \hat{z} \, \text{ CBT of } S(t)\},\end{equation*}

where a CBT the semigroup S(t) is a map

\begin{equation*} t \mapsto \hat z(t) = \hat u(t) \in H_0^1(\Omega) \end{equation*}

satisfying

\begin{equation*}\sup_{t\in \mathbb{R}} \|\hat z(t)\|_{H_0^1(\Omega) } < \infty \quad \text{ and } \quad \hat z(t) = S(t) \hat z(\tau) \,\, \forall t \ge 0, \forall \tau \in \mathbb{R}.\end{equation*}

On the other hand, according to Theorem 4.4 and [Reference Conti and Pata11, Theorem 3.2], the invariant time-dependent global attractor is characterized as the set of all CBT of the process, that is,

\begin{equation*} A = \{z(t )\,{:}\, z \text{ CBT of } U(t,\tau)\},\end{equation*}

where a CBT of $U(t,\tau)$ is a map

\begin{equation*} t \mapsto z(t) = (u(t),\partial_tu(t), \eta^t) \in \mathcal V_t\end{equation*}

satisfying

\begin{equation*}\sup_{t\in \mathbb{R}} \|z(t)\|_{\mathcal V_t} < \infty \quad \text{ and } \quad z(t) = U(t,\tau)z(\tau) \,\, \forall t \ge \tau \in \mathbb{R}.\end{equation*}

Using ideas as in [Reference Conti, Danese and Pata7], we have the following theorem which establishes the closeness of the long-term dynamics of (1.1) to the one of the “limit problem” (5.2) when $k_t \to m\delta_0$ .

Theorem 5.1. Assume that (5.1) hold. Then, for any sequence $(u_n, \eta^t_n)$ of CBT of $U(t,\tau)$ and any $t_n \to \infty$ , there exists a CBT $ \hat u $ of S(t) such that the convergence

(5.3) \begin{equation}\sup_{t \in [\!-T,T]} \|u_n(t+t_n)-\hat u(t)\|_{ H_0^1(\Omega)} \to 0\end{equation}

holds up to a subsequence as $n \to \infty$ for every $T > 0$ .

Defining the canonical projection $\mathbb{P}_t\,{:}\, \mathcal V_t \to H_0^1(\Omega)$ by $\mathbb P_t(u, \eta) = u$ , and denoting the Hausdorff semidistance of two (nonempty) sets $B, C \subset X$ by

\begin{equation*}dist_{X} (B,C)= \sup_{x\in B} \inf_{x\in C} \|x-y\|_{X},\end{equation*}

then we have an immediate corollary.

Corollary 5.1. Assume that (5.1) hold. Then, we have the convergence

\begin{equation*}\lim_{t\to \infty} \left[\text{dist}_{H^2(\Omega)\cap H_0^1(\Omega)} (\mathbb{P}_tA_t,\hat A)\right] = 0.\end{equation*}

Indeed, from (5.3), we deduce that for every $t_n \to \infty$ the convergence

\begin{equation*}\text{dist}_{H^2(\Omega)\cap H_0^1(\Omega)} (\mathbb{P}_tA_t,\hat A) \to 0\end{equation*}

holds (up to a sequence) as $n \to \infty$ . This is clearly enough to draw the desired conclusion.

Proof of Theorem 5.1.Let $w_n(\!\cdot\!)= u_n(\cdot+t_n)$ , for every n. Then the function $w_n$ fulfills the equation

(5.4) \begin{equation} \partial_{t}w_n(t) - \Delta \partial_t w_n(t) - \Delta w_n(t) - \int_0^\infty k_{t+t_n}(s)\Delta w_n(t-s)ds+f(w_n(t))=g.\end{equation}

From the estimates of Lemmas 4.2 and 4.3, we get that

(5.5) \begin{equation} w_n \text{ is bounded in } L^\infty(\mathbb{R};\ H^2(\Omega)\cap H_0^1(\Omega)).\end{equation}

Then, there exists $\hat u \in L^\infty(\mathbb{R};\ H^2(\Omega)\cap H_0^1(\Omega))$ such that, up to a subsequence,

\begin{equation*} w_n \rightharpoonup \hat u \text{ weakly-star in } L^\infty(\mathbb{R}, H^2(\Omega)\cap H_0^1(\Omega)).\end{equation*}

Since $\lim_{n\to\infty} k_{t+t_n} = m \delta_0$ in the sense of (5.1), we can see that, for every $T>0$ ,

\begin{equation*}\sup_{t\in [\!-T,T]} \int_0^\infty k_{t+t_n}(s)ds \leq 2m,\end{equation*}

for every n sufficiently large (depending on T).

Multiplying equation (5.4) by $w_n+ \partial_t w_n$ then integrating from $-T $ to T we conclude that

(5.6) \begin{equation} \partial_tw_n \text{ is bounded in } L^\infty(\!-T,T;\ H_0^1(\Omega)).\end{equation}

From (5.5) and (5.6) and applying the classical Simon-Aubin Theorem [Reference Simon23], the strong convergence

\begin{equation*} w_n \to \hat u \text{ in } \mathcal{C}([\!-T,T],H_0^1(\Omega))\end{equation*}

holds (up to a subsequence), implying in particular (5.3).

Now, we are left to prove that $ \hat u $ solves the nonclassical diffusion equation, namely

\begin{equation*}\partial_{t} \hat u -\Delta \partial_t \hat u -(1+m)\Delta \hat u + f(\hat u )=g.\end{equation*}

Indeed, we will prove that the equality above is recovered when passing to the limit as $ n \to \infty$ in (5.4), the only nonstandard convergence being

\begin{equation*} -\int_0^\infty k_{t+t_n} (s) \Delta w_n(t-s)ds \to -m \Delta \hat u(t).\end{equation*}

Let $\phi$ be fixed, and

\begin{equation*} \Lambda_n(t) = \int_0^\infty k_{t+t_n} (s) \langle w_n(t-s), -\Delta \phi\rangle ds.\end{equation*}

Then, we decompose the function $\Lambda_n(t)$ into the sum

\begin{equation*}\Lambda_n(t) = \Lambda^1_n(t)+\Lambda_n^2(t)+\Lambda_n^3(t),\end{equation*}

where

\begin{align*}&\Lambda^1_n(t) = \int_0^1 k_{t+t_n} (s) \langle \hat u(t-s), -\Delta \phi\rangle ds,\\ &\Lambda_n^2(t)= \int_0^1 k_{t+t_n} (s) \langle w_n(t-s)- \hat u(t-s), -\Delta \phi\rangle ds,\\ &\Lambda_n^3(t)= \int_1^\infty k_{t+t_n} (s) \langle w_n(t-s), -\Delta \phi\rangle ds.\end{align*}

Since $\langle w_n, -\Delta \phi\rangle \in L^\infty (\mathbb{R})$ uniformly with respect to n, and $\langle w_n, -\Delta \phi\rangle \to \langle \hat u, -\Delta \phi\rangle $ in C(I) for every closed interval I, and using (5.1) once again

\begin{align*}|\Lambda_n^2(t)+\Lambda_n^3(t)| \leq &\,\|\langle w_n - \hat u, -\Delta\phi\rangle \|_{\mathcal{C}([t-1,t])}\int_0^1 k_{t+t_n} (s) ds\\&+ \|\langle w_n, -\Delta \phi\rangle \|_{L^\infty(\mathbb{R})}\int_1^\infty k_{t+t_n} (s) ds \to 0,\end{align*}

while

\begin{equation*}\Lambda_n^1(t) \to m \langle \hat u, -\Delta \phi\rangle.\end{equation*}

Therefore, we conclude that

\begin{equation*} \Lambda_n(t) = \int_0^\infty k_{t+t_n} (s) \langle w_n(t-s), -\Delta \phi\rangle ds \to m \langle \hat u, -\Delta \phi\rangle, \end{equation*}

for any sufficiently regular $\phi$ . Thus

\begin{equation*} -\int_0^\infty k_{t+t_n} (s) \Delta w_n(t-s)ds \to -m\Delta \hat u(t).\end{equation*}

This completes the proof.