Continuity and pullback attractors for a semilinear heat equation on time-varying domains

Abstract

We consider dynamics of a semilinear heat equation on time-varying domains with lower regular forcing term. Instead of requiring the forcing term $f(\cdot )$ to satisfy $\int _{-\infty}^{t}e^{\lambda s}\|f(s)\|^{2}_{L^{2}}\,ds<\infty$ for all $t\in \mathbb{R}$, we show that the solutions of a semilinear heat equation on time-varying domains are continuous with respect to initial data in $H^{1}$ topology and the usual $(L^{2},L^{2})$ pullback $\mathscr{D}_{\lambda}$-attractor indeed can attract in the $H^{1}$-norm, provided that $\int _{-\infty}^{t}e^{\lambda s}\|f(s)\|^{2}_{H^{-1}(\mathcal{O}_{s})}\,ds< \infty$ and $f\in L^{2}_{\mathrm{loc}}(\mathbb{R},L^{2}(\mathcal{O}_{s}))$.

1 Introduction

Let $\mathcal{O}$ be a nonempty bounded open subset of $\mathbb{R}^{N}$ with $C^{2}$ boundary $\partial \mathcal{O}$, and let $r=r(y,t)$ be a vector function

$$ r\in C^{1}\bigl(\overline{\mathcal{O}}\times \mathbb{R}; \mathbb{R}^{N}\bigr) $$

(1.1)

such that

$$ r(\cdot ,t): \mathcal{O}\to \mathcal{O}_{t}\quad \text{is a }C^{2}\text{-diffeomorphism for all }t\in \mathbb{R}. $$

(1.2)

We consider the following initial boundary value problem for a semilinear parabolic equation:

$$ \textstyle\begin{cases} u_{t}-\Delta u+g(u)=f(t)\quad \text{in }Q_{\tau}, \\ u=0\quad \text{on } \Sigma _{\tau}, \\ u(\tau ,x)=u_{\tau}(x),\quad x\in \mathcal{O}_{\tau}, \end{cases} $$

(1.3)

where $\tau \in \mathbb{R}$, $u_{\tau}:\mathcal{O}_{\tau} \rightarrow \mathbb{R}$, $Q_{\tau ,T}:=\bigcup_{t\in (\tau ,T)}\mathcal{O}_{t}\times \{t\}$ for all $T>\tau$, $Q_{\tau}:=\bigcup_{t\in (\tau ,+\infty )}\mathcal{O}_{t}\times \{t \}$, $\Sigma _{\tau ,T}:=\bigcup_{t\in (\tau ,T)}\partial \mathcal{O}_{t} \times \{t\}$, $\Sigma _{\tau}:=\bigcup_{t\in (\tau ,+\infty )} \partial \mathcal{O}_{t} \times \{t\}$, and $f:Q_{\tau}\rightarrow \mathbb{R}$ are given. We assume that $g\in C^{1}(\mathbb{R},\mathbb{R})$ is a given function for which there exist nonnegative constants $\alpha _{1}$, $\alpha _{2}$, β, l, and $p\geqslant 2$ such that

$$ -\beta +\alpha _{1} \vert s \vert ^{p} \leqslant g(s)s\leqslant \beta +\alpha _{2} \vert s \vert ^{p}, \qquad g'(s)\geqslant -l\quad \forall s\in \mathbb{R}\mathbbm{,} $$

(1.4)

and, moreover, g satisfies the Lipschitz condition: there exists a positive constant $c_{0}$ such that

$$ \bigl\vert g(u)-g(v) \bigr\vert \leqslant c_{0} \bigl(1+ \vert u \vert ^{p-2}+ \vert v \vert ^{p-2}\bigr) \cdot \vert u-v \vert \quad \forall u,v\in \mathbb{R}. $$

(1.5)

About the diffeomorphism $r(\cdot ,\cdot )$, as in Límacoet al. [6] and Kloedenet al. [5], we assume that the function $\bar{r}=\bar{r}(x,t)$, where $\bar{r}(\cdot ,t)=r^{-1}(\cdot ,t)$ denotes the inverse of $r(\cdot ,t)$, satisfies

$$ \bar{r}\in C^{2,1}\bigl({\bar{Q}}_{\tau ,T}; \mathbb{R}^{N}\bigr)\quad \text{for all }\tau < T. $$

(1.6)

The reaction–diffusion equation with nonlinear term $g(\cdot )$ satisfying assumptions (1.4) is one of the classical example models in the theory of infinite-dimensional dynamical systems, especially regarding to the theory of attractors; e.g., see the classical monographs in this field like [1, 8, 11].

About the dynamics of reaction–diffusion equation (1.3)–(1.4), the known results mainly concentrate in the $L^{2}$ phase space; e.g., see [8, 11] for the fixed domain case (i.e., $r(\cdot ,t)\equiv Id$) and Kloeden et al. [5, 13] for time-varying domain case; and the corresponding mathematical analysis is standard to some extent. When we try to improve the corresponding results to a more regular phase space, say $H^{1}$, some essential difficulties arise, for example, the continuity with respect to the initial data and asymptotical compactness in $H^{1}$ topology. Indeed, even in the autonomous case, for any space dimension N and any growth power $p\geqslant 2$ (comes from (1.4)), the question about the continuity of solution with respect to initial data in $H^{1}$ remained open until 2008; see Robinson [8]. In 2008, for the autonomous case of (1.3) and with the same assumption (1.4) about the nonlinearity, Trujillo and Wang [12] used the method of differentiating the equation with respect to t to get the bounded estimate for $\|tu_{t}\|_{L^{2}}$ for $t\in [0,T]$ and then obtained the uniform boundedness of $tu(t)$ in $L^{\infty}(0, T ; H^{2})$ and, finally, obtained the continuity in $H^{1}$ for any space dimension N and any growth power $p\geqslant 2$ (to our knowledge, this is the first result). Later, Cao et al. [2] obtained such continuity for nonautonomous case by establishing some new a priori estimates for the difference of solutions near the initial time; see also [3, 13] for further discussion in this direction.

Note that to obtain the continuity with respect to the initial data and existence of attractors in the $H^{1}$ topology, to our knowledge, the known results always required the force term to belong to $L^{2}$; e.g., see [2, 3, 12] for autonomous and stochastic case; and in [13], to obtain similar results as in [2] in the nonautonomous case, they required $f(\cdot )$ to satisfy

$$ \int _{-\infty}^{t}e^{\lambda s} \bigl\Vert f(s) \bigr\Vert ^{2}_{L^{2}}\,ds< \infty \quad \text{for all }t\in \mathbb{R} $$

(1.7)

for some proper positive constant λ. On the other hand, it is well-known that when we consider system (1.3)–(1.4) in $H^{1}$, it is natural to require $f(\cdot )\in H^{-1}$ only.

The main aim of this paper is to establish the same continuity with respect to the initial data in the $H^{1}$ topology and $H^{1}$-attraction as that in [2, 3, 12, 13] and relax the assumption on the forcing term. To include the nonautonomous case, we consider systems (1.3)–(1.4) defined on a time-varying domain. Note that a semilinear heat equation on a time-varying domain is intrinsically nonautonomous even if the terms in the equation do not depend explicitly on time.

Assumption I

r and r̄ satisfy assumptions (1.1), (1.2), and (1.6); $\normalfont{\mathrm{\partial \mathcal{O}}}$ is $C^{2}$ and $N\leqslant 2p/(p-2)$, or $\partial \mathcal{O}$ is $C^{j}$ with $j\geqslant 2$ integer such that $j\geqslant N(p-2)/2p$; $g(\cdot )$ satisfies (1.4), and $f\in L^{2}_{\normalfont{\mathrm{loc}}}(\mathbb{R}; H^{-1}(\mathcal{O}_{t}))$.

Under Assumption I, the existence and uniqueness of strong solution and weak solution of (1.3) (see [5, 6] for the corresponding definition of solutions) were obtained by Kloeden et al. [5] and then defined the nonautonomous process $U(t,\tau ): L^{2}(\mathcal{O}_{\tau})\to L^{2}(\mathcal{O}_{t})$, $- \infty <\tau \leqslant t<\infty$ by $U(t,\tau )u_{\tau}:=u(t;\tau ,u_{\tau})=u(t)$. Moreover, if we assume further that r satisfies

$$ r\in C_{b}\bigl(\bar{\mathcal{O}}\times \mathbb{R}; \mathbb{R}^{N}\bigr) $$

(1.8)

and f satisfies

$$ \int _{-\infty}^{t}e^{\lambda s} \bigl\Vert f(s) \bigr\Vert ^{2}_{H^{-1}(\mathcal{O}_{s})}\,ds < \infty \quad \text{for all }t\in \mathbb{R}, $$

(1.9)

where $\lambda :=\min_{v\in H^{1}_{0}(\Omega ), v\ne 0} \frac{\|\nabla v\|^{2}_{(L^{2}(\Omega ))^{N}}}{\|v\|^{2}_{L^{2}(\Omega )}}$ is the first eigenvalue of −Δ on $H^{1}_{0}(\Omega )$ with $\Omega :=\bigcup_{t\in \mathbb{R}}\mathcal{O}_{t}$, then the process $U(t,\tau )$ has an $(L^{2},L^{2})$ pullback attractor $\hat{\mathscr{A}}=\{\mathscr{A}(t): t\in \mathbb{R}\}$; see [5] for more detail.

Our main result is the following theorem.

Theorem 1.1

Let Assumption I, (1.5), and (1.8)–(1.9) hold. If the forcing term $f\in L^{2}_{\mathrm{loc}}(\mathbb{R}, L^{2}(\mathcal{O}_{t}))$, then the process $U(t,\tau )$ is continuous with respect to the initial data in the $H^{1}$ topology; more precisely, for all $\tau \in \mathbb{R}$ and $t>\tau$, if $u_{n\tau}\in L^{2}(\mathcal{O}_{\tau})$ satisfy $u_{n\tau}\to u_{0\tau}$ in $L^{2}(\mathcal{O}_{\tau})$ as $n\to \infty$, then

$$ U(t,\tau )u_{n\tau}\to U(t,\tau )u_{0\tau} \quad\textit{in } H^{1}_{0}( \mathcal{O}_{t}) \textit{ as } n \to \infty . $$

(1.10)

Moreover, the $(L^{2},L^{2})$ pullback attractor $\hat{\mathscr{A}}=\{\mathscr{A}(t): t\in \mathbb{R}\}$ obtained in [5] can pullback attract in the topology of $H^{1}$, i.e., for all $t\in \mathbb{R}$ and $\hat{D}=\{D(t):t\in \mathbb{R}\}\in \mathscr{D}$,

$$ \operatorname{dist}_{H_{0}^{1}(\mathcal{O}_{t})}\bigl(U(t,\tau )D(\tau ), \mathscr{A}(t)\bigr) \to 0\quad \textit{as } \tau \to -\infty . $$

(1.11)

As mentioned previously, after the work [2], although (1.3) is defined on a time-varying domain, the continuity in (1.10) and attraction (1.11) is more or less expectable, in this paper, we give rigorous proofs about how to justify the approximation that is necessary due to relaxing the assumption on the forcing term. Note also that here we only additionally assume that $f\in L^{2}_{\mathrm{loc}}(\mathbb{R}, L^{2}(\mathcal{O}_{t}))$, but not (1.7), which was required in [3, 7, 9, 13] etc. for obtaining the boundedness in $L^{p}$ and $H^{1}$. However, in the nonautonomous case, the question whether we can remove further the additional condition $f\in L^{2}_{\mathrm{loc}}(\mathbb{R}, L^{2}(\mathcal{O}_{t}))$ remains open.

2 Preliminaries

2.1 Functional spaces

We first recall some functional spaces and notations.

For a fixed finite time interval $[\tau ,T]$, let $(X_{t}, \|\cdot \|_{X_{t}})$ $(t\in [\tau ,T])$ be a family of Banach spaces such that $X_{t}\subset L^{1}_{\mathrm{loc}}(\mathcal{O}_{t})$ for all $t\in [\tau ,T]$. For any $1\leqslant q\leqslant \infty$, we denote by $L^{q}(\tau ,T;X_{t})$ the vector space of all functions $u\in L^{1}_{\mathrm{loc}}(Q_{\tau ,T})$ such that $u(t)=u(\cdot ,t)\in X_{t}$ for a.e. $t\in (\tau ,T)$ and the function $\|u(\cdot )\|_{X_{\cdot}}$ defined by $t\mapsto \|u(t)\|_{X_{t}}$ belongs to $L^{q}(\tau ,T)$.

On $L^{q}(\tau ,T;X_{t})$, we consider the norm given by

$$ \Vert u \Vert _{L^{q}(\tau ,T;X_{t})}:= \big\Vert \big\Vert u(\cdot )\big\| _{X_{\cdot}} \big\| _{L^{q}( \tau ,T)}. $$

2.2 Definitions of solutions

For the readers’ convenience, in this subsection, we recall the definition of different solutions of equation (1.3); see Límaco et al. [6] and Kloeden et al. [5] for more detail.

For each $T>\tau$, consider the auxiliary problem

$$ \textstyle\begin{cases} {\frac {\partial u}{\partial t}}-\Delta u+g(u)=f(t)\quad \text{in }Q_{\tau ,T}, \\ u=0 \quad \text{on } \Sigma _{\tau ,T}, \\ u(\tau ,x)=u_{\tau}(x),\quad x\in \mathcal{O}_{\tau}, \end{cases} $$

(2.1)

where $\tau \in \mathbb{R}$ and $u_{\tau}:\mathcal{O}_{\tau} \rightarrow \mathbb{R}$.

Definition 2.1

(Strong solution)

A function $u=u(x,t)$ defined in $Q_{\tau ,T}$ is said to be a strong solution of problem (2.1) if

$$ u\in L^{2}\bigl(\tau ,T;H^{2}(\mathcal{O}_{t}) \bigr) \cap C\bigl([\tau ,T];H_{0}^{1}( \mathcal{O}_{t}) \bigr)\cap L^{\infty}\bigl(\tau ,T;L^{q}(\mathcal{O}_{t}) \bigr),\quad u'\in L^{2}\bigl(\tau ,T;L^{2}( \mathcal{O}_{t})\bigr), $$

and the three equations in (2.1) are satisfied almost everywhere in their corresponding domains.

Denote

$$\begin{aligned} \mathcal{U}_{\tau ,T}:={}& \bigl\{ \varphi \in L^{2}\bigl(\tau ,T;H_{0}^{1}( \mathcal{O}_{t})\bigr)\cap L^{q}\bigl(\tau ,T;L^{q}(\mathcal{O}_{t})\bigr): \varphi ' \in L^{2}\bigl(\tau ,T;L^{2}( \mathcal{O}_{t})\bigr),\\ &{} \varphi (\tau )=\varphi (T)=0 \bigr\} . \end{aligned}$$

Definition 2.2

Let $u_{\tau}\in L^{2}(\mathcal{O}_{\tau})$, $f\in L^{2}(\tau ,T;H^{-1}(\mathcal{O}_{t}))$, and $-\infty <\tau \leqslant T<\infty$. We say that a function u is a weak solution of (2.1) if

• (1) $u\in C([\tau ,T];L^{2}(\mathcal{O}_{t}))\cap L^{2}(\tau ,T; H_{0}^{1}( \mathcal{O}_{t}))\cap L^{q}(\tau ,T;L^{q}(\mathcal{O}_{t}))$ with $u(\tau )=u_{\tau}$;

• (2) there exists a sequence of regular data $u_{\tau m}\in H_{0}^{1}(\mathcal{O}_{\tau})\cap L^{q}(\mathcal{O}_{ \tau})$ and $f_{m}\in L^{2}(\tau ,T;L^{2}(\mathcal{O}_{t})$, $m=1,2,\ldots$ , such that

  $$ u_{\tau m} \to u_{\tau}\quad \text{in } L^{2}( \mathcal{O}_{\tau}),\qquad f_{m} \to f \quad \text{in } L^{2} \bigl(\tau ,T;H^{-1}(\mathcal{O}_{t})\bigr), $$

  and

  $$ u_{m} \to u \quad \text{in } C\bigl([\tau ,T];L^{2}( \mathcal{O}_{t})\bigr), $$

  where $u_{m}$ is the unique strong solution of (2.1) corresponding to $(u_{\tau m},f_{m})$;

• (3) for all $\varphi \in \mathcal{U}_{\tau ,T}$,

  $$\begin{aligned} &\int _{\tau}^{T} \int _{\mathcal{O}_{t}} u(x,t)\varphi '(x,t)\,dx \,dt + \int _{\tau}^{T} \int _{\mathcal{O}_{t}} \nabla _{x} u \cdot \nabla _{x} \varphi \,dx \,dt \\ &\quad = - \int _{\tau}^{T} \int _{\mathcal{O}_{t}} g\bigl(u(x,t)\bigr) \varphi (x,t)\,dx \,dt + \int _{\tau}^{T} \int _{\mathcal{O}_{t}} f(x,t) \varphi (x,t)\,dx \,dt. \end{aligned}$$

Definition 2.3

(Weak solution)

A function $u:\bigcup_{t\in [\tau ,\infty )}\mathcal{O}_{t}\times \{t\} \to \mathbb{R}$ is called a weak solution of (1.3) if for any $T> \tau$, the restriction of u on $\bigcup_{t\in [\tau ,T]}\mathcal{O}_{t}\times \{t\}$ is a weak solution of (2.1).

2.3 Preliminary lemmas

For later application, in the following, we collect some results for obtaining higher-order integrability, which can be proved by the standard methods; see [5, 10] for the detailed proofs.

Lemma 2.4

If $u\in L^{2}(\tau ,T;H_{0}^{1}(\mathcal{O}_{t}))\cap L^{\infty}(Q_{ \tau ,t})$ and $u'\in L^{2}(\tau ,T;L^{2}(\mathcal{O}_{t}))$, then for any $k\in [0,\infty )$,

$$ \vert u \vert ^{k}\cdot u\in L^{2}\bigl( \tau ,T;H_{0}^{1}(\mathcal{O}_{t})\bigr)\cap L^{ \infty}(Q_{\tau ,t}), $$

(2.2)

and the following energy equality is satisfied:

$$\begin{aligned}& \bigl\Vert u(t_{2}) \bigr\Vert _{L^{k+2}(\mathcal{O}_{t_{2}})}^{k+2}- \bigl\Vert u(t_{1}) \bigr\Vert _{L^{k+2}( \mathcal{O}_{t_{1}})}^{k+2} \\& \quad =(k+2) \int _{t_{1}}^{t_{2}}\bigl(u'(t), \bigl\vert u(t) \bigr\vert ^{k} \cdot u(t)\bigr)_{t}\,dt \quad \forall \tau \leqslant t_{1}\leqslant t_{2} \leqslant T. \end{aligned}$$

(2.3)

Lemma 2.5

For any $k>0$ and any $\phi \in H_{0}^{1}(\mathcal{O}_{s})\cap L^{\infty}(\mathcal{O}_{s})$ for some $s\in \mathbb{R}$, we the following equality:

$$ \int _{\mathcal{O}_{s}}\nabla \phi \cdot \nabla \bigl( \vert \phi \vert ^{k}\phi \bigr)\,dx= (k+1) \biggl(\frac{2}{k+2} \biggr)^{2} \int _{\mathcal{O}_{s}} \bigl\vert \nabla \vert \phi \vert ^{\frac{k+2}{2}} \bigr\vert ^{2}\,dx, $$

(2.4)

where ⋅ stands for the usual inner product in $\mathbb{R}^{N}$.

Lemma 2.6

Let $f\in L^{2}_{\mathrm{loc}}(\mathbb{R};L^{2}(\mathcal{O}_{s}))$ satisfy (1.9). Then, for each $T\in \mathbb{R}$, there is a family $\{f_{m}\}\subset L^{\infty}_{\mathrm{loc}}(Q_{-\infty ,T})$ such that

$$ \textit{for any (fixed)} \quad \tau \in (-\infty ,T),\qquad f_{m} \to f \quad \textit{in } L^{2}\bigl(\tau ,T;L^{2}(\mathcal{O}_{s}) \bigr) $$

(2.5)

and for any $t\in (-\infty , T)$,

$$ \int _{-\infty}^{t}e^{\lambda s} \bigl\Vert f_{m}(s) \bigr\Vert _{L^{2}(\mathcal{O}_{s})}^{2}\,ds \leqslant 2 \int _{-\infty}^{t}e^{\lambda s} \bigl\Vert f(s) \bigr\Vert _{L^{2}( \mathcal{O}_{s})}^{2}\,ds+\frac{1}{4} \quad \textit{for all } m=1,2, \ldots . $$

(2.6)

Recall that $Q_{-\infty ,T}=\bigcup_{t\in (-\infty ,T)}\mathcal{O}_{t}\times \{t\}$ and the family $\{f_{m}\}$ may depend on T.

In order the test function $|u|^{k}\cdot u$ to make sense, we also recall the following $L^{\infty}$-estimate on the nice initial data, which can be obtained by applying the standard Stampacchia’s truncation method; see [10] for a detailed proof.

Lemma 2.7

($L^{\infty}$-estimate)

Let Assumption Ibe satisfied. Then for any $-\infty <\tau \leqslant T<\infty$ and any initial data $(u_{\tau}, f)\in (H_{0}^{1}(\mathcal{O}_{\tau})\cap L^{\infty}( \mathcal{O}_{\tau}), L^{\infty}(Q_{\tau ,T}) )$, the unique strong solution u of (2.1) belongs to $L^{\infty}(Q_{\tau ,T})$.

3 Higher-order integrability

Along the ideas in [2], as the preliminaries, in this section, we obtain some higher-order integrability of the difference of two weak solutions near the initial time, which was firstly established in [2] for the (autonomous and fixed domain) stochastic case of (1.3), and later, similar results were obtained in [13] for (1.3)–(1.4) in the stochastic case (in time-varying case, but the forcing term was required to satisfy (1.7)).

3.1 A priori estimates for approximation solutions

To make our proof rigorous, we will use the approximation techniques.

For any (fixed) $T\in \mathbb{R}$, throughout this section, we choose (we can do this by Lemma 2.6) and fix a family $\{f_{m}\}\subset L^{\infty}_{\mathrm{loc}}(Q_{-\infty ,T})$ such that

$$ \text{the family }\{f_{m}\}\text{ satisfying conditions (2.5)--(2.6) in Lemma 2.6.} $$

(3.1)

Then, for any $\tau < T$ and any $u_{\tau},v_{\tau}\in L^{2}(\mathcal{O}_{\tau})$, according to the definition of a weak solution, we know that there are two sequences $\{(u_{\tau m},f_{m})\}$ and $\{(v_{\tau m},f_{m})\}$ satisfying

$$ u_{\tau m},v_{\tau m}\in H_{0}^{1}( \mathcal{O}_{\tau})\cap L^{\infty}( \mathcal{O}_{\tau}) \quad \text{and}\quad f_{m}\in L^{\infty}(Q_{\tau ,T}) $$

(3.2)

such that

$$ \begin{aligned} &u_{\tau m} \to u_{\tau},\qquad v_{\tau m} \to v_{\tau} \quad \text{in } L^{2}( \mathcal{O}_{\tau}) \quad \text{and} \\ & f_{m}\to f \quad \text{in } L^{2}\bigl( \tau ,T;L^{2}(\mathcal{O}_{t})\bigr) \text{ as } m\to \infty \end{aligned}$$

(3.3)

and

$$ u_{m}\to u \quad \text{and}\quad v_{m}\to v\quad \text{in } C^{0}\bigl([\tau ,T];L^{2}(\mathcal{O}_{t}) \bigr), $$

(3.4)

where $u_{m}$ and $v_{m}$ are the unique strong solution of (1.3) corresponding to the regular data $(u_{\tau m},f_{m})$ and $(v_{\tau m},f_{m})$, respectively.

Without loss of generality, by (3.3) we can require that

$$ \Vert u_{\tau m} \Vert ^{2}_{\tau } \leqslant \Vert u_{\tau} \Vert ^{2}_{\tau}+1 \quad \text{and}\quad \Vert v_{\tau m} \Vert ^{2}_{\tau} \leqslant \Vert v_{\tau} \Vert ^{2}_{ \tau }+1 \quad \text{for all } m=1,2,\ldots , $$

(3.5)

where and hereafter, $\|\cdot \|_{s}$ denotes the usual norm of $L^{2}(\mathcal{O}_{s})$ ($s\in \mathbb{R}$).

Denote

$$ w_{m}(t)=u_{m}(t)-v_{m}(t) \quad \text{for any } \tau \leqslant t \leqslant T. $$

(3.6)

Then $w_{m}(t)$ ($m=1,2,\ldots$) is the unique strong solution of the following equation:

$$ \textstyle\begin{cases} {\frac {\partial w_{m}}{\partial t}}-\Delta w_{m}+g(u_{m})-g(v_{m})=0 \quad \text{in }Q_{\tau ,T}, \\ w_{m}=0 \quad \text{on } \Sigma _{\tau ,T}, \\ w_{m}(\tau ,x)=u_{\tau m}-v_{\tau m}, \quad x\in \mathcal{O}_{ \tau}, \end{cases} $$

(3.7)

that is, $w_{m}\in L^{2}(\tau ,T;H^{2}(\mathcal{O}_{t})) \cap C([\tau ,T];H_{0}^{1}( \mathcal{O}_{t}))\cap L^{\infty}(\tau ,T;L^{q}(\mathcal{O}_{t}))$, $w_{m}'\in L^{2}(\tau ,T;L^{2}(\mathcal{O}_{t}))$, and the three equations in (3.7) are satisfied almost everywhere in their corresponding domains.

The main purpose of this subsection is to prove the following uniform (with respect to m) a priori estimates of $w_{m}$ defined in (3.6).

Theorem 3.1

Let Assumption Ihold. Then, for any $\tau \leqslant T$ and any $k=1,2,\ldots$ , there exists a positive constant $M_{k}=M(T-\tau , k, N,l,\|u_{\tau}\|_{\tau},\|v_{\tau}\|_{\tau})$, such that for all $m=1,2,\ldots$ ,

and

where $w_{m}(t)=u_{m}(t)-v_{m}(t)=U(t,\tau )u_{\tau m}-U(t,\tau )v_{\tau m}$,

$$ b_{1}=1+\frac{1}{2},\qquad b_{2}=1+ \frac{1}{2}+1, \quad \textit{and}\quad b_{k+1}=b_{k}+\frac{1+\frac{N}{N-2}}{2(\frac{N}{N-2})^{k+1}}\quad \textit{for }k=2,3,\ldots , $$

(3.8)

and all constants $M_{k}$ ($k=1,2,\ldots$) are independent of m.

Proof

By Lemma 2.7 we know that $u_{m},v_{m}\in L^{\infty}(Q_{\tau ,T})$ for each $m=1,2,\ldots$ , and so

$$ w_{m}=u_{m}-v_{m} \in C\bigl([\tau ,T];H_{0}^{1}(\mathcal{O}_{t})\bigr)\cap L^{ \infty}(Q_{\tau ,T}), $$

and for any $0\leqslant \theta <\infty$,

$$ \vert w_{m} \vert ^{\theta}\cdot w_{m} \in L^{2}\bigl(\tau ,T; H_{0}^{1}( \mathcal{O}_{t})\bigr) \cap L^{\infty}(Q_{\tau ,T}). $$

Consequently, we can multiply (3.7) by $|w_{m}|^{\theta}\cdot w_{m}$ for all $\theta \in [0,\infty )$.

In the following, we will separate our proof into two steps.

Step 1 $k=1$

At first, multiplying (3.7) by $w_{m}$, from the definition of a strong solution and (1.4), applying Lemmas 2.4 and 2.5, we obtain that

$$ \begin{aligned} \frac{1}{2}\frac{d}{dt} \Vert w_{m} \Vert _{t}^{2} + \int _{ \mathcal{O}_{t}} \bigl\vert \nabla w_{m}(t) \bigr\vert ^{2}\,dx &= - \int _{ \mathcal{O}_{t}} \bigl(g(u_{m})-g(v_{m}) \bigr)w_{m}\,dx \\ & \leqslant l \bigl\Vert w_{m}(t) \bigr\Vert _{t}^{2} \quad \text{a.e. } t\in (\tau ,T) \end{aligned} $$

(3.9)

(recall that $\|\cdot \|_{s}$ denotes the $L^{2}(\mathcal{O}_{s})$-norm), which implies that

$$ \bigl\Vert w_{m}(t) \bigr\Vert _{t}^{2}\leqslant e^{2l(t-\tau )} \bigl\Vert w_{m}(\tau ) \bigr\Vert _{\tau}^{2}, $$

(3.10)

and then

$$\begin{aligned} \int ^{T}_{\tau} \bigl\Vert \nabla w_{m}(t) \bigr\Vert _{t}^{2}\,dt \leqslant& l \int ^{T}_{ \tau} \bigl\Vert w_{m}(s) \bigr\Vert _{s}^{2}\,ds+\frac{1}{2} \bigl\Vert w_{m}(\tau ) \bigr\Vert _{\tau}^{2} \\ \leqslant& \frac{1}{2} \bigl(e^{2l(T-\tau )}+1 \bigr) \bigl\Vert w_{m}( \tau ) \bigr\Vert _{ \tau}^{2}. \end{aligned}$$

(3.11)

Consequently, combining with the embedding

$$ \biggl( \int _{\mathcal{O}_{s}} \vert v \vert ^{\frac{2N}{N-2}}\,dx \biggr)^{\frac{N-2}{N}} \leqslant c_{N,\tau ,T} \int _{\mathcal{O}_{s}} \vert \nabla v \vert ^{2}\,dx,\quad \forall v\in H^{1}(\mathcal{O}_{s})\ \forall s\in [\tau ,T], $$

(3.12)

we can deduce that

$$\begin{aligned}& \int ^{T}_{\tau} \biggl( \int _{\mathcal{O}_{t}} \bigl\vert (t-\tau )^{b_{1}} w_{m}(t) \bigr\vert ^{ \frac{2N}{N-2}}\,dx \biggr)^{\frac{N-2}{N}}\,dt \\& \quad \leqslant (T-\tau )^{2b_{1}} \frac{c_{N,\tau ,T}}{2} \bigl(e^{2l(T-\tau )}+1 \bigr) \bigl\Vert w_{m}(\tau ) \bigr\Vert _{ \tau}^{2}. \end{aligned}$$

(3.13)

Note that here the embedding constant $c_{N,\tau ,T}$ in (3.12) depends only on the domain $\bigcup_{s\in [\tau ,T]}\mathcal{O}_{s}$.

Secondly, multiplying (3.7) by $|w_{m}|^{\frac{2N}{N-2}-2}\cdot w_{m}$, and similarly to (3.9), we have that

$$ \begin{aligned} &\frac{1}{2} \biggl(\frac{N-2}{N} \biggr) \frac{d}{dt} \bigl\Vert w_{m}(t) \bigr\Vert ^{\frac{2N}{N-2}}_{L^{\frac{2N}{N-2}}(\mathcal{O}_{t})} + \frac{\frac{2N}{N-2}-1}{ ( \frac{N}{N-2} )^{2}} \int _{ \mathcal{O}_{t}} \bigl\vert \nabla \bigl\vert w_{m}(t) \bigr\vert ^{(\frac{N}{N-2})} \bigr\vert ^{2}\,dx \\ &\quad \leqslant l \bigl\Vert w_{m}(t) \bigr\Vert ^{\frac{2N}{N-2}}_{L^{\frac{2N}{N-2}}( \mathcal{O}_{t})}\quad \text{for a.e. } t\in (\tau ,T). \end{aligned} $$

To simplify the calculations, we denote by c, $c_{i}$ ($i=1,2,\ldots$) the constants that depend only on N, $T-\tau$, k, and l and may vary from line to line. Then the above inequality can be written as

$$ \frac{d}{dt} \bigl\Vert w_{m}(t) \bigr\Vert ^{\frac{2N}{N-2}}_{L^{\frac{2N}{N-2}}( \mathcal{O}_{t})}+c_{1} \int _{\mathcal{O}_{t}} \bigl\vert \nabla \bigl\vert w_{m}(t) \bigr\vert ^{ \frac{N}{N-2}} \bigr\vert ^{2}\,dx \leq c_{2} \bigl\Vert w_{m}(t) \bigr\Vert ^{\frac{2N}{N-2}}_{L^{ \frac{2N}{N-2}}(\mathcal{O}_{t})}, $$

(3.14)

and by multiplying both sides with $(t-\tau )^{\frac{3N}{N-2}}$ we obtain that

$$ \begin{aligned} &\frac{d}{dt} \bigl\Vert (t-\tau )^{b_{1}}w_{t}(t) \bigr\Vert ^{ \frac{2N}{N-2}}_{L^{\frac{2N}{N-2}}(\mathcal{O}_{t})}+c_{1} \int _{ \mathcal{O}_{t}} \bigl\vert \nabla \bigl\vert (t-\tau )^{b_{1}}w_{m}(t) \bigr\vert ^{ \frac{N}{N-2}} \bigr\vert ^{2}\,dx \\ &\quad \leqslant c_{2} \bigl\Vert (t-\tau )^{b_{1}}w_{m}(t) \bigr\Vert ^{\frac{2N}{N-2}}_{L^{ \frac{2N}{N-2}}(\mathcal{O}_{t})}+ c_{3} (t-\tau )^{\frac{3N}{N-2}-1} \bigl\Vert w_{m}(t) \bigr\Vert ^{ \frac{2N}{N-2}}_{L^{\frac{2N}{N-2}}(\mathcal{O}_{t})} \\ &\quad \leqslant c\biggl(1+\frac{1}{t-\tau}\biggr) \bigl\Vert (t-\tau )^{b_{1}}w_{m}(t) \bigr\Vert ^{ \frac{2N}{N-2}}_{L^{\frac{2N}{N-2}}(\mathcal{O}_{t})}. \end{aligned} $$

(3.15)

Recall that $b_{1}=1+\frac{1}{2}$ was defined in (3.2).

One direct result of (3.15) is that

$$ (t-\tau )\frac{d}{dt} \bigl\Vert (t-\tau )^{b_{1}}w_{m}(t) \bigr\Vert ^{\frac{2N}{N-2}}_{L^{ \frac{2N}{N-2}}(\mathcal{O}_{t})} \leqslant c \bigl\Vert (t-\tau )^{b_{1}}w_{m}(t) \bigr\Vert ^{\frac{2N}{N-2}}_{L^{\frac{2N}{N-2}}(\mathcal{O}_{t})}, $$

and so

$$ (t-\tau )\frac{d}{dt} \bigl\Vert (t-\tau )^{b_{1}}w_{m}(t) \bigr\Vert ^{2}_{L^{ \frac{2N}{N-2}}(\mathcal{O}_{t})} \leqslant c\frac{N-2}{N} \bigl\Vert (t-\tau )^{b_{1}}w_{m}(t) \bigr\Vert ^{2}_{L^{\frac{2N}{N-2}}(\mathcal{O}_{t})}. $$

(3.16)

Consequently, for any $t\in [\tau ,T]$, integrating (3.16) over $[\tau ,t]$, we obtain that

$$ \begin{aligned} (t-\tau ) \bigl\Vert (t-\tau )^{b_{1}}w_{m}(t) \bigr\Vert ^{2}_{L^{ \frac{2N}{N-2}}(\mathcal{O}_{t})}&\leqslant \biggl(c \frac{N-2}{N}+1\biggr) \int ^{T}_{ \tau} \bigl\Vert (s-\tau )^{b_{1}}w_{m}(s) \bigr\Vert ^{2}_{L^{\frac{2N}{N-2}}( \mathcal{O}_{s})}\,ds \\ &\leqslant c \bigl\Vert w_{m}(\tau ) \bigr\Vert _{\tau}^{2} \quad \text{(by (3.13))}, \end{aligned} $$

and hence

$$ (t-\tau )^{\frac{N}{N-2}} \bigl\Vert (t-\tau )^{b_{1}}w_{m}(t) \bigr\Vert _{L^{ \frac{2N}{N-2}}(\mathcal{O}_{t})}^{\frac{2N}{N-2}} \leqslant c \bigl\Vert w_{m}( \tau ) \bigr\Vert _{\tau}^{\frac{2N}{N-2}}\quad \text{for all }t\in [\tau ,T]. $$

(3.17)

Then, multiplying (3.15) by $(t-\tau )^{\frac{2N}{N-2}}$, we obtain that for a.e. $t\in (\tau ,T)$,

$$ \begin{aligned} &(t-\tau )^{\frac{2N}{N-2}}\frac{d}{dt} \bigl\Vert (t-\tau )^{b_{1}}w_{m}(t) \bigr\Vert ^{ \frac{2N}{N-2}}_{L^{\frac{2N}{N-2}}(\mathcal{O}_{t})}+c_{1} \int _{ \mathcal{O}_{t}} \bigl\vert \nabla \bigl\vert (t-\tau )^{b_{1}+1}w_{m}(t) \bigr\vert ^{ \frac{N}{N-2}} \bigr\vert ^{2}\,dx \\ &\quad \leqslant c(t-\tau )^{\frac{N+2}{N-2}} \bigl\Vert (t-\tau )^{b_{1}}w_{m}(t) \bigr\Vert _{L^{\frac{2N}{N-2}}(\mathcal{O}_{t})}^{\frac{2N}{N-2}} \\ &\quad \leqslant c(t-\tau )^{\frac{2}{N-2}} \bigl\Vert w_{m}(\tau ) \bigr\Vert _{\tau}^{ \frac{2N}{N-2}}\quad \text{(by (3.17))}. \end{aligned} $$

Integrating this inequality over $[\tau ,T]$ with respect to t, we obtain that

$$ \int ^{T}_{\tau} \int _{\mathcal{O}_{t}} \bigl\vert \nabla \bigl\vert (t-\tau )^{b_{2}}w_{m}(t) \bigr\vert ^{ \frac{N}{N-2}} \bigr\vert ^{2}\,dx\,dt\leqslant c \bigl\Vert w_{m}(\tau ) \bigr\Vert _{\tau}^{ \frac{2N}{N-2}}, $$

(3.18)

where we have used (3.17). Consequently, applying embedding (3.12) again, we can deduce that

$$ \int ^{T}_{0} \biggl( \int _{\Omega} \bigl\vert (t-\tau )^{b_{2}}w_{m}(t) \bigr\vert ^{2( \frac{N}{N-2})^{2}}\,dx \biggr)^{\frac{N-2}{N}}\,dt\leqslant c_{N,\tau ,T}c \bigl\Vert w_{m}(\tau ) \bigr\Vert _{\tau}^{\frac{2N}{N-2}}. $$

(3.19)

Therefore, noticing (3.3) and (3.5), from (3.17) and (3.19) we know that there is a positive constant $M_{1}$, which depends only on N, τ, T, l, $\|u_{\tau}\|_{\tau}$, $\|v_{\tau}\|_{\tau}$, such that ($A_{1}$) and ($B_{1}$) hold.

Step 2 Assuming that ($A_{k}$) and ($B_{k}$) hold for $k\geqslant 1$, we will show that ($A_{k+1}$) and ($B_{k+1}$) hold.

Multiplying (3.7) by $|w_{m}|^{2(\frac{N}{N-2})^{k+1}-2}\cdot w_{m}$, using (1.4), and applying Lemmas 2.4 and 2.5, we obtain that

$$ \begin{aligned} &\frac{d}{dt} \bigl\Vert w_{m}(t) \bigr\Vert ^{2(\frac{N}{N-2})^{k+1}}_{L^{2( \frac{N}{N-2})^{k+1}}(\mathcal{O}_{t})} + c \int _{\mathcal{O}_{t}} \bigl\vert \nabla \bigl\vert w_{m}(t) \bigr\vert ^{(\frac{N}{N-2})^{k+1}} \bigr\vert ^{2}\,dx \\ &\quad \leqslant c_{1} \bigl\Vert w_{m}(t) \bigr\Vert ^{2(\frac{N}{N-2})^{k+1}}_{L^{2( \frac{N}{N-2})^{k+1}}(\mathcal{O}_{t})} \quad \text{for a.e. } t\in ( \tau ,T). \end{aligned} $$

(3.20)

Multiplying both sides of (3.20) by $(t-\tau )^{2(\frac{N}{N-2})^{k+1}\cdot b_{k+1}}$, we deduce that

$$ \begin{aligned} &\frac{d}{dt} \bigl( (t-\tau )^{2(\frac{N}{N-2})^{k+1} \cdot b_{k+1}} \Vert w_{m} \Vert ^{2(\frac{N}{N-2})^{k+1}}_{L^{2(\frac{N}{N-2})^{k+1}}( \mathcal{O}_{t})} \bigr) + c \int _{\mathcal{O}_{t}} \bigl\vert \nabla \bigl\vert (t- \tau )^{b_{k+1}}\cdot w_{m}(t) \bigr\vert ^{(\frac{N}{N-2})^{k+1}} \bigr\vert ^{2}\,dx \\ &\quad \leqslant c_{1} \bigl\Vert (t-\tau )^{b_{k+1}}\cdot w_{m}(t) \bigr\Vert ^{2( \frac{N}{N-2})^{k+1}}_{L^{2(\frac{N}{N-2})^{k+1}}(\mathcal{O}_{t})} \\ &\qquad {}+ c_{2} (t-\tau )^{2(\frac{N}{N-2})^{k+1}\cdot b_{k+1}-1} \bigl\Vert w_{m}(t) \bigr\Vert ^{2( \frac{N}{N-2})^{k+1}}_{L^{2(\frac{N}{N-2})^{k+1}}(\mathcal{O}_{t})}, \end{aligned} $$

i.e.,

$$ \begin{aligned} &\frac{d}{dt} \bigl\Vert (t-\tau )^{b_{k+1}}\cdot w_{m}(t) \bigr\Vert ^{2( \frac{N}{N-2})^{k+1}}_{L^{2(\frac{N}{N-2})^{k+1}}(\mathcal{O}_{t})} + c \int _{\mathcal{O}_{t}} \bigl\vert \nabla \bigl\vert (t-\tau )^{b_{k+1}}\cdot w_{m}(t) \bigr\vert ^{( \frac{N}{N-2})^{k+1}} \bigr\vert ^{2}\,dx \\ &\quad \leqslant \biggl(c_{1} +\frac{c_{2}}{t-\tau}\biggr) \bigl\Vert (t- \tau )^{b_{k+1}}\cdot w_{m}(t) \bigr\Vert ^{2(\frac{N}{N-2})^{k+1}}_{L^{2(\frac{N}{N-2})^{k+1}}(\mathcal{O}_{t})}. \end{aligned} $$

(3.21)

At first, from (3.21) we have

$$\begin{aligned}& (t-\tau )\frac{d}{dt} \bigl\Vert (t-\tau )^{b_{k+1}}\cdot w_{m}(t) \bigr\Vert ^{2( \frac{N}{N-2})^{k+1}}_{L^{2(\frac{N}{N-2})^{k+1}}(\mathcal{O}_{t})} \\& \quad \leqslant c \bigl\Vert (t-\tau )^{b_{k+1}}\cdot w_{m}(t) \bigr\Vert ^{2(\frac{N}{N-2})^{k+1}}_{L^{2( \frac{N}{N-2})^{k+1}}(\mathcal{O}_{t})}, \end{aligned}$$

(3.22)

and so

$$\begin{aligned}& (t-\tau )\frac{d}{dt} \bigl\Vert (t-\tau )^{b_{k+1}} \cdot w_{m}(t) \bigr\Vert ^{2( \frac{N}{N-2})^{k}}_{L^{2(\frac{N}{N-2})^{k+1}}(\mathcal{O}_{t})} \\& \quad \leqslant c \frac{N-2}{N} \bigl\Vert (t-\tau )^{b_{k+1}}\cdot w_{m}(t) \bigr\Vert ^{2( \frac{N}{N-2})^{k}}_{L^{2(\frac{N}{N-2})^{k+1}}(\mathcal{O}_{t})}. \end{aligned}$$

(3.23)

Integrating (3.23) over $[\tau ,t]$ and applying ($B_{k}$), we deduce that

$$ \begin{aligned} &(t-\tau ) \bigl\Vert (t-\tau )^{b_{k+1}}\cdot w_{m}(t) \bigr\Vert ^{2( \frac{N}{N-2})^{k}}_{L^{2(\frac{N}{N-2})^{k+1}}(\mathcal{O}_{t})} \\ &\quad \leqslant \biggl(c \frac{N-2}{N}+1\biggr) \int _{\tau}^{T} \bigl\Vert (s-\tau )^{b_{k+1}} \cdot w_{m}(s) \bigr\Vert ^{2(\frac{N}{N-2})^{k}}_{L^{2(\frac{N}{N-2})^{k+1}}( \mathcal{O}_{s})}\,ds \\ &\quad \leqslant \biggl(c \frac{N-2}{N}+1\biggr)M_{k} \quad \text{for all } t \in [\tau , T], \end{aligned} $$

which implies that

$$ \begin{aligned} &(t-\tau )^{\frac{N}{N-2}} \bigl\Vert (t-\tau )^{b_{k+1}}\cdot w_{m}(t) \bigr\Vert ^{2(\frac{N}{N-2})^{k+1}}_{L^{2(\frac{N}{N-2})^{k+1}}(\mathcal{O}_{t})} \\ &\quad \leqslant \biggl[\biggl(c \frac{N-2}{N}+1\biggr)M_{k} \biggr]^{\frac{N}{N-2}}\quad \text{for all } t\in [\tau , T]. \end{aligned} $$

(3.24)

In the following, after obtained (3.24), we will return to (3.21) to deduce ($B_{k+1}$). Multiplying both sides of (3.21) by $(t-\tau )^{1+\frac{N}{N-2}}$, we obtain that

$$ \begin{aligned} &(t-\tau )^{1+\frac{N}{N-2}}\frac{d}{dt} \bigl\Vert (t-\tau )^{b_{k+1}} \cdot w_{m}(t) \bigr\Vert ^{2(\frac{N}{N-2})^{k+1}}_{L^{2(\frac{N}{N-2})^{k+1}}( \mathcal{O}_{t})} \\ &\qquad {} + c \int _{\mathcal{O}_{t}} \bigl\vert \nabla \bigl\vert (t-\tau )^{b_{k+1}+ \frac{1+\frac{N}{N-2}}{2(\frac{N}{N-2})^{k+1}}}\cdot w_{m}(t) \bigr\vert ^{( \frac{N}{N-2})^{k+1}} \bigr\vert ^{2}\,dx \\ &\quad \leqslant c_{3} (t-\tau )^{\frac{N}{N-2}} \bigl\Vert (t-\tau )^{b_{k+1}} \cdot w_{m}(t) \bigr\Vert ^{2(\frac{N}{N-2})^{k+1}}_{L^{2(\frac{N}{N-2})^{k+1}}( \mathcal{O}_{t})}. \end{aligned} $$

(3.25)

Then from (3.24) and the definition of $b_{k+2}$ we obtain that

$$ \begin{aligned} &(t-\tau )^{1+\frac{N}{N-2}}\frac{d}{dt} \bigl\Vert (t-\tau )^{b_{k+1}} \cdot w_{m}(t) \bigr\Vert ^{2(\frac{N}{N-2})^{k+1}}_{L^{2(\frac{N}{N-2})^{k+1}}( \mathcal{O}_{t})} \\ &\qquad {} + c \int _{\mathcal{O}_{t}} \bigl\vert \nabla \bigl\vert (t-\tau )^{b_{k+2}} \cdot w_{m}(t) \bigr\vert ^{(\frac{N}{N-2})^{k+1}} \bigr\vert ^{2}\,dx \\ &\quad \leqslant c_{3} \biggl[\biggl(c \frac{N-2}{N}+1 \biggr)M_{k} \biggr]^{ \frac{N}{N-2}} \quad \text{for all } t\in [\tau , T]. \end{aligned} $$

(3.26)

Integrating this inequality over $[\tau ,T]$ and using (3.24) again, we deduce that

$$ \begin{aligned} \int _{\tau}^{T} \int _{\mathcal{O}_{t}} \bigl\vert \nabla \bigl\vert (t- \tau )^{b_{k+2}}\cdot w_{m}(t) \bigr\vert ^{(\frac{N}{N-2})^{k+1}} \bigr\vert ^{2}\,dx\,dt \leqslant c_{4} \biggl[\biggl(c \frac{N-2}{N}+1\biggr)M_{k} \biggr]^{ \frac{N}{N-2}}. \end{aligned} $$

(3.27)

Consequently, using of the embedding inequality (3.12) again, we obtain that

$$ \int _{\tau}^{T} \biggl( \int _{\Omega} \bigl\vert (t-\tau )^{b_{k+2}}\cdot w_{m}(t) \bigr\vert ^{2( \frac{N}{N-2})^{k+2}}\,dx \biggr)^{\frac{N-2}{N}}\,dt \leqslant c_{5} \biggl[\biggl(c \frac{N-2}{N}+1 \biggr)M_{k} \biggr]^{\frac{N}{N-2}}. $$

(3.28)

Therefore by setting

$$ M_{k+1}=(1+c_{5}) \biggl[\biggl(c \frac{N-2}{N}+1 \biggr)M_{k} \biggr]^{ \frac{N}{N-2}}, $$

(3.24) and (3.28) imply that ($A_{k+1}$) and ($B_{k+1}$) hold, respectively. □

3.2 Higher-order integrability near the initial time

Based on the a priori estimate in Theorem 3.1 for the approximation solutions, we can obtain the following higher-order integrability near the initial time:

Theorem 3.2

Let Assumption Ihold, and let $u_{\tau},v_{\tau}\in L^{2}(\mathcal{O}_{\tau})$. Then for any $T\geqslant \tau$ and $k=1,2,\ldots$ , there exists a positive constant $M_{k}=M(T-\tau , k, N,l,\|u_{\tau}\|_{\tau},\|v_{\tau}\|_{\tau})$ such that

$$ (t-\tau )^{\frac{N}{N-2}} \bigl\Vert (t-\tau )^{b_{k}}w(t) \bigr\Vert _{L^{2( \frac{N}{N-2})^{k}}(\mathcal{O}_{t})}^{2(\frac{N}{N-2})^{k}} \leqslant M_{k} \quad \textit{for all } t\in [\tau , T], $$

where $w(t)=U(t,\tau )u_{\tau}-U(t,\tau )v_{\tau}$, and

$$ b_{1}=1+\frac{1}{2}, \qquad b_{2}=1+ \frac{1}{2}+1\quad \textit{and} \quad b_{k+1}=b_{k}+\frac{1+\frac{N}{N-2}}{2(\frac{N}{N-2})^{k+1}}\quad \textit{for } k=2,3,\ldots . $$

Proof

For any (fixed) $\tau \in \mathbb{R}$ and $T\geqslant \tau$, choose two sequences $(u_{\tau m}, f_{m})$ and $(v_{\tau m}, f_{m})$ satisfying all conditions (3.1)–(3.5).

Then from Theorem 3.1 we have that for any $k=1,2,\ldots$ , there exists a positive constant $M_{k}=M(T-\tau , k, N,l,\|u_{\tau}\|_{\tau}, \|v_{\tau}\|_{\tau})$ such that

$$ (t-\tau )^{\frac{N}{N-2}} \bigl\Vert (t-\tau )^{b_{k}} \bigl(u_{m}(t)-v_{m}(t) \bigr) \bigr\Vert _{L^{2(\frac{N}{N-2})^{k}}(\mathcal{O}_{t})}^{2(\frac{N}{N-2})^{k}} \leqslant M_{k}\quad \text{for all } t\in [ \tau , T], $$

(3.29)

where $u_{m}$ and $v_{m}$ are the unique strong solutions of (1.3) corresponding to the regular data $(u_{\tau m},f_{m})$ and $(v_{\tau m},f_{m})$ on the interval $[\tau , T]$, respectively.

From (3.4) we know that for each $t\in [\tau , T]$, there are two subsequences $\{u_{m_{j}}(t)\} \subset \{u_{m}(t)\}$ and $\{v_{m_{j}}(t)\} \subset \{v_{m}(t)\}$ satisfying

$$ u_{m_{j}}(t)\to u(t)=U(t,\tau )u_{\tau} \quad \text{and}\quad v_{m_{j}}(t) \to v(t)=U(t,\tau )v_{\tau}\quad \text{a.e. on } \mathcal{O}_{t} \text{ as } j\to \infty , $$

where the subindex $m_{j}$ may depend on t.

Hence, since estimate (3.29) is independent of m, we can finish our proof by applying the Fatou lemma:

$$ \begin{aligned} &(t-\tau )^{\frac{N}{N-2}} \bigl\Vert (t-\tau )^{b_{k}} \bigl(u(t)-v(t) \bigr) \bigr\Vert _{L^{2(\frac{N}{N-2})^{k}}(\mathcal{O}_{t})}^{2(\frac{N}{N-2})^{k}} \\ &\quad = (t-\tau )^{\frac{N}{N-2}} \int _{\mathcal{O}_{t}}\liminf_{j\to \infty} \bigl\vert (t- \tau )^{b_{k}} \bigl(u_{m_{j}}(t)-v_{m_{j}}(t) \bigr) \bigr\vert ^{2 ( \frac{N}{N-2} )^{k}}\,dx \\ &\quad \leqslant \liminf_{j\to \infty}(t-\tau )^{\frac{N}{N-2}} \int _{ \mathcal{O}_{t}} \bigl\vert (t-\tau )^{b_{k}} \bigl(u_{m_{j}}(t)-v_{m_{j}}(t) \bigr) \bigr\vert ^{2 (\frac{N}{N-2} )^{k}}\,dx \\ &\quad \leqslant M_{k}. \end{aligned} $$

 □

4 Proof of Theorem 1.1

We start with the following a priori estimates.

Lemma 4.1

Let Assumption Ihold, and let $f\in L^{2}_{\mathrm{loc}}(\mathbb{R}, L^{2}(\mathcal{O}_{t}))$. Then for all $\tau \in \mathbb{R}$ and $u_{\tau}\in L^{2}{(\mathcal{O}_{\tau})}$, the corresponding weak solution $u(t)=U(t,\tau )u_{\tau}$ ($t\geqslant \tau$) of equation (1.3) satisfies the following estimates: for any $T> \tau$,

$$ \begin{aligned} &\int _{\mathcal{O}_{s}} \bigl\vert u(s) \bigr\vert ^{p}\,dx \leqslant M \quad \textit{for all } s \in \biggl[\tau +\frac{T-\tau}{2}, T\biggr], \quad \textit{and} \\ &\int _{\frac{\tau +T}{2}}^{T} \int _{\mathcal{O}_{s}} \bigl\vert u(s) \bigr\vert ^{2p-2}\,dx\,ds \leqslant M \end{aligned}$$

(4.1)

with constant M depending only on $T-\tau$, $|\bigcup_{s\in [\tau , T]}\mathcal{O}_{s}|$, $\lambda _{\tau T}$, $\int _{\tau}^{T}\|f(s)\|^{2}_{L^{2}(\mathcal{O}_{s})}\,ds$, and $\|u_{\tau}\|_{\tau}$, where $\lambda _{\tau T}$ is the first eigenvalue of −Δ on $H_{0}^{1}(\bigcup_{s\in [\tau , T]}\mathcal{O}_{s})$.

Note that since we only assume that $f\in L^{2}_{\mathrm{loc}}(\mathbb{R}, L^{2}(\mathcal{O}_{t}))$, we cannot obtain the uniform boundedness of the solutions in the $L^{p}$ sense as that in [3, 9, 13], i.e., our constant M above depends on the time $t-\tau$. However, we will show further that such boundedness is sufficient for Theorem 1.1.

Proof

Since the results of the lemma are more or less standard, we restrict ourselves by only formal derivation of estimate (4.1), which can be easily justified using, e.g., the methods as in Sect. 3: first, deduce the a priori estimates for approximation solutions and then obtain (4.1) by Fatou’s lemma.

First, multiplying (1.3) by u and integrating with respect to $x\in \mathcal{O}_{t}$, we have that

$$ \begin{aligned} \frac{1}{2}\frac{d}{dt} \Vert u \Vert _{t}^{2} + \int _{ \mathcal{O}_{t}} \bigl\vert \nabla u(t) \bigr\vert ^{2}\,dx + \int _{\mathcal{O}_{t}}g(u)u\,dx \leqslant \bigl\Vert f(t) \bigr\Vert _{t} \bigl\Vert u(t) \bigr\Vert _{t}\quad \text{for a.e. } t\in (\tau ,T); \end{aligned} $$

(4.2)

recall that $\|\cdot \|_{t}$ denotes the $L^{2}(\mathcal{O}_{t})$-norm; Then using (1.4) and Cauchy’s inequality, we obtain that

$$ \begin{aligned} &\frac{d}{dt} \Vert u \Vert _{t}^{2}+2\lambda _{\tau T} \bigl\Vert u(t) \bigr\Vert _{t}^{2}+ 2\alpha _{1} \int _{\tau}^{t} \int _{\mathcal{O}_{s}} \bigl\vert u(s) \bigr\vert ^{p}\,dx\,ds-2 \beta \vert \mathcal{O}_{t} \vert \\ &\quad \leqslant \frac{1}{2\lambda _{\tau T}} \int _{\tau}^{t} \bigl\Vert f(s) \bigr\Vert ^{2}_{s}\,ds +2\lambda _{\tau T} \bigl\Vert u(t) \bigr\Vert _{t}^{2}\quad \text{for all } t\in [\tau ,T] \end{aligned} $$

(recall that $\lambda _{\tau T}$ is the first eigenvalue of −Δ on $H_{0}^{1}(\bigcup_{s\in [\tau , T]}\mathcal{O}_{s})$), which implies that

$$ \begin{aligned} & \bigl\Vert u(t) \bigr\Vert _{t}^{2}+2\alpha _{1} \int _{\tau}^{t} \int _{ \mathcal{O}_{s}} \bigl\vert u(s) \bigr\vert ^{p}\,dx\,ds \\ &\quad \leqslant \frac{1}{2\lambda _{\tau T}} \int _{\tau}^{t} \bigl\Vert f(s) \bigr\Vert ^{2}_{s}\,ds +2\beta \biggl\vert \bigcup _{s\in [\tau , T]}\mathcal{O}_{s} \biggr\vert + \Vert u_{\tau} \Vert _{ \tau}^{2}\quad \text{for all } t\in [\tau ,T]. \end{aligned} $$

(4.3)

Secondly, multiplying in (1.3) by $|u|^{p-2}\cdot u$ and integrating with respect to $x\in \mathcal{O}_{t}$, we have that

$$ \begin{aligned} &\frac{1}{p}\frac{d}{dt} \int _{\mathcal{O}_{t}} \bigl\vert u(t) \bigr\vert ^{p}\,dx + \alpha _{1} \int _{\mathcal{O}_{t}} \bigl\vert u(t) \bigr\vert ^{2p-2}\,dx \\ &\quad \leqslant \beta \int _{\mathcal{O}_{t}} \bigl\vert u(t) \bigr\vert ^{p-2}\,dx+ \bigl\Vert f(t) \bigr\Vert _{L^{2}( \mathcal{O}_{t})} \bigl\Vert u(t) \bigr\Vert _{L^{2p-2}(\mathcal{O}_{t})}^{p-1} \quad \text{a.e. } t\in (\tau ,T), \end{aligned} $$

where we have used Lemmas 2.4 and 2.5 and (1.4). Consequently, using Cauchy’s inequality, we have that

$$ \begin{aligned} &\frac{d}{dt} \int _{\mathcal{O}_{t}} \bigl\vert u(t) \bigr\vert ^{p}\,dx +c_{1} \int _{\mathcal{O}_{t}} \bigl\vert u(t) \bigr\vert ^{2p-2}\,dx & \\ &\quad \leqslant c_{2} + c_{3} \bigl\Vert f(t) \bigr\Vert ^{2}_{L^{2}(\mathcal{O}_{t})} \quad \text{for a.e. } t\in (\tau ,T), \end{aligned} $$

(4.4)

where the constants $c_{1}$, $c_{2}$, $c_{3}$ depend only on β, $\alpha _{1}$, and p.

Now from (4.3) we know that there is $t_{0}\in [\tau ,\frac{\tau +T}{2}]$ such that

$$ u(t_{0})\in L^{p}(\mathcal{O}_{t_{0}}) $$

(4.5)

and

$$ \begin{aligned} & \bigl\Vert u(t_{0}) \bigr\Vert ^{p}_{L^{p}(\mathcal{O}_{t_{0}})}\leqslant \frac{1}{\alpha _{1}(T-\tau )} \biggl( \frac{1}{2\lambda _{\tau T}} \int _{\tau}^{T} \bigl\Vert f(s) \bigr\Vert ^{2}_{s}\,ds +2\beta \biggl\vert \bigcup _{s\in [\tau , T]} \mathcal{O}_{s} \biggr\vert + \Vert u_{\tau} \Vert _{\tau}^{2} \biggr). \end{aligned} $$

(4.6)

Therefore, for any $t\in [\frac{T+\tau}{2},T]$, integrating (4.4) with respect to time from $t_{0}$ to t, we deduce that

$$ \begin{aligned} & \bigl\Vert u(t) \bigr\Vert ^{p}_{L^{p}(\mathcal{O}_{t})} +c_{1} \int _{t_{0}}^{t} \int _{\mathcal{O}_{s}} \bigl\vert u(s) \bigr\vert ^{2p-2}\,dx\,ds \\ &\quad \leqslant c_{2}(t-t_{0}) + c_{3} \int _{t_{0}}^{t} \bigl\Vert f(s) \bigr\Vert ^{2}_{L^{2}( \mathcal{O}_{s})}\,ds+ \bigl\Vert u(t_{0}) \bigr\Vert ^{p}_{L^{p}(\mathcal{O}_{t_{0}})}, \end{aligned} $$

(4.7)

which, combined with (4.6) and (4.3), immediately implies (4.1). □

Now we are ready to prove our main results.

Proof of Theorem 1.1

It suffices to prove the following claim: For any $u_{\tau},v_{\tau}\in L^{2}(\mathcal{O}_{\tau})$, we have the following estimate for $t>\tau$:

$$ \begin{aligned} \bigl\Vert U(t,\tau )u_{\tau} - U(t,\tau )v_{\tau} \bigr\Vert ^{2}_{H_{0}^{1}( \mathcal{O}_{t})} \leqslant c_{1} \Vert u_{\tau}-v_{\tau} \Vert ^{2}_{\tau} + c_{2} \Vert u_{\tau}-v_{\tau} \Vert ^{2\theta}_{\tau}, \end{aligned} $$

(4.8)

where the constants $c_{i}>0$ and $\theta \in (0,1)$ depend only on $t-\tau$, $\|u_{\tau}\|_{\tau}$, and $\|v_{\tau}\|_{\tau}$.

Indeed, the $H^{1}$-continuity (1.10) immediately follows from (4.8).

To see the $H^{1}$-pullback attraction (1.11), for each $t\in \mathbb{R}$, we denote by $B(t)$ the 1-neighborhood of $\mathscr{A}(t)$ with respect to the $L^{2}(\mathcal{O}_{t})$-norm. Then $B(t)$ is bounded in $L^{2}(\mathcal{O}_{t})$, and by (4.8) there are two positive constants $c'_{i}>0$ and $\theta \in (0,1)$ that depend only on t and $\|B(t)\|_{t}$ such that, for all $u_{\tau},v_{\tau}\in B(t-1)$,

$$ \begin{aligned} \bigl\Vert U(t,t-1)u_{\tau} - U(t,t-1)v_{\tau} \bigr\Vert ^{2}_{H_{0}^{1}( \mathcal{O}_{t})} \leqslant c'_{1} \Vert u_{\tau}-v_{\tau} \Vert ^{2}_{\tau} + c'_{2} \Vert u_{\tau}-v_{\tau} \Vert ^{2\theta}_{\tau}. \end{aligned} $$

(4.9)

Now by the definition of the $(L^{2},L^{2})$ pullback $\mathscr{D}_{\lambda}$-attractor $\mathscr{A}$, for any $\varepsilon >0$ and any $\hat{D}=\{D(t):t\in \mathbb{R}\}\in \mathscr{D}$, there is a time $\tau _{1}(< t-1)$, which depends only on t, ε, and D̂, such that

$$ \operatorname{dist}_{L^{2}(\mathcal{O}_{t-1})}\bigl(U(t-1,\tau )D(\tau ), \mathscr{A}(t-1)\bigr) \leqslant \varepsilon\quad \text{for all } \tau \leqslant \tau _{1} $$

(4.10)

and

$$ U(t-1,\tau )D(\tau )\subset B(t-1) \quad \text{for all } \tau \leqslant \tau _{1}. $$

(4.11)

Then from (4.9)–(4.11) we have that for $\tau \leqslant \tau _{1}$,

$$ \begin{aligned} &\operatorname{dist}^{2}_{H_{0}^{1}(\mathcal{O}_{t})} \bigl(U(t,\tau )D(\tau ), \mathscr{A}(t)\bigr) \\ &\quad = \operatorname{dist}^{2}_{H_{0}^{1}(\mathcal{O}_{t})}\bigl(U(t,t-1)U(t-1,\tau )D( \tau ), U(t,t-1)\mathscr{A}(t-1)\bigr) \\ &\quad \leqslant c'_{1} \operatorname{dist}^{2}_{L^{2}(\mathcal{O}_{t-1})} \bigl(U(t-1,\tau )D( \tau ), \mathscr{A}(t-1)\bigr) \\ & \qquad {}+ c'_{2} \operatorname{dist}^{2\theta}_{L^{2}(\mathcal{O}_{t-1})} \bigl(U(t-1, \tau )D(\tau ), \mathscr{A}(t-1)\bigr)\quad (\text{by (4.9)}) \\ &\quad \leqslant c'_{1} \varepsilon ^{2} + c'_{2} \varepsilon ^{2\theta}\quad (\text{by (4.10)}). \end{aligned} $$

Consequently, we obtain the $H^{1}$-pullback attraction (1.11) by the arbitrariness of ε and D̂.

In the following, we give the proof of the above claim. To make our proof rigorous, as in Sect. 3, we will prove the claim firstly for approximation solutions and then take the limit.

Fix T such that $T\geqslant t>\tau$. Then, for the initial data $u_{\tau}$ and $v_{\tau}$, take $\{u_{\tau m}\}_{m=1}^{\infty}$, $\{v_{\tau m}\}_{m=1}^{\infty}$, and $\{f_{m}\}_{m=1}^{\infty}$ satisfying (3.1)–(3.5).

Denote

$$ w_{m}(s)=u_{m}(s)-v_{m}(s)\quad \text{for } \tau \leqslant s \leqslant T. $$

(4.12)

Then $w_{m}(s)$ ($m=1,2,\ldots$) is the unique strong solution of (3.7).

First, multiplying (3.7) by $w_{m}$ and integrating with respect to $x\in \mathcal{O}_{s}$ and time, we obtain that

$$ \bigl\Vert w_{m}(s) \bigr\Vert _{s}^{2}\leqslant e^{2l(s-\tau )} \bigl\Vert w_{m}(\tau ) \bigr\Vert _{\tau}^{2}\quad \forall s\in [\tau ,T] $$

(4.13)

and

$$ \int _{\tau}^{t} \bigl\Vert \nabla w_{m}(s) \bigr\Vert _{s}^{2}\,ds\leqslant \frac{1}{2} \bigl\Vert w_{m}( \tau ) \bigr\Vert _{\tau}^{2}+ \int _{\tau}^{t} \bigl\Vert w_{m}(s) \bigr\Vert _{s}^{2}\,ds\quad \forall t\in [\tau ,T], $$

(4.14)

where we have used (1.4); recall that $\|\cdot \|_{s}$ denotes the usual $L^{2}(\mathcal{O}_{s})$-norm and the constant l comes from (1.4).

Secondly, applying Lemma 4.1 to the initial data $u_{\tau m}$ and $v_{\tau m}$, we obtain that there is a constant $M_{0}$, which depends only on $t-\tau$, $|\bigcup_{s\in [\tau , t]}\mathcal{O}_{s}|$, $\lambda _{\tau t}$, $\int _{\tau}^{t}\|f(s)\|^{2}_{L^{2}(\mathcal{O}_{s})}\,ds$, β, $\alpha _{1}$, p, $\|u_{\tau m}\|_{\tau}$, and $\|v_{\tau m}\|_{\tau}$, such that

$$ \begin{aligned} \int _{\frac{\tau +t}{2}}^{t} \int _{\Omega} \bigl\vert u_{m}(s) \bigr\vert ^{2p-2}\,dx\,ds + \int _{\frac{\tau +t}{2}}^{t} \int _{\Omega} \bigl\vert v_{m}(s) \bigr\vert ^{2p-2}\,dx\,ds \leqslant M_{0}, \end{aligned} $$

(4.15)

and from (3.5) we know that $M_{0}$ depends indeed only on $\|u_{\tau}\|_{L^{2}(\mathcal{O}_{\tau})}$ and $\|v_{\tau}\|_{L^{2}(\mathcal{O}_{\tau})}$ regarding to the initial data.

We now multiply (3.7) by $-\Delta w_{m}$ (since $w_{m}\in L^{2}(\tau ,T;H^{2}(\mathcal{O}_{t}))$). We then have

$$ - \int _{\mathcal{O}_{s}}w_{m}'\Delta w_{m}\,dx+ \int _{\mathcal{O}_{s}} \bigl\vert \Delta w_{m}(s) \bigr\vert ^{2} = \int _{\mathcal{O}_{s}} (g(u_{m}(s)-g\bigl(v_{m}(s) \bigr)\Delta w_{m}(s)\,dx. $$

(4.16)

Moreover, as in Límaco, Medeiros, and Zuazua [6], we have

$$ - \int _{\mathcal{O}_{s}}w_{m}'\Delta w_{m}\,dx =\frac{1}{2}\frac{d}{ds} \int _{\mathcal{O}_{s}} \bigl\vert \nabla w_{m}(s) \bigr\vert ^{2}\,dx - \int _{\Gamma _{s}} \bigl\vert \nabla w_{m}(s) \bigr\vert ^{2}\psi \cdot n_{s}\,d\sigma , $$

(4.17)

where $n_{s}$ denotes the unit outward normal vector to $\mathcal{O}_{s}$, and ψ is the velocity field $\psi =[\partial _{s}r](\bar{r}(x,s))$. Then, according to (1.1), (1.2), and (1.6), by classical trace results and interpolation we have (e.g., see Duvaut [4]) that

$$ \biggl\vert \int _{\Gamma _{s}} \bigl\vert \nabla w_{m}(s) \bigr\vert ^{2}\psi \cdot n_{s}\,d\sigma \biggr\vert \leqslant c_{\nu} \biggl( \int _{\mathcal{O}_{s}} \bigl\vert \Delta w_{m}(s) \bigr\vert ^{2}\,dx \biggr)^{\nu} \biggl( \int _{\mathcal{O}_{s}} \bigl\vert \nabla w_{m}(s) \bigr\vert ^{2}\,dx \biggr)^{1- \nu} $$

(4.18)

for all $\nu \geqslant \frac{1}{2}$. In particular, taking $\nu =\frac{1}{2}$ in (4.18) and using Cauchy’s inequality, we have that

$$\begin{aligned}& \biggl\vert \int _{\Gamma _{s}} \bigl\vert \nabla w_{m}(s) \bigr\vert ^{2}\psi \cdot n_{s}\,d\sigma \biggr\vert \\& \quad \leqslant \frac{1}{4} \int _{\mathcal{O}_{s}} \bigl\vert \Delta w_{m}(s) \bigr\vert ^{2}\,dx + 2c_{\frac{1}{2}} \int _{\mathcal{O}_{s}} \bigl\vert \nabla w_{m}(s) \bigr\vert ^{2}\,dx \quad \text{for all } s\in [\tau ,T]. \end{aligned}$$

(4.19)

At the same time, from (1.4) we have that

$$ \begin{aligned} & \biggl\vert \int _{\mathcal{O}_{s}} \bigl(g\bigl(u_{m}(s)\bigr)-g \bigl(v_{m}(s)\bigr) \bigr)\Delta w_{m}(s)\,dx \biggr\vert \\ &\quad \leqslant c \int _{\mathcal{O}_{s}} \bigl(1+ \bigl\vert u_{m}(s) \bigr\vert ^{p-2}+ \bigl\vert v_{m}(s) \bigr\vert ^{p-2}\bigr) \bigl\vert w_{m}(s) \bigr\vert \bigl\vert \Delta w_{m}(s) \bigr\vert \,dx \\ &\quad \leqslant c \int _{\mathcal{O}_{s}} \bigl\vert w_{m}(s) \bigr\vert \bigl\vert \Delta w_{m}(s) \bigr\vert \,dx \\ &\qquad {} + c \int _{\mathcal{O}_{s}} \bigl( \bigl\vert u_{m}(s) \bigr\vert ^{p-2}+ \bigl\vert v_{m}(s) \bigr\vert ^{p-2}\bigr) \bigl\vert w_{m}(s) \bigr\vert \bigl\vert \Delta w_{m}(s) \bigr\vert \,dx \\ &\quad \leqslant \frac{1}{4} \int _{\mathcal{O}_{s}} \bigl\vert \Delta w_{m}(s) \bigr\vert ^{2}\,dx +c \bigl\Vert w_{m}(s) \bigr\Vert ^{2}_{s} \\ &\qquad {} +c\bigl( \bigl\Vert u_{m}(s) \bigr\Vert ^{2p-4}_{L^{2p-2}(\mathcal{O}_{s})}+ \bigl\Vert v_{m}(s) \bigr\Vert ^{2p-4}_{L^{2p-2}(\mathcal{O}_{s})} \bigr) \bigl\Vert w_{m}(s) \bigr\Vert ^{2}_{L^{2p-2}( \mathcal{O}_{s})}, \end{aligned} $$

(4.20)

where, for the last inequality, we used the Hölder inequality with power $\frac{p-2}{2p-2}+\frac{1}{2p-2}+\frac{1}{2}=1$.

Therefore, inserting (4.17)–(4.20) into (4.16), we finally obtain that

$$ \begin{aligned} &\frac{d}{ds} \int _{\mathcal{O}_{s}} \bigl\vert \nabla w_{m}(s) \bigr\vert ^{2}\,dx \\ &\quad \leqslant 4c_{\frac{1}{2}} \int _{\mathcal{O}_{s}} \bigl\vert \nabla w_{m}(s) \bigr\vert ^{2}\,dx + 2c \bigl\Vert w_{m}(s) \bigr\Vert _{s}^{2} +2c \bigl( \bigl\Vert u_{m}(s) \bigr\Vert ^{2p-4}_{L^{2p-2}( \mathcal{O}_{s})} \\ &\qquad {} + \bigl\Vert v_{m}(s) \bigr\Vert ^{2p-4}_{L^{2p-2}(\mathcal{O}_{s})} \bigr) \bigl\Vert w_{m}(s) \bigr\Vert ^{2}_{L^{2p-2}(\mathcal{O}_{s})}. \end{aligned} $$

(4.21)

Since $2(\frac{N}{N-2})^{k}\to \infty$ as $k\to \infty$, there is $k_{0}\in \mathbb{N}$ such that

$$ 2\biggl(\frac{N}{N-2}\biggr)^{k_{0}}>2p-2. $$

For this $k_{0}$, by interpolation we have

$$ \Vert w \Vert _{L^{2p-2}(\mathbb{R}^{N})}\leqslant \Vert w \Vert _{L^{2(\frac{N}{N-2})^{k_{0}}}( \mathbb{R}^{N})}^{1-\theta}\cdot \Vert w \Vert _{L^{2}(\mathbb{R}^{N})}^{ \theta}, $$

where the power $\theta \in (0,1)$ depends only on p, $k_{0}$.

Hence from (4.21) we have that for a.e. $s\in [\tau ,T]$,

$$ \begin{aligned} \frac{d}{ds} \int _{\mathcal{O}_{s}} \bigl\vert \nabla w_{m}(s) \bigr\vert ^{2}\,dx\leqslant{}& c \int _{\mathcal{O}_{s}} \bigl\vert \nabla w_{m}(s) \bigr\vert ^{2}\,dx + c \bigl\Vert w_{m}(s) \bigr\Vert _{s}^{2} +c \bigl( \bigl\Vert u_{m}(s) \bigr\Vert ^{2p-4}_{L^{2p-2}(\mathcal{O}_{s})} \\ &{} + \bigl\Vert v_{m}(s) \bigr\Vert ^{2p-4}_{L^{2p-2}(\mathcal{O}_{s})} \bigr) \bigl\Vert w_{m}(s) \bigr\Vert _{L^{2(\frac{N}{N-2})^{k_{0}}}(\mathcal{O}_{s})}^{1-\theta} \cdot \bigl\Vert w_{m}(s) \bigr\Vert _{L^{2}(\mathcal{O}_{s})}^{\theta}. \end{aligned} $$

(4.22)

In the following, we will apply Theorem 3.2 to control the terms in (4.22).

Denoting $r_{0}= (\frac{N}{N-2})\frac{2-2\theta}{2(\frac{N}{N-2})^{k_{0}}}+(2-2 \theta )b_{k_{0}}$ and multiplying (4.22) by $(s-\frac{t+\tau}{2})^{r_{0}}$, we obtain that

$$\begin{aligned}& \biggl(s-\frac{t+\tau}{2} \biggr)^{r_{0}}\frac{d}{ds} \bigl\Vert \nabla w_{m}(s) \bigr\Vert _{s}^{2} \\& \quad \leqslant c\biggl(s-\frac{t+\tau}{2}\biggr)^{r_{0}} \bigl( \bigl\Vert \nabla w_{m}(s) \bigr\Vert _{s}^{2}+ \bigl\Vert w_{m}(s) \bigr\Vert _{s}^{2} \bigr) \\& \qquad {}+ c \bigl( \bigl\Vert u_{m}(s) \bigr\Vert ^{2p-4}_{L^{2p-2}( \mathcal{O}_{s})}+ \bigl\Vert v_{m}(s) \bigr\Vert ^{2p-4}_{L^{2p-2}(\mathcal{O}_{s})} \bigr) \\& \qquad {} \cdot \bigl((s-\tau )^{\frac{N}{N-2}} \bigl\Vert (s-\tau )^{b_{k_{0}}}w_{m}(s) \bigr\Vert ^{2(\frac{N}{N-2})^{k_{0}}}_{L^{2(\frac{N}{N-2})^{k_{0}}}( \mathcal{O}_{s})} \bigr)^{ \frac{2-2\theta}{2(\frac{N}{N-2})^{k_{0}}}}\cdot \bigl\Vert w_{m}(s) \bigr\Vert _{s}^{2 \theta}, \end{aligned}$$

(4.23)

where $b_{k_{0}}$ is given by (3.2) corresponding to $k_{0}$.

Then applying Theorem 3.2 to the initial data $u_{\tau m}$, $v_{\tau m}$, times τ, t, and $k_{0}$, we get that there is a constant $M_{k_{0}}$, which depends only on $t-\tau$, N, l, $k_{0}$, and $\|u_{m\tau}\|_{\tau}$, $\|v_{m\tau}\|_{\tau}$, such that

$$ \bigl((s-\tau )^{\frac{N}{N-2}} \bigl\Vert (s-\tau )^{b_{k_{0}}}w_{m}(s) \bigr\Vert ^{2( \frac{N}{N-2})^{k_{0}}}_{L^{2(\frac{N}{N-2})^{k_{0}}}(\mathcal{O}_{s})} \bigr)^{\frac{2-2\theta}{2(\frac{N}{N-2})^{k_{0}}}} \leqslant M_{k_{0}}^{2-2 \theta} \quad \text{for all } s \in [\tau , t]; $$

(4.24)

Noting (3.5) again, we see that $M_{k_{0}}$ also depends only on $\|u_{\tau}\|_{\tau}$ and $\|v_{\tau}\|_{\tau}$ regarding to the initial data.

Therefore we have the following estimate: for a.e. $s\in [ \frac{t+\tau}{2}, t]$,

$$ \begin{aligned} &\biggl(s-\frac{t+\tau}{2} \biggr)^{r_{0}}\frac{d}{ds} \bigl\Vert \nabla w_{m}(s) \bigr\Vert _{s}^{2} \\ &\quad \leqslant c\biggl(s-\frac{t-\tau}{2} \biggr)^{r_{0}} \bigl( \bigl\Vert \nabla w_{m}(s) \bigr\Vert _{s}^{2}+ \bigl\Vert w_{m}(s) \bigr\Vert _{s}^{2} \bigr) \\ &\qquad {} +cM_{k_{0}}^{2-2\theta} \bigl( \bigl\Vert u_{m}(s) \bigr\Vert ^{2p-4}_{L^{2p-2}( \mathcal{O}_{s})}+ \bigl\Vert v_{m}(s) \bigr\Vert ^{2p-4}_{L^{2p-2}(\mathcal{O}_{s})} \bigr) \cdot \bigl\Vert w_{m}(s) \bigr\Vert _{s}^{2\theta}. \end{aligned} $$

(4.25)

To ensure the power of $(s-\frac{t+\tau}{2})$ to be strictly greater than 1, we may multiply both sides by $(s-\frac{t+\tau}{2})$ and then obtain that

$$ \begin{aligned} &\biggl(s-\frac{t+\tau}{2} \biggr)^{r_{0}+1}\frac{d}{ds} \bigl\Vert \nabla w_{m}(s) \bigr\Vert _{s}^{2} \\ &\quad \leqslant c\biggl(s-\frac{t+\tau}{2} \biggr)^{r_{0}+1} \bigl( \bigl\Vert \nabla w_{m}(s) \bigr\Vert _{s}^{2}+ \bigl\Vert w_{m}(s) \bigr\Vert _{s}^{2} \bigr) \\ & \qquad {}+c\biggl(s-\frac{t+\tau}{2}\biggr)M_{k_{0}}^{2-2\theta} \bigl( \bigl\Vert u_{m}(s) \bigr\Vert ^{2p-4}_{L^{2p-2}( \mathcal{O}_{s})}+ \bigl\Vert v_{m}(s) \bigr\Vert ^{2p-4}_{L^{2p-2}(\mathcal{O}_{s})} \bigr) \cdot \bigl\Vert w_{m}(s) \bigr\Vert _{s}^{2\theta}. \end{aligned} $$

(4.26)

Integrating (4.26) from $\frac{\tau +t}{2}$ to t, we obtain that

$$\begin{aligned}& \biggl(\frac{t-\tau}{2} \biggr)^{1+r_{0}} \bigl\Vert \nabla w_{m}(t) \bigr\Vert ^{2} \\& \quad \leqslant (1+r_{0}) \biggl(\frac{t-\tau}{2}\biggr)^{r_{0}} \int _{ \frac{\tau +t}{2}}^{t} \bigl\Vert \nabla w_{m}(s) \bigr\Vert ^{2}\,ds \\& \qquad {} + c\biggl(\frac{t-\tau}{2}\biggr)^{r_{0}+1} \int _{\frac{\tau +t}{2}}^{t} \bigl( \bigl\Vert \nabla w_{m}(s) \bigr\Vert _{s}^{2}+ \bigl\Vert w_{m}(s) \bigr\Vert _{s}^{2} \bigr)\,ds \\& \qquad {} + c\frac{t-\tau}{2}M_{k_{0}}^{2-2\theta} \int _{ \frac{\tau +t}{2}}^{t} \bigl( \bigl\Vert u_{m}(s) \bigr\Vert ^{2p-4}_{L^{2p-2}(\mathcal{O}_{s})}+ \bigl\Vert v_{m}(s) \bigr\Vert ^{2p-4}_{L^{2p-2}(\mathcal{O}_{s})} \bigr) \cdot \bigl\Vert w_{m}(s) \bigr\Vert _{s}^{2\theta}\,ds \\& \quad :=I_{1}+I_{2}+I_{3}. \end{aligned}$$

(4.27)

From (4.13) and (4.14) we have that

$$ \begin{aligned} &I_{1}+I_{2} \\ &\quad \leqslant \biggl(\frac{t-\tau}{2} \biggr)^{r_{0}} \biggl( \biggl(1+r_{0}+c \frac{t-\tau}{2} \biggr) \biggl(\frac{1}{2} + \frac{1}{2l}e^{2l(t-\tau )} \biggr)+c\frac{t-\tau}{2} \frac{1}{2l}e^{2l(t-\tau )} \biggr) \\ &\qquad {}\times \bigl\Vert w_{m}(\tau ) \bigr\Vert ^{2}_{\tau} \\ &\quad \leqslant c_{r_{0},t-\tau ,l} \bigl\Vert w_{m}(\tau ) \bigr\Vert ^{2}_{\tau}. \end{aligned} $$

(4.28)

For $I_{3}$, using the Hölder inequality and (4.15), we have that

$$ \begin{aligned} I_{3} & \leqslant c \frac{t-\tau}{2}M_{k_{0}}^{2-2\theta}2M_{0}^{ \frac{2p-4}{2p-2}} \biggl( \int _{\frac{\tau +t}{2}}^{t} \bigl\Vert w_{m}(s) \bigr\Vert _{s}^{2 \theta (p-1)}\,ds \biggr)^{\frac{2}{2p-2}} \\ &\leqslant c_{M_{k_{0}},p,M_{0},t-\tau ,\theta} \biggl( \int _{ \frac{\tau +t}{2}}^{t} e^{2l(s-\tau )\theta (p-1)}\,ds \biggr)^{ \frac{2}{2p-2}} \bigl\Vert w_{m}(\tau ) \bigr\Vert ^{2\theta}_{\tau} \quad (\text{by (4.13)}) \\ &\leqslant c_{M_{k_{0}},p,M_{0},t-\tau ,\theta ,l} \bigl\Vert w_{m}(\tau ) \bigr\Vert ^{2 \theta}_{\tau}. \end{aligned} $$

(4.29)

Putting (4.28) and (4.29) into (4.27), we finally obtain that

$$ \begin{aligned} \bigl\Vert \nabla w_{m}(t) \bigr\Vert ^{2}\leqslant c_{r_{0},t-\tau ,l} \bigl\Vert w_{m}( \tau ) \bigr\Vert ^{2}_{\tau} + c_{r_{0},M_{k_{0}},p,M_{0},t-\tau ,\theta ,l} \bigl\Vert w_{m}( \tau ) \bigr\Vert ^{2\theta}_{\tau}, \end{aligned} $$

(4.30)

and all the constants contained in the above inequality depend only on $\|u_{\tau}\|_{\tau}$, $\|v_{\tau}\|_{\tau}$ about initial data, and, consequently, they are independent of m.

From (4.30) we know that $\{w_{m}(t)\}_{m=1}^{\infty}$ is bounded in $H_{0}^{1}(\mathcal{O}_{t})$, and therefore there is a subsequence $\{w_{m_{j}}(t)\}_{j=1}^{\infty}$ such that

$$ w_{m_{j}}(t) \to \chi \quad \text{weakly in } H_{0}^{1}(\mathcal{O}_{t}) \text{ as } j\to \infty . $$

(4.31)

On the other hand, from (3.4) we know that

$$ w_{m_{j}}(t) \to u(t)-v(t) \quad \text{in } L^{2}( \mathcal{O}_{t}) \text{ as } j\to \infty . $$

Hence

$$ u(t)-v(t)=\chi \in H_{0}^{1}(\mathcal{O}_{t}), $$

and using (4.30), (4.31), and (3.3), we deduce that

$$ \begin{aligned}\bigl\Vert \nabla \bigl(u(t)-v(t)\bigr) \bigr\Vert _{t}^{2} &\leqslant \liminf_{j\to \infty} \bigl\Vert \nabla w_{m_{j}}(t) \bigr\Vert ^{2}_{t} \\ &\leqslant c_{r_{0},t-\tau ,l} \Vert u_{\tau}-v_{ \tau} \Vert ^{2}_{\tau} + c_{r_{0},M_{k_{0}},p,M_{0},t-\tau ,\theta ,l} \Vert u_{ \tau}-v_{\tau} \Vert ^{2\theta}_{\tau}. \end{aligned}$$

This finishes the proof of the claim and thus the proof of the theorem. □