Attractors for the nonclassical reaction–diffusion equations on time-dependent spaces

Abstract

In this paper, based on the notation of time-dependent attractors introduced by Conti, Pata and Temam in (J. Differ. Equ. 255:1254–1277, 2013), we prove the existence of time-dependent global attractors in $\mathcal{H}_{t}$ for a class of nonclassical reaction–diffusion equations with the forcing term $g(x)\in H^{-1}(\varOmega )$ and the nonlinearity f satisfying the polynomial growth of arbitrary $p-1$ ($p\geq 2$) order, which generalizes the results obtained in (Appl. Anal. 94:1439–1449, 2015) and (Bound. Value Probl. 2016: 10, 2016).

1 Introduction

Let Ω be a bounded domain in $\mathbb{R}^{n}$$(n\geq 3)$ with smooth boundary, we consider the long-time behavior of the solutions for the following nonclassical reaction–diffusion equation:

$$ \textstyle\begin{cases} u_{t}-\varepsilon (t)\Delta u_{t}-\Delta u+f(u)=g(x) &\text{in } \varOmega \times (\tau,\infty ), \\ u=0 &\text{on } \partial \varOmega \times (\tau,\infty ), \\ u(x,\tau )=u_{\tau }, &x\in \varOmega, \end{cases} $$

(1.1)

where $t>\tau$, $\tau \in \mathbb{R}$ is the initial time, $g(x)\in H^{-1}(\varOmega )$ is an external force term, $\varepsilon (t)\in C^{1}(\mathbb{R})$ is a decreasing bounded function satisfying

$$\begin{aligned} \lim_{t\rightarrow +\infty }\varepsilon (t)=0, \end{aligned}$$

(1.2)

and there exists $L>0$ such that

$$\begin{aligned} \sup_{t\in \mathbb{R}} \bigl( \bigl\vert \varepsilon (t) \bigr\vert + \bigl\vert \varepsilon '(t) \bigr\vert \bigr)\leq L. \end{aligned}$$

(1.3)

For the nonlinear term $f\in C(\mathbb{R},\mathbb{R})$, similar to that in [3, 20, 24], we make the following classical assumptions:

$$ f'(u)\geq -l, \quad\forall u\in \mathbb{R,} $$

(1.4)

and

$$ -c_{0}+c_{1} \vert u \vert ^{p} \leq f(u)u\leq c_{0}+c_{2} \vert u \vert ^{p},\quad p\geq 2, $$

(1.5)

for some positive constants $c_{0}, c_{1}, c_{2}$.

Let $\mathcal{F}(u)=\int _{0}^{u}f(r)\,dr$, then there are constants $\tilde{c}_{i}>0$$(i=0,1,2)$ such that

$$\begin{aligned} -\tilde{c}_{0}+\tilde{c}_{1} \vert u \vert ^{p}\leq \mathcal{F}(u)\leq \tilde{c}_{0}+ \tilde{c}_{2} \vert u \vert ^{p},\quad \forall u\in \mathbb{R}. \end{aligned}$$

(1.6)

For Eq. (1.1), when $\varepsilon (t)>0$ is a constant, the existence and long-time behavior of solutions have been extensively studied by several authors; see, e.g., [1, 4, 5, 23, 25–27, 29, 30, 32]. In [4, 5, 29], the authors main considered the existence of solutions for this type of equations. In [1, 23, 25–27, 30], the authors main considered the existence of the global attractors (see [23, 25–27]) and the pullback (or the uniform) attractors (see [1, 23, 30]) in $H_{0}^{1}(\varOmega )$ (or $H^{1}(\mathbb{R}^{N})$). In particular, in [32], we obtained the existence of the pullback attractors in $C_{H_{0}^{1}(\varOmega )}$ (rather than in $H_{0}^{1}(\varOmega )$) for the nonclassical reaction–diffusion equations with delays.

When $\varepsilon (t)=0$, Eq. (1.1) becomes the classical reaction–diffusion equation. The existence and the long-time behavior of solutions have also been extensively investigated by several authors; see, e.g., [2, 11, 12, 17, 21, 28, 31]. In [2, 11, 12, 28], the authors mainly considered the existence (or the blowup), uniqueness and the long-time decay of the solutions for the semilinear parabolic equation [11, 12], the nonlinear parabolic equation [2] and the coupled parabolic systems [28]. In [17, 21, 31], the authors have proved the existence of the global attractors in $L^{p}(\varOmega )$, $H_{0}^{1}(\varOmega )$, $L^{2p-2}(\varOmega )$, $H^{2}(\varOmega )$ (see [31]) and the existence of the pullback attractors in $L^{p}(\varOmega )$ and $H_{0}^{1}(\varOmega )$ (see [17] and [21], respectively).

When $\varepsilon (t)\in C^{1}(\mathbb{R})$ satisfies (1.2)–(1.3), the long-time behavior of solutions for Eq. (1.1) has been considered by some researchers; see, e.g., [16, 18]. In [16], the authors have proved the existence of the time-dependent global attractors in $\mathcal{H}_{t}$ with the nonlinearity f satisfying $|f''(u)|\leq c(1+|u|)$ (see Theorem 3.4 in [16] for details). Furthermore, in [18], the authors have considered the case of the nonlinearity f satisfying the critical exponent growth and proved the existence of the time-dependent global attractors in $\mathcal{H}_{t}$ (see Theorem 3.3 in [18] for details).

In this paper, we consider Eq. (1.1) with the nonlinearity f satisfying polynomial growth of arbitrary $p-1$$(p\geq 2)$ order, which makes that the Sobolev compact embedding is no longer valid and brings more difficulty for verifying the corresponding asymptotic compactness of the solutions process $\{U(t,\tau )\}_{t\geq \tau }$. In order to overcome the difficulty mentioned above, we verify the existence of the time-dependent global attractors $\hat{\mathcal{A}}$ in $\mathcal{H}_{t}$ for the process $\{U(t,\tau )\}_{t\geq \tau }$ by applying the contractive function methods as in [6, 13, 14, 19, 22, 27] (see Theorem 3.8).

2 Preliminaries

In this section, we firstly review briefly some notations, basic definitions and results about processes on time-dependent spaces (see [7–9, 19] for details).

2.1 Notations

Let $\{X_{t}\}_{t\in \mathbb{R}}$ be a family of normed spaces, we introduce the R-ball of $X_{t}$ as

$$ \mathbb{B}_{t}(R)=\bigl\{ z\in X_{t}: \Vert z \Vert _{X_{t}}\leq R\bigr\} . $$

For any given $\epsilon >0$, the ϵ-neighborhood of a set $B\subset X_{t}$ is defined as

$$ \mathcal{O}_{t}^{\epsilon }(B)=\bigcup _{x\in B}\bigl\{ y\in X_{t}: \Vert x-y \Vert _{X_{t}}< \epsilon \bigr\} =\bigcup_{x\in B}\bigl\{ x+\mathbb{B}_{t}(\epsilon )\bigr\} . $$

We denote the Hausdorff semidistance of two (nonempty) sets $B,C\subset X_{t}$ by

$$ \delta _{t}(B,C)=\sup_{x\in B}\inf _{y\in C} \Vert x-y \Vert _{X_{t}}. $$

Moreover, we introduce the time-dependent space $\mathcal{H}_{t}$ endowed with the norms

$$ \Vert u \Vert ^{2}_{\mathcal{H}_{t}}= \Vert u \Vert ^{2}_{2}+\varepsilon (t) \Vert \nabla u \Vert ^{2}_{2}, $$

where $\|\cdot \|_{2}$ denotes the usual norm in $L^{2}(\varOmega )$.

2.2 Some concepts

In this subsection, we give some concepts about the time-dependent global attractors.

Definition 2.1

Let $\{X_{t}\}_{t\in \mathbb{R}}$ be a family of normed spaces. A process is a two-parameter family of mappings $U(t,\tau ): X_{\tau }\rightarrow X_{t}$, $t\geq \tau$, $\tau \in \mathbb{R}$ with properties

• (i) $U(\tau,\tau )=\mathrm{Id}$ is the identity operator on $X_{\tau }$, $\tau \in \mathbb{R}$;

• (ii) $U(t,s)U(s,\tau )=U(t,\tau )$, $\forall t\geq s\geq \tau$, $\tau \in \mathbb{R}$.

Definition 2.2

A family $\hat{C}=\{C_{t}\}_{t\in \mathbb{R}}$ of bounded sets $C_{t}\subset X_{t}$ is called uniformly bounded if there exists a constant $R>0$ such that $C_{t}\subset \mathbb{B}_{t}(R)$ for all $t\in \mathbb{R}$.

Definition 2.3

A family $\hat{B}=\{B_{t}\}_{t\in \mathbb{R}}$ is called pullback absorbing if it is uniformly bounded and for every $R>0$, there exists a constant $t_{0}=t_{0}(t,R)\leq t$ such that $U(t,\tau )\mathbb{B}_{\tau }(R)\subset B_{t}$ for all $\tau \leq t_{0}$.

The process $\{U(t,\tau )\}_{t\geq \tau }$ is called dissipative whenever it admits a pullback absorbing family.

Definition 2.4

A time-dependent absorbing set for the process $\{U(t,\tau )\}_{t\geq \tau }$ is a uniformly bounded family $\hat{B}=\{B_{t}\}_{t\in \mathbb{R}}$ with the following property: for every $R\geq 0$ there exists a $t_{0}=t_{0}(R)\geq 0$ such that

$$ U(t,\tau )\mathbb{B}_{\tau }(R)\subset B_{t} \quad\text{for all } \tau \leq t-t_{0}. $$

Definition 2.5

The process $\{U(t,\tau )\}_{t\geq \tau }$ is said to be pullback asymptotically compact if for any $t\in \mathbb{R}$, any bounded sequence $\{x_{n}\}_{n=1}^{\infty }\subset X_{\tau _{n}}$ and any sequence $\{\tau _{n}\}_{n=1}^{\infty }$ with $\tau _{n}\rightarrow -\infty$ as $n\rightarrow \infty$, the sequence $\{U(t,\tau _{n})x_{n}\}_{n=1}^{\infty }$ is precompact in $\{X_{t}\}_{t\in \mathbb{R}}$.

Definition 2.6

The time-dependent global attractor for the process $\{U(t,\tau )\}_{t\geq \tau }$ is the smallest family $\hat{\mathcal{A}}=\{\mathcal{A}_{t}\}_{t\in \mathbb{R}}$ such that

• (i) $\mathcal{A}_{t}$ is compact in $X_{t}$;

• (ii) $\hat{\mathcal{A}}$ is invariant, i.e., $U(t,\tau )\mathcal{A}_{\tau }=\mathcal{A}_{t}, \forall t\geq \tau$;

• (iii) $\hat{\mathcal{A}}$ is pullback attracting, i.e., it is uniformly bounded and the limit

  $$ \lim_{\tau \rightarrow -\infty }\delta _{t}\bigl(U(t,\tau )C_{\tau }, \mathcal{A}_{t}\bigr)=0 $$

  holds for every uniformly bounded family $\hat{C}=\{C_{t}\}_{t\in \mathbb{R}}$ and every fixed $t\in \mathbb{R}$.

Remark 2.7

The attracting property can be equivalently stated in terms of pullback absorbing: a (uniformly bounded) family $\mathcal{K}=\{K_{t}\}_{t\in \mathbb{R}}$ is called pullback attracting if for any $\epsilon >0$ the family $\{\mathcal{O}_{t}^{\epsilon }(K_{t})\}_{t\in \mathbb{R}}$ is pullback absorbing.

Similarly to Theorem 4.2 in [8], we have the following theorem.

Theorem 2.8

The time-dependent global attractor$\hat{\mathcal{A}}$exists and it is unique if and only if the process$\{U(t,\tau )\}_{t\geq \tau }$is asymptotically compact, namely, the set

$$ \mathbb{K}=\bigl\{ \mathcal{K}=\{K_{t}\}_{t\in \mathbb{R}}: K_{t}\subset X_{t} \textit{ is compact }, \mathbb{K} \textit{ is pullback attracting}\bigr\} $$

is not empty.

2.3 Some results

In order to obtain the time-dependent global attractors of Eq. (1.1), we need the following definitions and conclusions, which are similar to those in [6, 13, 14, 19, 22, 27].

Definition 2.9

Let $\{X_{t}\}_{t\in \mathbb{R}}$ be a family of Banach spaces and $\hat{C}=\{C_{t}\}_{t\in \mathbb{R}}$ be a family of uniformly bounded subset of $\{X_{t}\}_{t\in \mathbb{R}}$. We call a function $\psi _{\tau }^{t}(\cdot,\cdot )$, defined on $\{X_{t}\}_{t\in \mathbb{R}}\times \{X_{t}\}_{t\in \mathbb{R}}$, a contractive function on $C_{\tau }\times C_{\tau }$ if for fixed $t\in \mathbb{R}$ and any sequence $\{x_{n}\}_{n=1}^{\infty }\subset C_{\tau }$, there is a subsequence $\{x_{n_{k}}\}_{k=1}^{\infty }\subset \{x_{n}\}_{n=1}^{\infty }$ such that

$$\begin{aligned} \lim_{k\rightarrow \infty }\lim_{l\rightarrow \infty }\psi _{\tau }^{t}(x_{n_{k}},x_{n_{l}})=0\quad \text{for all } t\geq \tau. \end{aligned}$$

We denote the set of all contractive functions on $C_{\tau }\times C_{\tau }$ by $Contr(C_{\tau})$.

Theorem 2.10

Let$\{U(t,\tau )\}_{t\geq \tau }$be a process on Banach spaces$\{X_{t}\}_{t\in \mathbb{R}}$and have a pullback absorbing set$\hat{B}=\{B_{t}\}_{t\in \mathbb{R}}$. Moreover, assume that, for any$\epsilon >0$, there exist$\tau _{0}=\tau _{0}(\epsilon )< t$and$\psi _{\tau _{0}}^{t}(\cdot,\cdot )\in \hat{C}(B_{\tau _{0}})$such that

$$\begin{aligned} \bigl\Vert U(t,\tau _{0})x-U(t,\tau _{0})y \bigr\Vert _{X_{t}}\leq \epsilon +\psi _{ \tau _{0}}^{t}(x,y),\quad \forall x,y\in B_{\tau _{0}}, \end{aligned}$$

for any$t\in \mathbb{R}$. Then$\{U(t,\tau )\}_{t\geq \tau }$is pullback asymptotically compact in$\{X_{t}\}_{t\in \mathbb{R}}$.

Proof

We need to prove that, for any $\{x_{n}\}_{n=1}^{\infty }\subset B_{\tau _{n}}$ and any $\tau _{n}\rightarrow -\infty$ as $n\rightarrow \infty$,

$$ \text{the sequence } \bigl\{ U(t,\tau _{n})x_{n}\bigr\} _{n=1}^{\infty } \text{ is precompact in } \{X_{t} \}_{t\in \mathbb{R}}. $$

In the following, we will show that $\{U(t,\tau _{n})x_{n}\}_{n=1}^{\infty }$ has a convergent subsequence via diagonal methods.

Taking $\epsilon _{m}>0$ with $\epsilon _{m}\rightarrow 0$ as $m\rightarrow \infty$.

Then, for $\epsilon _{1}>0$, by the assumptions, there exist $\tau _{0}=\tau _{0}(\epsilon _{1})< t$ and $\psi ^{t}_{\tau _{0}}(\cdot,\cdot )\in \hat{C}(B_{\tau _{0}})$ such that

$$\begin{aligned} \bigl\Vert U(t,\tau _{0})x-U(t,\tau _{0})y \bigr\Vert _{X_{t}}\leq \epsilon _{1}+\psi ^{t}_{ \tau _{0}}(x,y), \quad\forall x,y\in B_{\tau _{0}}, \end{aligned}$$

(2.1)

for any $t\in \mathbb{R}$, where $\psi ^{t}_{\tau _{0}}$ depends on $\tau _{0}$.

Since $\tau _{n}\rightarrow -\infty$, without loss of generality, we assume that $\tau _{n}\leq \tau _{0}$ such that $U(\tau _{0},\tau _{n})x_{n}\in B_{\tau _{0}}$ for each $n\in \mathbb{N}$. Set $y_{n}=U(\tau _{0},\tau _{n})x_{n}$, then from (2.1) we have

$$\begin{aligned} \bigl\Vert U(t,\tau _{n})x_{n}-U(t,\tau _{m})x_{m} \bigr\Vert _{X_{t}} &= \bigl\Vert U(t,\tau _{0})U( \tau _{0},\tau _{n})x_{n}-U(t, \tau _{0})U(\tau _{0},\tau _{m})x_{m} \bigr\Vert _{X_{t}} \\ &= \bigl\Vert U(t,\tau _{0})y_{n}-U(t,\tau _{0})y_{m} \bigr\Vert _{X_{t}} \\ &\leq \epsilon _{1}+\psi ^{t}_{\tau _{0}}(y_{n},y_{m}). \end{aligned}$$

(2.2)

By the definition of $\hat{C}(B_{\tau _{0}})$ and $\psi ^{t}_{\tau _{0}}\in \hat{C}(B_{\tau _{0}})$, we know that $\{y_{n}\}_{n=1}^{\infty }$ have a subsequence $\{y_{n_{k}}^{(1)}\}_{k=1}^{\infty }$ such that

$$\begin{aligned} \lim_{k\rightarrow \infty }\lim_{l\rightarrow \infty } \psi ^{t}_{ \tau _{0}}\bigl(y_{n_{k}}^{(1)},y_{n_{l}}^{(1)} \bigr)\leq \epsilon _{1}. \end{aligned}$$

(2.3)

Similarly to [13, 22, 27], we have

$$\begin{aligned} &\lim_{k\rightarrow \infty }\sup_{q\in \mathbb{N}} \bigl\Vert U\bigl(t, \tau _{n_{k+q}}^{(1)}\bigr)x_{n_{k+q}}^{(1)}-U\bigl(t, \tau _{n_{k}}^{(1)}\bigr)x_{n_{k}}^{(1)} \bigr\Vert _{X_{t}} \\ & \quad\leq \lim_{k\rightarrow \infty }\sup_{q\in \mathbb{N}} \limsup _{l\rightarrow \infty } \bigl\Vert U\bigl(t,\tau _{n_{k+q}}^{(1)} \bigr)x_{n_{k+q}}^{(1)}-U\bigl(t, \tau _{n_{l}}^{(1)} \bigr)x_{n_{l}}^{(1)} \bigr\Vert _{X_{t}} \\ &\qquad{} +\limsup_{k\rightarrow \infty }\limsup_{l\rightarrow \infty } \bigl\Vert U \bigl(t,\tau _{n_{k}}^{(1)}\bigr)x_{n_{k}}^{(1)}-U \bigl(t,\tau _{n_{l}}^{(1)}\bigr)x_{n_{l}}^{(1)} \bigr\Vert _{X_{t}} \\ & \quad\leq \epsilon _{1}+\lim_{k\rightarrow \infty }\sup _{q\in \mathbb{N}}\limsup_{l\rightarrow \infty } \psi ^{t}_{\tau _{0}} \bigl(y_{n_{k+q}}^{(1)},y_{n_{l}}^{(1)}\bigr) + \epsilon _{1}+\lim_{k\rightarrow \infty }\lim_{l\rightarrow \infty } \psi ^{t}_{\tau _{0}}\bigl(y_{n_{k}}^{(1)},y_{n_{l}}^{(1)} \bigr), \end{aligned}$$

which, combining with (2.2) and (2.3), implies that

$$\begin{aligned} \lim_{k\rightarrow \infty }\sup_{q\in \mathbb{N}} \bigl\Vert U\bigl(t, \tau _{n_{k+q}}^{(1)}\bigr)x_{n_{k+q}}^{(1)}-U\bigl(t, \tau _{n_{k}}^{(1)}\bigr)x_{n_{k}}^{(1)} \bigr\Vert _{X_{t}}\leq 4\epsilon _{1}. \end{aligned}$$

Therefore, there exists a $K_{1}\in \mathbb{N}$ such that

$$\begin{aligned} \lim_{k\rightarrow \infty }\sup_{q\in \mathbb{N}} \bigl\Vert U\bigl(t, \tau _{n_{k}}^{(1)}\bigr)x_{n_{k}}^{(1)}-U\bigl(t, \tau _{n_{l}}^{(1)}\bigr)x_{n_{l}}^{(1)} \bigr\Vert _{X_{t}}\leq 5\epsilon _{1},\quad \text{for all } k,l\geq K_{1}. \end{aligned}$$

By induction, we can obtain that, for each $m\geq 1$, there exists a subsequence $\{U(t, \tau _{n_{k}}^{(m+1)})x_{n_{k}}^{(m+1)}\}_{k=1}^{\infty }$ of $\{U(t,\tau _{n_{k}}^{(m)})x_{n_{k}}^{(m)}\}_{k=1}^{\infty }$ and certain $K_{m+1}$ such that

$$\begin{aligned} \lim_{k\rightarrow \infty }\sup_{q\in \mathbb{N}} \bigl\Vert U\bigl(t, \tau _{n_{k}}^{(m+1)}\bigr)x_{n_{k}}^{(m+1)}-U\bigl(t, \tau _{n_{l}}^{(m+1)}\bigr)x_{n_{l}}^{(m+1)} \bigr\Vert _{X_{t}}\leq 5\epsilon _{m+1},\quad \text{for all } k,l\geq K_{m+1}. \end{aligned}$$

Now, we consider the diagonal subsequence $\{U(t,\tau _{n_{k}}^{(k)})x_{n_{k}}^{(k)}\}_{k=1}^{\infty }$. Since for each $m\in \mathbb{N}$, $\{U(t,\tau _{n_{k}}^{(k)})x_{n_{k}}^{(k)}\}_{k=m}^{\infty }$ is a subsequence of $\{U(t,\tau _{n_{k}}^{(k)})x_{n_{k}}^{(k)}\}_{k=1}^{\infty }$, then

$$\begin{aligned} \lim_{k\rightarrow \infty }\sup_{q\in \mathbb{N}} \bigl\Vert U\bigl(t, \tau _{n_{k}}^{(k)}\bigr)x_{n_{k}}^{(k)}-U\bigl(t, \tau _{n_{l}}^{(l)}\bigr)x_{n_{l}}^{(l)} \bigr\Vert _{X_{t}}\leq 6\epsilon _{m}, \quad\text{for all } k,l\geq \max \{m,K_{m}\}, \end{aligned}$$

which combining with $\epsilon _{m}\rightarrow 0$ as $m\rightarrow \infty$, implies that $\{U(t,\tau _{n_{k}}^{(k)})x_{n_{k}}^{(k)}\}_{k=1}^{\infty }$ is a Cauchy sequence in $\{X_{t}\}_{t\in \mathbb{R}}$. This shows that $\{U(t,\tau _{n})x_{n}\}_{n=1}^{\infty }$ is precompact in $\{X_{t}\}_{t\in \mathbb{R}}$. □

Similarly to Theorem 3.3 in [19], we have the following conclusion, which will be used to verify the existence of the time-dependent global attractor.

Theorem 2.11

Let$\{U(t,\tau )\}_{t\geq \tau }$be a process on Banach space$\{X_{t}\}_{t\in \mathbb{R}}$, then$\{U(t,\tau )\}_{t\geq \tau }$has a time-dependent global attractor in$\{X_{t}\}_{t\in \mathbb{R}}$if the following conditions hold:

• (i) $\{U(t,\tau )\}_{t\geq \tau }$has a pullback absorbing set$\hat{B}=\{B_{t}\}_{t\in \mathbb{R}}$in$\{X_{t}\}_{t\in \mathbb{R}}$;

• (ii) $\{U(t,\tau )\}_{t\geq \tau }$is pullback asymptotically compact in$\hat{B}=\{B_{t}\}_{t\in \mathbb{R}}$.

3 Time-dependent global attractors

In this section, we will establish the existence of the time-dependent global attractors.

3.1 Existence and uniqueness of solutions

In this subsection, we consider the well-posedness of the solutions for Eq. (1.1) with (1.4)–(1.5). At first, we define the weak solutions as follows.

Definition 3.1

A weak solution of Eq. (1.1) is a function $u\in C([\tau, T]; \mathcal{H}_{t})\cap L^{2}(\tau, T; H_{0}^{1}( \varOmega ))\cap L^{p}(\tau, T; L^{p}(\varOmega ))$ for all $T>\tau$, with $u(\tau )=u_{\tau }$ and such that, for all $\varphi \in H_{0}^{1}(\varOmega )$, it satisfies

$$\begin{aligned} &\frac{d}{dt}\bigl[\bigl(u(t),\varphi \bigr)+\varepsilon (t) \bigl(\nabla u(t),\nabla \varphi \bigr)\bigr] +\bigl(1-\varepsilon '(t)\bigr) \bigl(\nabla u(t),\nabla \varphi \bigr)+\bigl(f\bigl(u(t)\bigr), \varphi \bigr) \\ &\quad =\bigl(g(x),\varphi \bigr),\quad \text{in } \mathcal{D}'(\tau,+\infty ). \end{aligned}$$

Remark 3.2

We notice that, if $u(t)$ is a weak solution of Eq. (1.1), then it satisfies the energy equality

$$\begin{aligned} & \bigl\Vert u(t) \bigr\Vert _{2}^{2}+\varepsilon (t) \bigl\Vert \nabla u(t) \bigr\Vert _{2}^{2}+ \int _{s}^{t}\bigl(2- \varepsilon '(r) \bigr) \bigl\Vert \nabla u(r) \bigr\Vert _{2}^{2}\,dr +2 \int _{s}^{t}\bigl(f\bigl(u(r)\bigr),u(r)\bigr)\,dr \\ &\quad= \bigl\Vert u(s) \bigr\Vert _{2}^{2}+\varepsilon (s) \bigl\Vert \nabla u(s) \bigr\Vert _{2}^{2} +2 \int _{s}^{t}\bigl(g(r),u(r)\bigr)\,dr \quad\text{for all } \tau \leq s\leq t. \end{aligned}$$

The following theorem gives the existence of the weak solutions, which is similar to that in [10] and can be obtained by the Faedo–Galerkin methods.

Theorem 3.3

Letfsatisfy (1.4)–(1.5), $g\in H^{-1}(\varOmega )$and$u_{\tau }\in \mathcal{H}_{\tau }$. Then, for any$\tau \in \mathbb{R}$and$t>\tau$, there exists a weak solution$u(t)$to Eq. (1.1), which satisfies$u\in C([\tau,t]; \mathcal{H}_{t})\cap L^{2}(\tau,t;H_{0}^{1}( \varOmega )) \cap L^{p}(\tau,t; L^{p}(\varOmega ))$, $u_{t}\in L^{2}(\tau,t;\mathcal{H}_{t})$.

Proof

Let $\{w_{j}\}_{j\geq 1}\subset H_{0}^{1}(\varOmega )\cap L^{p}(\varOmega )$ be a Hilbert basis of $L^{2}(\varOmega )$ such that $\operatorname{span} \{w_{j}\}_{j\geq 1}$ is dense in $H_{0}^{1}(\varOmega )\cap L^{p}(\varOmega )$. In order to establish the existence of the weak solutions, we need the approximate system for any $m\geq n$ seeking $\tilde{u}^{m}(t, x)=\varSigma _{j=1}^{m}\gamma _{mj}(t)\omega _{j}(x)$ that satisfies

$$\begin{aligned} \textstyle\begin{cases} \frac{d}{dt}[(\tilde{u}^{m}(t), \omega _{j})+\varepsilon (t)(\nabla \tilde{u}^{m}(t),\nabla \omega _{j})] +(1-\varepsilon '(t))(\nabla \tilde{u}^{m}(t), \nabla \omega _{j}) +(f(\tilde{u}^{m}(t)),\omega _{j})\\ \quad=(g(x), \omega _{j}), \\ \tilde{u}_{\tau }^{m}=u_{\tau }, \end{cases}\displaystyle \end{aligned}$$

for a.e. $t>\tau, 1\leq j\leq m$.

We will provide a priori estimates that show that these solutions are well-defined in the interval $[\tau,t]$ for any $t>\tau$.

Step 1: First a priori estimates. Multiplying each equation in the above system by $\gamma _{mj}(t)$, respectively, and summing from $j=1$ to m, we obtain

$$\begin{aligned} &\frac{1}{2}\frac{d}{dt} \bigl( \bigl\Vert \tilde{u}^{m}(t) \bigr\Vert _{2}^{2} + \varepsilon (t) \bigl\Vert \nabla \tilde{u}^{m}(t) \bigr\Vert _{2}^{2} \bigr) + \bigl(1- \varepsilon '(t)\bigr) \bigl\Vert \nabla \tilde{u}^{m}(t) \bigr\Vert _{2}^{2} \\ &\quad{} +\bigl(f\bigl(\tilde{u}^{m}(t)\bigr),\tilde{u}^{m}(t) \bigr)=\bigl(g(x),\tilde{u}^{m}(t)\bigr) \leq \frac{1}{2} \Vert g \Vert _{H^{-1}}^{2} +\frac{1}{2} \bigl\Vert \nabla \tilde{u}^{m}(t) \bigr\Vert _{2}^{2},\quad \text{a.e. }t> \tau, \end{aligned}$$

where we have used the Hölder and Young inequalities.

Furthermore, by (1.5), we know that

$$\begin{aligned} &\frac{d}{dt} \bigl( \bigl\Vert \tilde{u}^{m}(t) \bigr\Vert _{2}^{2} +\varepsilon (t) \bigl\Vert \nabla \tilde{u}^{m}(t) \bigr\Vert _{2}^{2} \bigr)+ \bigl(1-2\varepsilon '(t)\bigr) \bigl\Vert \nabla \tilde{u}^{m}(t) \bigr\Vert _{2}^{2}+2c_{1} \bigl\Vert \tilde{u}^{m}(t) \bigr\Vert _{p}^{p} \\ &\quad\leq 2c_{0} \vert \varOmega \vert + \Vert g \Vert _{H^{-1}}^{2},\quad \text{a.e. } t>\tau. \end{aligned}$$

Integrating it in $[\tau,t]$, we have

$$\begin{aligned} & \bigl\Vert \tilde{u}^{m}(t) \bigr\Vert _{2}^{2}+ \varepsilon (t) \bigl\Vert \nabla \tilde{u}^{m}(t) \bigr\Vert _{2}^{2} + \int _{\tau }^{t}\bigl(1-2\varepsilon '(s) \bigr) \bigl\Vert \nabla \tilde{u}^{m}(s) \bigr\Vert _{2}^{2}\,ds +2c_{1} \int _{\tau }^{t} \bigl\Vert \tilde{u}^{m}(s) \bigr\Vert _{p}^{p}\,ds \\ & \quad\leq \bigl\Vert \tilde{u}^{m}(\tau ) \bigr\Vert _{2}^{2}+\varepsilon (\tau ) \bigl\Vert \nabla \tilde{u}^{m}(\tau ) \bigr\Vert _{2}^{2} + \bigl(2c_{0} \vert \varOmega \vert + \Vert g \Vert _{H^{-1}}^{2}\bigr) (t- \tau )\quad \text{for all } t\geq \tau. \end{aligned}$$

Hence,

$$\begin{aligned} & \bigl\Vert \tilde{u}^{m}(t) \bigr\Vert _{2}^{2}+\varepsilon (t) \bigl\Vert \nabla \tilde{u}^{m}(t) \bigr\Vert _{2}^{2} + \int _{\tau }^{t} \bigl\Vert \nabla \tilde{u}^{m}(s) \bigr\Vert _{2}^{2}\,ds +2c_{1} \int _{\tau }^{t} \bigl\Vert \tilde{u}^{m}(s) \bigr\Vert _{p}^{p}\,ds \\ & \quad\leq \bigl\Vert \tilde{u}^{m}(\tau ) \bigr\Vert _{2}^{2}+\varepsilon (\tau ) \bigl\Vert \nabla \tilde{u}^{m}(\tau ) \bigr\Vert _{2}^{2} + \bigl(2c_{0} \vert \varOmega \vert + \Vert g \Vert _{H^{-1}}^{2}\bigr) (t- \tau )\quad \text{for all } t\geq \tau. \end{aligned}$$

(3.1)

So, from (3.1), we can get

$$\begin{aligned} \bigl\{ \tilde{u}^{m}\bigr\} _{m\geq n}\quad \text{is bounded in } L^{\infty }(\tau,t; \mathcal{H}_{t})\cap L^{2}\bigl(\tau,t;H_{0}^{1}(\varOmega )\bigr)\cap L^{p}\bigl( \tau,t;L^{p}(\varOmega )\bigr) \end{aligned}$$

(3.2)

for all $t>\tau$.

Moreover, combining with (1.5) and (3.2), we obtain

$$\begin{aligned} \bigl\{ f\bigl(\tilde{u}^{m}\bigr)\bigr\} _{m\geq n} \quad\text{is bounded in } L^{q}\bigl(\tau,t;L^{q}( \varOmega )\bigr) \text{ for all } t>\tau, \end{aligned}$$

where $q=p/(p-1)$.

Then there exist functions $\tilde{u}\in L^{\infty }(\tau,t; \mathcal{H}_{t}) \cap L^{2}(\tau,t;H_{0}^{1}( \varOmega ))\cap L^{p}(\tau,t;L^{p}(\varOmega ))$ and $\tilde{\chi }\in L^{q}(\tau, t; L^{q}(\varOmega ))$ for all $t>\tau$, and a subsequence such that

$$\begin{aligned} \textstyle\begin{cases} \tilde{u}^{m}\rightarrow \tilde{u} &\text{weakly-star in } L^{ \infty }(\tau,t;\mathcal{H}_{t}), \\ \tilde{u}^{m}\rightarrow \tilde{u} &\text{weakly in } L^{2} (\tau, t; H_{0}^{1}( \varOmega )), \\ \tilde{u}^{m}\rightarrow \tilde{u} &\text{weakly in } L^{p}(\tau, t; L^{p}( \varOmega )), \\ f(\tilde{u}^{m})\rightarrow \tilde{\chi } &\text{weakly in } L^{q}( \tau, t; L^{q}(\varOmega )). \end{cases}\displaystyle \end{aligned}$$

(3.3)

Step 2: Uniform estimate for the time derivatives. Multiplying each equation of the approximate system by $\gamma '_{mj}(t)$ and summing from $j=1$ to m, we arrive at

$$\begin{aligned} &\bigl\Vert \bigl(\tilde{u}^{m}\bigr)'(t) \bigr\Vert _{2}^{2}+\varepsilon (t) \bigl\Vert \bigl(\nabla \tilde{u}^{m}\bigr)'(t) \bigr\Vert _{2}^{2} +\frac{1}{2}\frac{d}{dt} \bigl\Vert \nabla \tilde{u}^{m}(t) \bigr\Vert _{2}^{2} \\ &\quad{}+\bigl(f\bigl(\tilde{u}^{m}\bigr),\bigl(\tilde{u}^{m} \bigr)'(t)\bigr)=\bigl(g(x),\bigl(\tilde{u}^{m} \bigr)'(t)\bigr),\quad \text{a.e. } t>\tau. \end{aligned}$$

By the Hölder and Young inequalities, we have

$$\begin{aligned} &\bigl\Vert \bigl(\tilde{u}^{m}\bigr)'(t) \bigr\Vert _{2}^{2}+2\varepsilon (t) \bigl\Vert \bigl(\nabla \tilde{u}^{m}\bigr)'(t) \bigr\Vert _{2}^{2} +\frac{d}{dt} \bigl\Vert \nabla \tilde{u}^{m}(t) \bigr\Vert _{2}^{2} \\ &\quad +2\frac{d}{dt} \int _{\varOmega }\mathcal{F}\bigl(\tilde{u}^{m}(t,x)\bigr) \,dx \leq \Vert g \Vert _{2}^{2},\quad \text{a.e. }t>\tau. \end{aligned}$$

Integrating it from τ to t, and from (1.6) we can get

$$\begin{aligned} & \bigl\Vert \nabla \tilde{u}^{m}(t) \bigr\Vert _{2}^{2} +2\tilde{c}_{1} \bigl\Vert \tilde{u}^{m}(t) \bigr\Vert _{p}^{p}+ \int _{\tau }^{t} \bigl( \bigl\Vert \bigl( \tilde{u}^{m}\bigr)'(s) \bigr\Vert _{2}^{2} + \varepsilon (t) \bigl\Vert \bigl(\nabla \tilde{u}^{m} \bigr)'(s) \bigr\Vert _{2}^{2} \bigr)\,ds \\ &\quad \leq 4\tilde{c}_{0} \vert \varOmega \vert + \bigl\Vert \nabla \tilde{u}^{m}(\tau ) \bigr\Vert _{2}^{2} +2 \tilde{c}_{2} \bigl\Vert \tilde{u}^{m}(\tau ) \bigr\Vert _{p}^{p}+ \Vert g \Vert _{2}^{2}(t- \tau ) \end{aligned}$$

(3.4)

for all $t\geq \tau$ and any $m\geq n$.

Since $\tilde{u}_{\tau }^{m}=u_{\tau }$ for all $m\geq n$ and $\tilde{u}^{m}_{\tau }\in H_{0}^{1}(\varOmega )\cap L^{p}(\varOmega )$, by (3.4), we obtain

$$\begin{aligned} \bigl\{ \tilde{u}^{m}(t)\bigr\} _{m\geq n} \quad\text{is bounded in } L^{\infty }\bigl(\tau, t; H_{0}^{1}( \varOmega )\cap L^{p}(\varOmega )\bigr) \end{aligned}$$

(3.5)

and

$$\begin{aligned} \bigl\{ \bigl(\tilde{u}^{m}\bigr)'(t)\bigr\} _{m\geq n}\quad \text{is bounded in } L^{2}(\tau, t; \mathcal{H}_{t}) \end{aligned}$$

(3.6)

for all $t>\tau$. Then there exist functions $\tilde{u}\in L^{\infty }(\tau, t; H_{0}^{1}(\varOmega ) \cap L^{p}( \varOmega ))$ and $\tilde{u}_{t}\in L^{2}(\tau, t; \mathcal{H}_{t})$ for all $t>\tau$, which improve the regularity of ũ obtained in Step 1.

For any fixed $t>\tau$, since

$$\begin{aligned} \bigl\Vert \tilde{u}^{m}(t_{2})- \tilde{u}^{m}(t_{1}) \bigr\Vert _{\mathcal{H}_{t}}^{2} &= \bigl\Vert \tilde{u}^{m}(t_{2})-\tilde{u}^{m}(t_{1}) \bigr\Vert _{2}^{2}+\varepsilon (t) \bigl\Vert \nabla \tilde{u}^{m}(t_{2})-\nabla \tilde{u}^{m}(t_{1}) \bigr\Vert _{2}^{2} \\ &= \biggl\Vert \int _{t_{1}}^{t_{2}}\bigl(\tilde{u}^{m} \bigr)'(s)\,ds \biggr\Vert _{2}^{2} + \varepsilon (t) \biggl\Vert \int _{t_{1}}^{t_{2}}\bigl(\nabla \tilde{u}^{m} \bigr)'(s)\,ds \biggr\Vert _{2}^{2} \\ &\leq \bigl( \bigl\Vert \bigl(\tilde{u}^{m}\bigr)' \bigr\Vert _{L^{2}(\tau,t;L^{2}(\varOmega ))}^{2} + \varepsilon (t) \bigl\Vert \bigl( \nabla \tilde{u}^{m}\bigr)' \bigr\Vert _{L^{2}(\tau,t;L^{2}( \varOmega ))}^{2}\bigr) \vert t_{2}-t_{1} \vert \\ &= \bigl\Vert \bigl(\tilde{u}^{m}\bigr)' \bigr\Vert _{L^{2}(\tau,t;\mathcal{H}_{t})}^{2} \vert t_{2}-t_{1} \vert , \end{aligned}$$

(3.7)

for all $t_{1},t_{2}\in [\tau,t]$, from (3.5), (3.6) and (3.7), by the Ascoli–Arzelà Theorem, and taking into account the initial data for all the sequence, we deduce that there is a subsequence such that

$$\begin{aligned} \tilde{u}^{m}\rightarrow \tilde{u} \quad\text{in } C\bigl([ \tau,t]; \mathcal{H}_{t}\bigr) \end{aligned}$$

(3.8)

for all $t>\tau$ and a.e. in $\varOmega \times (\tau,\infty )$.

Since $f\in C(\mathbb{R}, \mathbb{R})$, we conclude that $f(\tilde{u}^{m})\rightarrow f(\tilde{u}) a.e$. in $\varOmega \times (\tau, \infty )$. So, combining with (3.3) and [15] (Lemma 1.3, p. 12) we obtain $\tilde{\chi }=f(\tilde{u})$.

Thus, together with (3.3) and (3.8), by taking the limit in the equations satisfied by $\{\tilde{u}^{m}\}$ and, thanks to the fact that $\operatorname{span} \{\omega _{j}\}_{j\geq 1}$ is dense in $H_{0}^{1}(\varOmega )\cap L^{p}(\varOmega )$, we conclude that ũ is a weak solution of Eq. (1.1).

Step 3: Proof of the general statement by density. For each $n\in \mathbb{N}$, we define $u^{n}_{\tau }=\varSigma _{j=1}^{n}(u_{\tau }, \omega _{j})\omega _{j}$. (Due to the fact that $\{\omega _{j}\}_{j\geq 1}$ is a Hilbert basis of $L^{2}(\varOmega )$, it is easy to check that $u^{n}_{\tau }\rightarrow u_{\tau }$ in $\mathcal{H}_{\tau }$.)

Let also consider a sequence $\{g^{n}\}_{n=1}^{\infty }\subset L^{2}(\varOmega )$ converging to $g\in H^{-1}(\varOmega )$.

Denote by $u^{n}$ the corresponding solution to Eq. (1.1) with g replaced by $g^{n}$ and initial data $u_{\tau }^{n}$.

Then, by the energy equality for each $u^{n}$, we have

$$\begin{aligned} & \bigl\Vert u^{n}(t) \bigr\Vert _{2}^{2}+ \varepsilon (t) \bigl\Vert \nabla u^{n}(t) \bigr\Vert _{2}^{2} +2 \int _{\tau }^{t} \bigl\Vert \nabla u^{n}(s) \bigr\Vert _{2}^{2}\,ds+2 \int _{\tau }^{t}\bigl(f\bigl(u^{n}(s)\bigr), u^{n}(s)\bigr)\,ds \\ &\quad = \bigl\Vert u^{n}(\tau ) \bigr\Vert _{2}^{2}+ \varepsilon (\tau ) \bigl\Vert \nabla u^{n}(\tau ) \bigr\Vert _{2}^{2} +2 \int _{\tau }^{t}\bigl(g^{n}(x), u^{n}(s)\bigr)\,ds,\quad \forall t\geq \tau. \end{aligned}$$

Similar to the reasoning process in Step 1, we get

$$\begin{aligned} \bigl\{ u^{n}\bigr\} \quad \text{is bounded in } L^{\infty }(\tau,t;\mathcal{H}_{t}) \cap L^{2}\bigl( \tau,t;H_{0}^{1}(\varOmega )\bigr) \cap L^{p}\bigl( \tau,t;L^{p}( \varOmega )\bigr) \end{aligned}$$

(3.9)

for all $t>\tau$.

Now, combining with (1.5) and (3.9), we see that $\{f(u^{n})\}$ is bounded in $L^{q}(\tau,t;L^{q}(\varOmega ))$ for all $t>\tau$.

Therefore, there exist functions $u\in L^{\infty }(\tau,t;\mathcal{H}_{t})\cap L^{2}(\tau,t;H_{0}^{1}( \varOmega )) \cap L^{p}(\tau,t;L^{p}(\varOmega ))$ and $\chi \in L^{q}(\tau,t;L^{q}(\varOmega ))$ for all $t>\tau$, and a subsequence such that

$$\begin{aligned} \textstyle\begin{cases} u^{n}\rightarrow u &\text{weakly-star in } L^{\infty }( \tau, t; \mathcal{H}_{t}), \\ u^{n}\rightarrow u &\text{weakly in } L^{2}(\tau, t; H_{0}^{1}(\varOmega )), \\ u^{n}\rightarrow u &\text{weakly in } L^{p}(\tau, t; L^{p}(\varOmega )), \\ f(u^{n})\rightarrow \chi &\text{weakly in } L^{q}(\tau, t; L^{q}( \varOmega )), \end{cases}\displaystyle \end{aligned}$$

(3.10)

for all $t> \tau$.

Moreover, we may improve some of the above convergence. Taking into account the energy equality for $u^{n}-u^{m}$, we have

$$\begin{aligned} &\bigl\Vert u^{n}(t)-u^{m}(t) \bigr\Vert _{2}^{2}+\varepsilon (t) \bigl\Vert \nabla u^{n}(t)- \nabla u^{m}(t) \bigr\Vert _{2}^{2} + \int _{\tau }^{t} \bigl\Vert \nabla u^{n}(s)-\nabla u^{m}(s) \bigr\Vert _{2}^{2} \,ds \\ &\quad \leq \bigl\Vert u^{n}(\tau )-u^{m}(\tau ) \bigr\Vert _{2}^{2} +\varepsilon (\tau ) \bigl\Vert \nabla u^{n}(\tau )-\nabla u^{m}(\tau ) \bigr\Vert _{2}^{2} +2l \int _{\tau }^{t} \bigl\Vert u^{n}(s)-u^{m}(s) \bigr\Vert _{2}^{2}\,ds \\ &\qquad{}+ \bigl\Vert g^{n}-g^{m} \bigr\Vert _{H^{-1}}^{2}(t-\tau ),\quad \forall t\geq \tau. \end{aligned}$$

(3.11)

By (3.11), we know that

$$\begin{aligned} \bigl\{ u^{n}\bigr\} \quad \text{is a Cauchy sequence in } C\bigl([\tau,t]; \mathcal{H}_{t}\bigr) \cap L^{2}\bigl(\tau,t; H_{0}^{1}(\varOmega )\bigr) \text{ for all } t>\tau. \end{aligned}$$

Thus, we have $u^{n}\rightarrow u$ a.e. in $\varOmega \times (\tau, \infty )$.

Therefore, as before, combining with (3.10) and [15] (Lemma 1.3, p. 12) we obtain $\chi =f(u)$; and from (3.10) we may take the limit in the equations satisfied by $u^{n}$ and conclude that u is a weak solution of Eq. (1.1). □

For the solutions of Eq. (1.1), the following theorem shows the uniqueness and continuity with respect to initial data.

Theorem 3.4

Letfsatisfy (1.4)–(1.5), $g\in H^{-1}(\varOmega )$and$u_{\tau }\in \mathcal{H}_{\tau }$, then the weak solution of Eq. (1.1) is unique. Moreover, for every two solutions$u^{1}(t)$and$u^{2}(t)$ (with different initial data), the following Lipschitz continuity holds:

$$\begin{aligned} \bigl\Vert \omega (t) \bigr\Vert _{2}^{2}+\varepsilon (t) \bigl\Vert \nabla \omega (t) \bigr\Vert _{2}^{2} \leq \bigl( \Vert \omega _{\tau } \Vert _{2}^{2}+ \varepsilon (\tau ) \Vert \nabla \omega _{ \tau } \Vert _{2}^{2}\bigr)e^{2l(t-\tau )}, \quad\forall t\geq \tau, \end{aligned}$$

where$\omega (t)=u^{1}(t)-u^{2}(t)$.

Proof

Let $\omega (t)=u^{1}(t)-u^{2}(t)$, then $\omega (t)$ satisfies the following equation:

$$\begin{aligned} \textstyle\begin{cases} \omega _{t}-\varepsilon (t)\Delta \omega _{t}-\Delta \omega =f(u^{1})-f(u^{2}) &\text{in } \varOmega \times (\tau,\infty ), \\ \omega (x, t)=0 &\text{on } \partial \varOmega \times (\tau,\infty ), \\ \omega (x, \tau )=u^{1}_{\tau }-u^{2}_{\tau }, &x\in \varOmega. \end{cases}\displaystyle \end{aligned}$$

(3.12)

Taking the $L^{2}$-inner product between (3.12) and ω, and using (1.4), we have

$$\begin{aligned} \frac{d}{dt}\bigl( \Vert \omega \Vert _{2}^{2}+ \varepsilon (t) \Vert \nabla \omega \Vert _{2}^{2}\bigr) + \bigl(2-\varepsilon '(t)\bigr) \Vert \nabla \omega \Vert _{2}^{2} \leq 2l \Vert \omega \Vert _{2}^{2}. \end{aligned}$$

Then

$$\begin{aligned} \frac{d}{dt}\bigl( \Vert \omega \Vert _{2}^{2}+ \varepsilon (t) \Vert \nabla \omega \Vert _{2}^{2}\bigr) \leq 2l\bigl( \Vert \omega \Vert _{2}^{2}+\varepsilon (t) \Vert \nabla \omega \Vert _{2}^{2}\bigr). \end{aligned}$$

By the Gronwall lemma, it yields

$$\begin{aligned} \bigl\Vert \omega (t) \bigr\Vert _{2}^{2}+\varepsilon (t) \bigl\Vert \nabla \omega (t) \bigr\Vert _{2}^{2} \leq \bigl( \Vert \omega _{\tau } \Vert _{2}^{2}+ \varepsilon (\tau ) \Vert \nabla \omega _{ \tau } \Vert _{2}^{2}\bigr)e^{2l(t-\tau )}, \end{aligned}$$

and the uniqueness holds. □

Thus, we define the solution processes $\{U(t,\tau )\}_{t\geq \tau }$ in the spaces $\mathcal{H}_{t}$ as:

$$\begin{aligned} U(t,\tau ): \mathcal{H}_{\tau }\rightarrow \mathcal{H}_{t},\qquad U(t, \tau )u_{\tau }=u(t),\quad \forall t\geq \tau. \end{aligned}$$

(3.13)

Moreover, Theorem 3.4 shows that the process $\{U(t,\tau )\}_{t\geq \tau }$ is Lipschitz in $\mathcal{H}_{t}$:

$$ \bigl\Vert U(t,\tau )u_{\tau }^{1}-U(t,\tau )u_{\tau }^{2} \bigr\Vert _{\mathcal{H}_{t}} \leq \bigl\Vert u_{\tau }^{1}-u_{\tau }^{2} \bigr\Vert _{\mathcal{H}_{\tau }}e^{2l(t- \tau )}, \quad\forall t\geq \tau. $$

3.2 Time-dependent global attractors

In this subsection, we will verify the existence of the time-dependent global attractors in $\mathcal{H}_{t}$ for the process $\{U(t, \tau )\}_{t\geq \tau }$ defined by (3.13).

3.2.1 Time-dependent absorbing sets

In the following, we will obtain the time-dependent global absorbing sets.

Lemma 3.5

Letfsatisfy (1.4)–(1.5), $g\in H^{-1}(\varOmega )$and$u_{\tau }\in \mathbb{B}_{\tau }(R)\subset \mathcal{H}_{\tau }$. Then there exists a$R_{0}>0$such that the family$\hat{B}=\{B_{t}(R_{0})\}_{t\in \mathbb{R}}$is a time-dependent absorbing set for the process$\{U(t, \tau )\}_{t\geq \tau }$.

Proof

Multiplying (1.1) by $u(t)$ and integrating over $x\in \varOmega$, we arrive at

$$\begin{aligned} \frac{1}{2}\frac{d}{dt}\bigl( \Vert u \Vert _{2}^{2}+ \varepsilon (t) \Vert \nabla u \Vert _{2}^{2}\bigr) + \biggl(1-\frac{1}{2}\varepsilon '(t)\biggr) \Vert \nabla u \Vert _{2}^{2}+\bigl(f(u), u\bigr)= \bigl\langle g(x), u \bigr\rangle . \end{aligned}$$

Thanks to (1.5) and the Hölder inequality, we have

$$\begin{aligned} \frac{d}{dt}\bigl( \Vert u \Vert _{2}^{2}+ \varepsilon (t) \Vert \nabla u \Vert _{2}^{2}\bigr) + \bigl(1- \varepsilon '(t)\bigr) \Vert \nabla u \Vert _{2}^{2}+2c_{1} \Vert u \Vert _{p}^{p}\leq 2c_{0} \vert \varOmega \vert + \Vert g \Vert _{H^{-1}}^{2}. \end{aligned}$$

Furthermore, by (1.3), we can get

$$\begin{aligned} \frac{d}{dt}\bigl( \Vert u \Vert _{2}^{2}+ \varepsilon (t) \Vert \nabla u \Vert _{2}^{2}\bigr) + \frac{1}{1+L}\bigl(\lambda _{1} \Vert u \Vert _{2}^{2}+\varepsilon (t) \Vert \nabla u \Vert _{2}^{2}\bigr) \leq 2c_{0} \vert \varOmega \vert + \Vert g \Vert _{H^{-1}}^{2}. \end{aligned}$$

Setting $\lambda =\min \{\lambda _{1},1\}$ and $\beta =\frac{\lambda }{1+L}$, we deduce that

$$\begin{aligned} \frac{d}{dt}\bigl( \Vert u \Vert _{2}^{2}+ \varepsilon (t) \Vert \nabla u \Vert _{2}^{2}\bigr) + \beta \bigl( \Vert u \Vert _{2}^{2}+\varepsilon (t) \Vert \nabla u \Vert _{2}^{2}\bigr)\leq 2c_{0} \vert \varOmega \vert + \Vert g \Vert _{H^{-1}}^{2}. \end{aligned}$$

(3.14)

Multiplying (3.14) by $e^{\beta t}$ and integrating it in $[\tau, t]$, we obtain

$$\begin{aligned} &\bigl( \bigl\Vert u(t) \bigr\Vert _{2}^{2}+\varepsilon (t) \bigl\Vert \nabla u(t) \bigr\Vert _{2}^{2} \bigr)e^{\beta t} \\ & \quad\leq \bigl( \bigl\Vert u(\tau ) \bigr\Vert _{2}^{2}+ \varepsilon (\tau ) \bigl\Vert \nabla u(\tau ) \bigr\Vert _{2}^{2} \bigr)e^{\beta \tau } +\bigl(2c_{0} \vert \varOmega \vert + \Vert g \Vert _{H^{-1}}^{2}\bigr) \int _{ \tau }^{t}e^{\beta s}\,ds,\quad \forall t\geq \tau. \end{aligned}$$

Therefore,

$$\begin{aligned} \bigl( \bigl\Vert u(t) \bigr\Vert _{2}^{2}+\varepsilon (t) \bigl\Vert \nabla u(t) \bigr\Vert _{2}^{2}\bigr) \leq {}&\bigl( \bigl\Vert u( \tau ) \bigr\Vert _{2}^{2}+ \varepsilon (\tau ) \bigl\Vert \nabla u(\tau ) \bigr\Vert _{2}^{2} \bigr)e^{- \beta (t-\tau )} +\frac{1}{\beta }\bigl(2c_{0} \vert \varOmega \vert + \Vert g \Vert _{H^{-1}}^{2}\bigr) \\ \leq {}&1+\frac{1}{\beta }\bigl(2c_{0} \vert \varOmega \vert + \Vert g \Vert _{H^{-1}}^{2}\bigr)=R_{0}, \end{aligned}$$

provided that $t-\tau \geq t_{0}$ with $t_{0}=\frac{1}{\beta }\ln (\|u_{\tau }\|_{2}^{2}+\varepsilon (\tau ) \|\nabla u_{\tau }\|_{2}^{2})$, from which we obtain the existence of the time-dependent absorbing set. □

3.2.2 Time-dependent global attractors

At first, we have the following lemma, which is similar to that in [15].

Lemma 3.6

Letfsatisfy (1.4)–(1.5), $g\in H^{-1}(\varOmega )$, $u_{\tau }\in \mathcal{H}_{\tau }$and$\{u^{n}(t)\}_{n=1}^{\infty }$be a sequence of solutions for Eq. (1.1) with initial data$u^{n}_{\tau }\in \mathcal{H}_{\tau }$$(n=1,2,\ldots )$, then there exists a subsequence of$\{u^{n}(t)\}_{n=1}^{\infty }$that converges strongly in$L^{2}(\tau,t; L^{2}(\varOmega ))$.

Proof

By (1.5) and Theorem 3.3, we know that there exists a sequence $\{u^{n}(t)\}_{n=1}^{\infty }\subset L^{2}(\tau,T;H_{0}^{1}(\varOmega ))$, $\{f(u^{n}(t))\}_{n=1}^{\infty }\subset L^{q}(\tau,T;L^{q}(\varOmega ))$. Then, from Eq. (1.1), we obtain $\partial _{t}u^{n}-\varepsilon (t)\partial _{t}\Delta u^{n}=\Delta u^{n}-f(u^{n})+g(x) \in L^{2}(\tau,T;H^{-1}(\varOmega ))+L^{q}(\tau,T;L^{q}(\varOmega )) \subset L^{2}(\tau,T;H^{-2}(\varOmega ))$. By the regularization theory for elliptic equations, we know that $\partial _{t}u^{n}\in L^{2}(\tau,T;L^{2}(\varOmega ))$. As in [15], there exists a subsequence of $\{u^{n}(t)\}_{n=1}^{\infty }$ (still denoted by $\{u^{n}(t)\}_{n=1}^{\infty }$) that converges strongly in $L^{2}(\tau,T; L^{2}(\varOmega ))$. □

Then we have the following theorem, which will obtain the pullback asymptotic compactness for the process $\{U(t,\tau )\}_{t\geq \tau }$ defined by (3.13).

Theorem 3.7

Letfsatisfy (1.4)–(1.5), $g\in H^{-1}(\varOmega )$and$u_{\tau }\in \mathbb{B}_{\tau }(R)\subset \mathcal{H}_{\tau }$, then$\{U(t, \tau )\}_{t\geq \tau }$is pullback asymptotically compact in$\mathcal{H}_{t}$.

Proof

Let $u^{i}(t)$ ($i=1,2$) be the solutions corresponding to initial data $u^{i}_{\tau}\in\mathbb{B}_{\tau}(R)\subset\mathcal{H}_{\tau}$, that is, $u^{i}(t)$ satisfies the following equation:

$$\begin{aligned} u_{t}-\varepsilon (t)\Delta u_{t}-\Delta u+f(u)=g(x),\quad \text{in } \varOmega \times (\tau,\infty ), \end{aligned}$$

with initial data

$$\begin{aligned} u^{i}(x,\tau )=u^{i}_{\tau },\quad x\in \varOmega. \end{aligned}$$

Denoting $\omega (t)=u^{1}(t)-u^{2}(t)$, then $\omega (t)$ satisfies the following equation:

$$\begin{aligned} \omega _{t}-\varepsilon (t)\Delta \omega _{t}-\Delta \omega +f\bigl(u^{1}\bigr)-f \bigl(u^{2}\bigr)=0,\quad \text{in } \varOmega \times (\tau,\infty ), \end{aligned}$$

(3.15)

with initial data

$$\begin{aligned} \omega (x,\tau )=u^{1}_{\tau }-u^{2}_{\tau },\quad x\in \varOmega. \end{aligned}$$

Multiplying (3.15) by $\omega (t)$ and integrating it in Ω, then, by (1.4), we obtain

$$\begin{aligned} \frac{d}{dt}\bigl( \Vert \omega \Vert _{2}^{2}+ \varepsilon (t) \Vert \nabla \omega \Vert _{2}^{2}\bigr) + \bigl(2-\varepsilon '(t)\bigr) \Vert \nabla \omega \Vert _{2}^{2}\leq 2l \Vert \omega \Vert _{2}^{2}. \end{aligned}$$

By the Poincaré inequality, we have

$$\begin{aligned} \frac{d}{dt}\bigl( \Vert \omega \Vert _{2}^{2}+ \varepsilon (t) \Vert \nabla \omega \Vert _{2}^{2}\bigr) +\beta _{1}\bigl( \Vert \omega \Vert _{2}^{2}+ \varepsilon (t) \Vert \nabla \omega \Vert _{2}^{2}\bigr) \leq 2l \Vert \omega \Vert _{2}^{2}, \end{aligned}$$

where $\beta _{1}=2\beta$, β is given by (3.14).

Thanks to the Gronwall lemma, we get

$$\begin{aligned} &\bigl\Vert \omega (t) \bigr\Vert _{2}^{2}+\varepsilon (t) \bigl\Vert \nabla \omega (t) \bigr\Vert _{2}^{2}\\ &\quad \leq \bigl( \bigl\Vert \omega (\tau ) \bigr\Vert _{2}^{2}+ \varepsilon (\tau ) \bigl\Vert \nabla \omega ( \tau ) \bigr\Vert _{2}^{2}\bigr) e^{-\beta _{1}(t-\tau )}+2l \int _{\tau }^{t} \bigl\Vert \omega (s) \bigr\Vert _{2}^{2}\,ds,\quad \forall t\geq \tau. \end{aligned}$$

Setting

$$\begin{aligned} \psi_{\tau}^{t}(u_{\tau}^{1},u_{\tau}^{2})=&2l \int _{\tau }^{t} \bigl\Vert \omega (s) \bigr\Vert _{2}^{2}\,ds, \end{aligned}$$

combining with Definition 2.9 and Lemma 3.6, we know that $\psi_{\tau}^{t}(\cdot,\cdot)$ is a contractive function. Then, for any $\epsilon >0$ and any fixed $t\in \mathbb{R}$, let $\tau _{0}=t-\frac{1}{\beta _{1}}\ln \frac{\|w_{\tau }\|_{2}^{2} +\varepsilon (\tau )\|\nabla w_{\tau }\|_{2}^{2}}{\epsilon }$, we easily see that $\{U(t,\tau )\}_{t\geq \tau }$ is pullback asymptotically compact in $\mathcal{H}_{t}$ by Theorem 2.10. □

Combining with Lemma 3.5 and Theorem 3.7, we have the main result of this paper.

Theorem 3.8

Letfsatisfy (1.4)–(1.5), $g\in H^{-1}(\varOmega )$and$u_{\tau }\in \mathbb{B}_{\tau }(R)\subset \mathcal{H}_{\tau }$, then$\{U(t,\tau )\}_{t\geq \tau }$possesses a time-dependent global attractor$\hat{\mathcal{A}}=\{\mathcal{A}_{t}\}_{t\in \mathbb{R}}$in$\mathcal{H}_{t}$; that is, $\mathcal{A}_{t}$is compact, $\hat{\mathcal{A}}$is nonempty, invariant in$\mathcal{H}_{t}$and pullback attracts every bounded subset of$\mathcal{H}_{t}$with respect to the$\mathcal{H}_{t}$-norm.

Remark 3.9

In Theorem 3.8, we have obtained the time-dependent global attractor $\hat{\mathcal{A}}=\{\mathcal{A}_{t}\}_{t\in \mathbb{R}}$ in $\mathcal{H}_{t}$. From (1.2) we know that $\varepsilon (t)\rightarrow 0$ as $t\rightarrow +\infty$, then Eq. (1.1) becomes the classical reaction–diffusion equation $u_{t}-\Delta u+f(u)=g(x)$. An interesting question is about the limitation of $\mathcal{A}_{t}$ as $t\rightarrow +\infty$, that is, how to describe $\lim_{t\rightarrow +\infty }\mathcal{A}_{t}$? We will consider this problem in our next work.