\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{mathrsfs}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 329, pp. 1--22.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/329\hfil Cahn-Hilliard/Allen-Cahn equation]
{Existence of solutions to the Cahn-Hilliard/Allen-Cahn equation
 with degenerate mobility}

\author[X. Zhang, C. Liu \hfil EJDE-2016/329\hfilneg]
{Xiaoli Zhang, Changchun Liu}

\address{Xiaoli Zhang \hfill\break
Department of Mathematics,
Jilin University, Changchun 130012, China}
\email{mathche@sina.com}

\address{Changchun Liu (corresponding author)\hfill\break
Department of Mathematics,
Jilin University, Changchun 130012, China}
\email{liucc@jlu.edu.cn}

\thanks{Submitted September 2, 2016. Published December 24, 2016.}
\subjclass[2010]{35G25, 35K55, 35K65}
\keywords{Cahn-Hilliard/Allen-Cahn equation; existence; Galerkin method;
\hfill\break\indent degenerate mobility}

\begin{abstract}
 This article we study the Cahn-Hilliard/Allen-Cahn equation with
 degenerate mobility. Under suitable assumptions on the degenerate mobility
 and the double well potential, we prove existence of weak solutions,
 which can be obtained by considering the limits of Cahn-Hilliard/Allen-Cahn
 equations with non-degenerate mobility.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\allowdisplaybreaks

\section{Introduction}

In this article, we consider a scalar Cahn-Hilliard/Allen-Cahn equation
with degenerate mobility
\begin{equation}
\label{1-1}
u_t = -\nabla [D(u)\nabla (\Delta u -f(u))]+(\Delta u -f(u)),\quad \text{in } Q_T,
\end{equation}
where $Q_T=\Omega\times(0, T)$, $\Omega$ is a bounded domain in
$\mathbb{R}^{n}$ with a $C^{3}$-boundary $\partial\Omega$ and $f(u)$
is the derivative of a double-well potential $F(u)$ with wells $\pm 1$.
The mobility $D(u)\in C(\mathbb{R}; [0, \infty))$ is in the form
\begin{equation}\label{1-2}
\begin{gathered}
D(u)= |u|^m, \quad \text{if } |u| < \delta,\\
C_0\leq D(u)\leq C_1|u|^m, \quad \text{if } |u| \geq \delta,
\end{gathered}
\end{equation}
for some constants $C_0, C_1, \delta> 0$, where $0 < m < \infty$ if
$n = 1, 2$ and $\frac{4}{n}< m < \frac{4}{n-2}$ if $n \geq 3$.

Equation \eqref{1-1} is supplemented by the boundary conditions
\begin{equation}\label{1-3}
u|_{\partial \Omega}= \Delta u|_{\partial \Omega}=0, \quad  t>0,
\end{equation}
and the initial condition
\begin{equation} \label{1-4}
u(x,0)=u_0(x).
\end{equation}
Equation \eqref{1-1} was introduced as a simplification of multiple
microscopic mechanisms model \cite{GM} in cluster interface evolution.
Equation \eqref{1-1} with constant mobility has been intensively studied.
Karali and Nagase \cite{GY} investigated existence of weak solution to
 \eqref{1-1} with $D(u)\equiv D$
and a quartic bistable potential $F(u) =(1-u^2)^2$.
Karali and Nagase \cite{GY} only provided existence of the solution for
the deterministic case. Then
Antonopoulou, Karali and Millet \cite{AKM} studied the stochastic case.
The main result of this paper
is the existence of a global solution, under a specific sub-linear growth
condition for the diffusion coefficient. Path regularity in time and in space
is also studied. In addition,
Karali and Ricciardi \cite{GK} constructed special sequences of solutions
to a fourth order nonlinear parabolic equation of the Cahn-Hilliard/Allen-Cahn
equation, converging to the second order Allen-Cahn equation.
They studied the equivalence of the fourth order equation with a system of
two second order elliptic equations. Karali
and Katsoulakis \cite{GM} focus on a mean field partial differential equation,
which contains qualitatively microscopic information on particle-particle
interactions and multiple particle dynamics, and rigorously derive the
macroscopic cluster evolution laws and transport structure. They show that
the motion by mean curvature is given by $V =\mu\sigma\kappa$, where $\kappa$
is the mean curvature, $\sigma$ is the surface tension and $\mu$ is an effective
mobility that depends on the presence of the multiple mechanisms and speeds up
the cluster evolution. This is in contrast with the Allen-Cahn equation where
the velocity equals the mean curvature. Tang, Liu and Zhao \cite{TLZ} proved the
existence of global attractor.
Liu and Tang \cite{LT} obtained the existence of periodic solution for a
Cahn-Hilliard/Allen-Cahn equation in two space dimensions.

During the past few years, many authors have paid much attention to the Cahn-
Hilliard equation with degenerate mobility. An existence result for the
Cahn-Hilliard equation with a degenerate mobility in
a one-dimensional situation has been established by Yin \cite{Y}.
Elliott and Garcke \cite{CMEHG} considered
the Cahn-Hilliard equation with non-constant mobility for arbitrary
space dimensions. Based on Galerkin approximation, they proved the global
existence of weak solutions. Dai and Du \cite {SDQD} improved the results
of the paper \cite{CMEHG}.
Liu \cite{L} proved the existence of weak solutions for the convective
Cahn-Hilliard equation with degenerate mobility.
The relevant equations or inequalities have also been studied in
\cite{Li, Li2, L1, L2}.

Motivated by the above works, we prove the existence of weak solution
to \eqref{1-1}-\eqref{1-4} under a more general range of the double-well
potential $F$. In particular, we assume that for $s\in\mathbb{R}$, 
$F \in {C^2(R)}$ satisfies
\begin{gather}
\label{1-6}
k_0(|s|^{r+1}-1)\leq F(s)\leq k_1(|s|^{r+1}+ 1),\\
\label{1-7}
|F'(s)|\leq k_2(|s|^{r}+1), \\
\label{1-8}
|F''(s)|\leq k_3(|s|^{r-1}+1),
\end{gather}
for some constants $k_0, k_1, k_2, k_3 > 0$ where
$1\leq r <\infty$ if $n=1, 2$ and $1 \leq r\leq \frac{n}{n-2}$ if $n\geq 3$.
What's more, we need the assumption on the boundary of $f(u)$,
\begin{equation}
\label{1-9}
f(u)|_{\partial \Omega} =0, \quad t>0.
\end{equation}
We can give examples satisfying the condition \eqref{1-9}, such as
$F(u)=(1-u^2)^2$ studied by Karali and Nagase \cite{GY},
the logarithmic function
$f(u)=-\theta_{c}u+\frac{\theta}{2}\ln \frac{1+u}{1-u}$,
$u\in (-1, 1)$, $0< \theta <\theta_{c}$ \cite{LC}.

Concerning the Allen-Cahn structure, we rewrite
\eqref{1-1}, \eqref{1-3}, \eqref{1-4} and \eqref{1-9} to the form
\begin{equation}\label{1-10}
\begin{gathered}
u_t=\nabla(D(u)\nabla v)-v, \quad \text{in } Q_T,\\
v=-{\Delta}u+f(u), \quad \text{in } Q_T,\\
u(x,0)=u_0(x), \quad \text{in } \Omega, \\
u=v=0, \quad \text{on } \partial\Omega.
\end{gathered}
\end{equation}
We consider the free energy functional $E(u)$ defined in \cite {GY} given by
\begin{equation} \label{1-11}
E(u):= \int_\Omega\Big(\frac12|\nabla u|^2 + F(u)\Big)\,dx.
\end{equation}
For a pair of solution $(u, v)$ of \eqref{1-10} it holds that
\[
\frac{d}{dt}E(u) = \int_{\Omega} v u_t\,dx
 = \int_{\Omega} v [\nabla (D(u)\nabla v)- v]\,dx
 = -\int_\Omega \left(D(u)|\nabla v|^2+v^2\right)dx \leq 0.
\]

\noindent{\bf Notation.} Define the usual Lebesgue norms and the
$L^2$-inner-product
$$
\|u\|_{p} = \|u\|_{L^{p}(\Omega)} \quad \text{and}\quad
(u, v) = (u, v)_{L^2(\Omega)}.
$$
The duality pairing between the space $H^2(\Omega)$ and its dual
$(H^2(\Omega))'$ will be denoted using the
form $\langle\cdot, \cdot\rangle$. For simplicity, $2^{*}:= \frac{2n}{n-2}$.
 $\chi_{B}$ denotes the characteristic function of $B$.

This paper is organized as follows.
In Section $2$, we use a Galerkin method to give
a existence of weak solution for a positive mobility.
Section $3$ uses a sequence of non-degenerate solutions to approximate
the degenerate case \eqref{1-10}.

\section{Existence for positive mobility}

In this section, we study the Cahn-Hilliard/Allen-Cahn equation with a
non-degenerate mobility $D_\varepsilon(u)$ defined for an $\varepsilon$
satisfying $0<\varepsilon < \delta$ by
\begin{equation} \label{2-1}
D_\varepsilon(u):= \begin{cases}
 |u|^m, &\text{if } |u| > \varepsilon, \\
 \varepsilon^m, & \text{if } |u| \leq \varepsilon.
 \end{cases}
\end{equation}
So we consider the  problem
\begin{equation}
\label{2-2}
\begin{gathered}
u_t=\nabla(D_\varepsilon(u)\nabla v)-v, \quad \text{in } Q_T,\\
v=-\Delta u+f(u), \quad \text{in } Q_T,\\
u(x,0)=u_0(x), \quad \text{in } \Omega,\\
u=v=0, \quad \text{on } \partial\Omega.
\end{gathered}
\end{equation}

\begin{theorem}\label{th2.1}
Suppose $u_0\in H^1(\Omega)$, under assumptions \eqref{1-2} and
\eqref{1-6}--\eqref{1-8}, for any $T > 0$, there
exists a pair of functions $(u_\varepsilon, v_\varepsilon)$ such that
\begin{enumerate}
\item $u_\varepsilon\in L^\infty(0, T; H^1_0(\Omega))\cap C([0, T]; 
L^p(\Omega))\cap L^2(0, T; H^3(\Omega))$, where 
$1\leq p <\infty$ if $n= 1, 2$ and $2\leq p <\frac{2n}{n-2}$ if $n\geq 3$,

\item $\partial_t u_\varepsilon\in {L^2(0, T; (H^2(\Omega))')}$,

\item $u_\varepsilon(x,0)= u_0(x)$ for all $x\in\Omega$,

\item $v_\varepsilon \in {L^2(0, T; H^1_0(\Omega))}$,
\end{enumerate}
which satisfies equation \eqref{2-2} in the following weak sense
\begin{equation} \label{2-3}
\begin{aligned}
&\int^T_0\langle \partial_t u_\varepsilon, \phi\rangle dt +
\iint_{Q_T}\big(-\Delta u_\varepsilon + f(u_\varepsilon)\big)\phi \,dx\,dt
\\
&=-\iint_{Q_T}D_\varepsilon(u_\varepsilon)\big(-\nabla\Delta u_\varepsilon
 +F''(u_\varepsilon)\nabla u_\varepsilon\big) \cdot\nabla\phi \,dx\,dt
\end{aligned}
\end{equation}
for all test functions $\phi\in L^2(0, T; H^2(\Omega)\cap H^1_0(\Omega))$.
In addition, $u_\varepsilon$ satisfies the  energy inequality
\begin{equation}
\label{2-4}
E(u_\varepsilon)+\int^t_0\int_\Omega
\left(D_\varepsilon(u_\varepsilon(x,\tau))|\nabla v_\varepsilon(x,\tau)|^2
+ |v_\varepsilon(x,\tau)|^2\right) \,dx\,d\tau \leq E(u_0),
\end{equation}
for all $t > 0$.
\end{theorem}

 To prove the above theorem, we apply a Galerkin approximation.
 Let $\{\phi_J\}_{j \in N}$ be the eigenfunctions of the Laplace operator on
$L^2 (\Omega)$ with Dirichlet boundary condition, i.e.,
\begin{equation}\label{2-5}
\begin{gathered}
-\Delta\phi_J = \lambda_J\phi_J, \quad \text{in }\Omega, \\
\phi_J = 0,  \quad \text{on }\partial\Omega .
\end{gathered}
\end{equation}
The eigenfunctions $\{\phi_J\}^{\infty}_{j=1}$ form an orthogonal
basis for $L^2(\Omega), H^1(\Omega) \text{ and } H^2(\Omega)$.
Hence, for initial data $u_0 \in H^1(\Omega)$, we can find sequences of
scalars $(u^0_{N, j}$; $j=1, 2, \dots, N)^{\infty}_{N=1}$ such that
\begin{equation}\label{2-6}
\lim_{N \to \infty}{\sum^{N}_{j=1} u^0_{N,j}\phi_J} = u_0,  \quad
\text{in }H^1(\Omega).
\end{equation}
Let $V_{N}$ denote the linear span of $(\phi_1, \dots, \phi_{N})$ and
${\mathscr {P}_{N}}$ be the orthogonal projection from $L^2(\Omega)$
to $V_{N}$, that is 
\[
\mathscr {P}_{N}\phi:= \sum^{N}_{j=1}\Big(\int_{\Omega}\phi\phi_J\,dx\Big)\phi_J.
\]
Let $u^{N}(x,t) = \sum^N_{j=1} c^{N}_J(t)\phi_J(x)$,
$v^{N}(x,t) = \sum^{N}_{j=1} d^{N}_J(t)\phi_J(x)$ be the
approximate solution of \eqref{2-2} in $V_{N}$; that is, $u^{N}$, $v^{N}$ 
satisfy the g system of equations
\begin{gather}\label{2-7}
\int_{\Omega}\partial_tu^{N}\phi_J\,dx
= -\int_{\Omega} D_{\varepsilon}(u^{N})\nabla v^{N}\cdot\nabla\phi_J\,dx
-\int_{\Omega} v^{N}\phi_J\,dx, \\
\label{2-8}
\int_{\Omega}v^{N}\phi_J\,dx
 = \int_{\Omega}\nabla u^{N}\cdot\nabla\phi_J + f(u^{N})\phi_J\,dx,\\
\label{2-9}
u^{N}(x,0)= \sum^{N}_{j=1} u^0_{N,j}\phi_J(x),
\end{gather}
for $j=1, \dots, N$ and $u^0_{N,j} = \int_{\Omega} u_0\phi_J\,dx$.

This gives an initial value problem for a system of ordinary differential 
equations for $(c_1, \dots, c_{N})$
\begin{gather}\label{2-10}
\partial_t c^{N}_J(t)=-\sum^{N}_{k = 1}d^{N}_{k}(t)
 \int_\Omega D_{\varepsilon} \big(\sum^{N}_{i = 1} c^{N}_i(t)\phi_i(x)\big)
 \nabla\phi_{k}\nabla\phi_J\,dx - d^{N}_J(t),\\
\label{2-11}
d^{N}_J(t)= \lambda_Jc^{N}_J(t)
 + \int_\Omega f \big(\sum^{N}_{i= 1} c^{N}_i(t)\phi_i(x)\big)\phi_J\,dx,\\
\label{2-12}
c^{N}_J(0)= u^0_{N,j}= (u_0, \phi_J),
\end{gather}
which has to hold for $j=1, \dots, N$.

Define $\mathbf{X}(t)=\big(c^{N}_1(t),\dots,c^{N}_{N}(t)\big)$, 
$\mathbf{F}(t,\mathbf{X}(t))=\big(f_1(t,\mathbf{X}(t)),\dots,
f_{N}(t,\mathbf{X}(t))\big)$, where
\begin{align*}
f_J(t,\mathbf{X}(t))&= -\sum^{N}_{k = 1}\int_\Omega D_{\varepsilon}
\Big(\sum^{N}_{i = 1} c^{N}_i(t)\phi_i(x)\Big)\nabla\phi_{k}\nabla\phi_J\,dx
\\
&\quad\times \Big(\lambda_{k}c^{N}_{k}(t)
 +\int_\Omega f \big(\sum^{N}_{i= 1} c^{N}_i(t)\phi_i(x)\big)\phi_{k}\,dx\Big)
\\
&\quad -\lambda_Jc^{N}_J(t)-\int_\Omega f \Big(\sum^{N}_{k = 1}
c^{N}_{k}(t)\phi_{k}(x)\Big)\phi_J \,dx
\end{align*}
for $j=1, \dots, N$. Then  problem \eqref{2-10}-\eqref{2-12} is equivalent to the 
 problem
$$
\mathbf{X'}(t)=\mathbf{F}(t,\mathbf{X}(t)), \quad
\mathbf{X}(0)=(u^0_{N,1},\dots,u^0_{N,N}).
$$
Since the right hand side of the above equation is continuous, it follows from
the Cauchy-Peano Theorem \cite {TCS} that the problem \eqref{2-10}-\eqref{2-12} 
has a solution $\mathbf{X}(t)\in C^1[0, T_{N}]$,  for some 
$T_{N} > 0$, i. e., the system \eqref{2-7}-\eqref{2-9} has a local solution.

To prove the existence of solutions, we need some a priori estimates on $u^N$.

\begin{lemma}
For any $T > 0$, we have
\begin{gather*}
\| u^{N}\|_{ L^{\infty}(0, T; H^1_0(\Omega))}\leq C, \quad \text{for all } N,
\\
\|\partial_tu^{N}\|_{L^2(0,T;(H^2(\Omega))')}\leq C,  \quad \text{for all } N,
\end{gather*}
where $C$ independent of $N$.
\end{lemma}

\begin{proof}
 For any fixed $N \in N^{+}$, we multiply \eqref{2-7} by
$d^{N}_J(t)$ and sum over $j=1, \dots, N$ to obtain
\begin{equation} \label{2-13}
\int_{\Omega}\partial_tu^{N}v^{N}\,dx 
= -\int_{\Omega} D_{\varepsilon}(u^{N})|\nabla v^{N}|^2\,dx 
-\int_{\Omega} |v^{N}|^2 dx.
\end{equation}
Multiply \eqref{2-8} by $\partial_tc^{N}_J(t)$ and sum over $j=1, \dots, N$
to obtain
\begin{align*}
\int_{\Omega}v^{N}\partial_tu^{N}\,dx 
&= \int_{\Omega}\big(\nabla u^{N}\partial_t\nabla u^{N} + f(u^{N})\partial_tu^{N}
 \big)dx,
\\
&=\frac{d}{dt}\int_\Omega\big(\frac{1}{2}|\nabla u^{N}|^2+ F(u^{N})\big) dx.
\end{align*}
By \eqref{2-13} and the above identity, we have
\begin{equation}
\label{2-14}
\frac{d}{dt}\int_\Omega\big(\frac12|\nabla u^{N}|^2+ F(u^{N})\big) dx
=-\int_{\Omega}D_{\varepsilon}(u^{N})|\nabla v^{N}|^2 dx
-\int_{\Omega}|v^{N}|^2\,dx.
\end{equation}
Replacing $t$ by $\tau$ in \eqref{2-14} and integrating over $\tau\in[0, t]$,
by \eqref{1-6} and the Sobolev embedding theorem we obtain
\begin{align*}
&\int_{\Omega}\left(\frac{1}{2}|\nabla u^{N}(x,t)|^2 + F(u^{N}(x,t))\right) dx
\\
&+\int^{t}_0 \int_{\Omega}\left(D_{\varepsilon}(u^{N}(x,\tau))|\nabla v^{N}(x,\tau)|^2 +|v^{N}(x,\tau)|^2\right) \,dx\,d\tau
\\
&=\int_{\Omega}\left(\frac{1}{2}|\nabla u^{N}(x,0)|^2 + F(u^{N}(x,0))\right) dx
\\
&\leq \frac{1}{2}\|\nabla u^{N}(x,0)\|^2_2+ k_1\|u^{N}(x,0)\|^{r+1}_{r+1}
 +k_1|\Omega|.
\\
&\leq \frac{1}{2}\|\nabla u_0\|^2_2
 + k_1 C\|u_0\|^{r+1}_{H^1(\Omega)}+ k_1 |\Omega|\leq C.
\end{align*}
The last inequality follows from $u_0 \in H^1(\Omega)$. This implies
\begin{equation} \label{2-15}
\begin{aligned}
&\int_{\Omega}\big(\frac{1}{2}|\nabla u^{N}(x,t)|^2+k_0|u^{N}|^{r+1}\big)dx\\
&+\int^{t}_0\int_{\Omega}\big(D_{\varepsilon}(u^{N}(x,\tau))|\nabla v^{N}(x,\tau)|^2
 +|v^{N}(x,\tau)|^2\big) \,dx\,d\tau \leq C.
\end{aligned}
\end{equation}
By \eqref{2-15} and Poincar\'{e}'s inequality we have
$$
\| u^{N}\|_{H^1(\Omega)}\leq C,  \quad \text{for }  t>0.
$$
This estimate implies that the coefficients $\{c^{N}_J: j=1, \dots, N\}$
are bounded in time and therefore a global solution to the system
\eqref{2-7}-\eqref{2-9} exists. In addition, for any $T > 0$, we have
\begin{equation} \label{2-16}
u^{N} \in L^{\infty}(0, T; H^1_0(\Omega)), \quad
\| u^{N}\|_{ L^{\infty}(0, T; H^1_0(\Omega))}\leq C, \quad \text{for all } N.
\end{equation}
Inequality \eqref{2-15} implies
\begin{gather}
\label{2-17}
\|\sqrt{D_{\varepsilon}(u^{N})}\nabla v^{N} \|_{L^2(Q_{T})} \leq C,
 \quad \text{for all } N,\\
\label{2-18}
\|v^{N} \|_{L^2(Q_{T})} \leq C, \;\; \quad \text{for all} \;N.
\end{gather}
By the Sobolev embedding theorem, the growth condition \eqref{1-2} and \eqref{2-1},
for $|u|> \varepsilon$, we obtain
\[
\int_{\Omega}|D_{\varepsilon}(u^{N})|^{n/2}\,dx
\leq (C_1+1)\int_{\Omega}|u^{N}|^{m\cdot\frac{n}{2}}\,dx,
\leq C\|u^{N}\|^{mn/2}_{H^1(\Omega)} \leq C.
\]
If $|u|\leq \varepsilon$, obviously we obtain the above estimate. This implies
\begin{equation} \label{2-19}
\|D_{\varepsilon}(u^{N})\|_{ L^{\infty}(0, T; L^{n/2}(\Omega))}\leq C,
 \quad \text{for all } N.
\end{equation}
For any $\phi \in L^2(0, T; H^2(\Omega))$, we obtain
$\mathscr {P}_{N}\phi= \sum^{N}_{j=1}a_J(t)\phi_J$, where
$a_J(t)=\int_{\Omega}\phi\phi_Jdx$. Multiplying \eqref{2-7} by $a_J(t) $,
summing over $j=1, 2,\dots, N$, by H\"{o}lder's
inequality, \eqref{2-17}-\eqref{2-19} and the Sobolev embedding theorem, we have
\begin{align*}
&\big|\int^T_0\int_{\Omega}\partial_tu^{N}\phi\,dx\,dt\big| \\
&= \big|\int^T_0\int_{\Omega} \partial_tu^{N} \mathscr {P}_{N}\phi \,dx\,dt\big|\\
&= \big|\int^T_0\int_{\Omega}\big( D_{\varepsilon}(u^{N})\nabla v^{N}
\nabla\mathscr {P}_{N}\phi + v^{N} \mathscr {P}_{N}\phi\big) \,dx\,dt \big|\\
&\leq\int^T_0\|\sqrt {D_{\varepsilon}(u^{N})}\|_{n} \|
\sqrt {D_{\varepsilon}(u^{N})}\nabla v^{N}\|_2\|\nabla
\mathscr {P}_{N}\phi\|_{2^{*}}\,dt
+\int^T_0\| v^{N}\|_2\| \mathscr {P}_{N}\phi\|_2\,dt\\
&\leq C\int^T_0\| \sqrt {D_{\varepsilon}(u^{N})}\nabla v^{N}\|_2 \|\phi\|_{H^2}
+ \| v^{N}\|_2\| \phi\|_{H^2}\,dt\\
&\leq C\big( \| \sqrt {D_{\varepsilon}(u^{N})}\nabla v^{N}\|_{L^2(Q_{T})}
 + \|v^{N}\|_{L^2(Q_{T})}\big)\|\phi\|_{L^2(0,T;H^2(\Omega))}
\\
&\leq C \|\phi\|_{L^2(0,T;H^2(\Omega))}.
\end{align*}
Hence,
\begin{equation}\label{2-20}
\|\partial_tu^{N}\|_{L^2(0,T;(H^2(\Omega))')}\leq C  \quad \text{for all } N.
\end{equation}
The proof is complete.
\end{proof}

\begin{lemma}
Suppose $u_0\in H^1(\Omega)$, under assumptions \eqref{1-2} and 
\eqref{1-6}-\eqref{1-8}, for any $T > 0$, there
exists a pair of functions $(u_\varepsilon, v_\varepsilon)$ such that
\begin{enumerate}
\item $u_\varepsilon\in L^\infty(0, T; H^1_0(\Omega))\cap C([0, T]; L^p(\Omega))$, 
where $1\leq p <\infty$ if $n= 1, 2$ and $2\leq p <\frac{2n}{n-2}$ if $n\geq 3$,

\item $\partial_t u_\varepsilon\in {L^2(0, T; (H^2(\Omega))')}$,

\item $u_\varepsilon(x,0)= u_0(x)$ for all $x\in\Omega$,

\item $v_\varepsilon \in {L^2(0, T; H^1_0(\Omega))}$,
\end{enumerate}
which satisfies
$$
\int^T_0\langle\partial_tu_{\varepsilon}, \phi\rangle\,dt
=-\int^T_0\int_{\Omega} D_{\varepsilon}(u_{\varepsilon})
\nabla v_{\varepsilon}\cdot \nabla\phi \,dx\,dt
 -\int^T_0\int_{\Omega}v_{\varepsilon}\phi \,dx\,dt.
$$
\end{lemma}

\begin{proof}
Since the embedding $H^1_0(\Omega)\hookrightarrow L^{p}(\Omega)$ is compact for
 $1 \leq p < \infty$ if $n = 1, 2$ and $1 \leq p < \frac{2n}{n-2}$ 
if $n \geq 3$, $L^{p}(\Omega)\hookrightarrow (H^2(\Omega))'$ is continuous for 
$p \geq 1$ if $n \leq 3$, $p > 1$ if $n = 4$ and $p \geq \frac {2n}{n+4}$ 
if $n\geq 5$. Using the Aubin-Lions lemma (Lions \cite {JS}), we can find
 a subsequence which we still denote by $u^{N}$ and 
$u_{\varepsilon} \in L^{\infty}(0, T; H^1_0(\Omega))$, such that as 
$N \to \infty $
\begin{gather} \label{2-21}
u^{N} \rightharpoonup u_{\varepsilon}, \quad\text{weak-*} 
\text{ in }L^{\infty}(0, T; H^1_0(\Omega)),\\
\label{2-22}
u^{N} \to u_{\varepsilon}, \quad\text{strongly in } C([0, T]; L^{p}(\Omega)),\\
\label{2-23}
u^{N} \to u_{\varepsilon}, \quad \text{strongly in } L^2(0, T; L^{p}(\Omega))
 \text{ and almost everywehre in }Q_{T},\\
\label{2-24}
\partial_t u^{N} \rightharpoonup \partial_t u_{\varepsilon}, \quad
\text{weakly in } L^2(0, T; (H^2(\Omega))'),
\end{gather}
where $2 \leq p < 2^{*}$ if $n \geq 3$ and $1 \leq p < \infty$ if $n = 1, 2$.

By multiplying \eqref{2-7} by $a_J(t) $ and integrating \eqref{2-7} over
 $t \in [0, T]$, we obtain
\begin{equation} \label{2-25}
\begin{aligned}
&\int^T_0\int_{\Omega}\partial_tu^{N}a_J(t)\phi_J \,dx\,dt\\
&=-\int^T_0\int_{\Omega} D_{\varepsilon}(u^{N})\nabla v^{N}\cdot a_J(t)\nabla\phi_J \,dx\,dt - \int^T_0\int_{\Omega}v^{N}a_J(t)\phi_J \,dx\,dt.
\end{aligned}
\end{equation}
To pass to the limit in \eqref{2-25}, we  need the convergence
of $v^{N}$ and $ D_{\varepsilon}(u^{N})\nabla v^{N}$.
 By \eqref{2-17} and $D_{\varepsilon}(u^{N}) \geq \varepsilon^m$, then
\begin{equation}\label{2-26}
\|\nabla v^{N}\|_{L^2(Q_{T})} \leq C\varepsilon^{-\frac {m}{2}} < \infty, \quad
 \text{for any } \varepsilon > 0.
\end{equation}
This implies that $\{\nabla v^{N}\} $ is a bounded sequence in $L^2(Q_{T})$,
thus there exists a subsequence, not relabeled, and
$\zeta_{\varepsilon} \in L^2(Q_{T})$ such that
\begin{equation} \label{2-27}
\nabla v^{N} \rightharpoonup \zeta_{\varepsilon}, \quad \text{weakly in }
L^2(Q_{T}).
\end{equation}
By \eqref{2-26} and Poincar\'{e}'s inequality, we have
$$
\|v^{N}\|_{L^2(0,T;H^1_0(\Omega))} \leq C\varepsilon^{-\frac {m}{2}}
< \infty,\quad \text{for any }\varepsilon > 0.
$$
Hence we can find a subsequence of $v^{N}$, not relabeled, and
$v_{\varepsilon} \in L^2(0, T; H^1_0(\Omega))$ such that
\begin{equation} \label{2-28}
v^{N} \rightharpoonup v_{\varepsilon}, \quad\text{weakly in }
L^2(0, T; H^1_0(\Omega)).
\end{equation}
For any $ g \in L^2(0, T; H^1_0(\Omega))$, by \eqref{2-26} and \eqref{2-27} we have
\begin{align*}
&\lim_{N \to \infty}\int^T_0\int_{\Omega}\nabla v^{N}g \,dx\,dt
 = \int^T_0\int_{\Omega}\zeta_{\varepsilon}g \,dx\,dt
\\
&=\lim_{N \to \infty}\int^T_0\int_{\Omega}v^{N}\nabla g \,dx\,dt
 =\int^T_0\int_{\Omega}\nabla v_{\varepsilon} g \,dx\,dt.
\end{align*}
Hence $\zeta_{\varepsilon} = \nabla v_{\varepsilon} $ almost all
in $Q_{T}$ and
\begin{equation} \label{2-29}
\nabla v^{N} \rightharpoonup \nabla v_{\varepsilon}, \quad \text{weakly in }
 L^2(Q_{T}).
\end{equation}
By \eqref{2-18}, we can extract a further sequence of $v^{N}$, not relabeled,
and $\eta_{\varepsilon} \in L^2(Q_{T})$
such that
\begin{equation} \label{2-30}
v^{N} \rightharpoonup \eta_{\varepsilon}, \quad \text{weakly in } L^2(Q_{T}).
\end{equation}
By \eqref{2-28} and \eqref{2-30} for any
$g \in L^2(Q_{T})\subset L^2(0, T; H^{-1}(\Omega))$, we have
$$
\lim_{N\to \infty}\int^T_0\int_{\Omega}v^{N}g \,dx\,dt
= \int^T_0\int_{\Omega}v_{\varepsilon}g \,dx\,dt
= \int^T_0\int_{\Omega}\eta_{\varepsilon}g \,dx\,dt.
$$
This implies $\eta_{\varepsilon} = v_{\varepsilon}$ almost all $Q_{T}$ and
\begin{equation}\label{2-31}
v^{N} \rightharpoonup v_{\varepsilon}, \quad \text{weakly in } L^2(Q_{T}).
\end{equation}
Consequently we have the bound
\begin{equation} \label{2-32}
 \int_{Q_{T}} |v_{\varepsilon}|^2 \,dx\,dt \leq C.
\end{equation}
For any $t \in [0, T]$, by $D_{\varepsilon}(u^{N}) \leq C(1+|u^{N}|^m)$, we have
$$
\big (D_{\varepsilon}(u^{N})\big)^{n/2}
 \leq C(1+|u^{N}|^m)^{n/2} \leq (C(1+|u^{N}|))^{mn/2},
$$
where $2\leq \frac{mn}{2} < 2^{*}$. By \eqref{2-22},
$C(1+|u^{N}|) \to C(1+|u_{\theta}|) \;\text{in } L^{mn/2}(\Omega)$.
Since $D_{\varepsilon}$ is continuous and \eqref{2-23}, we obtain
$$
D_{\varepsilon}(u^{N}) \to D_{\varepsilon}(u_{\varepsilon}), \quad \text{a.e. in }
\Omega.
$$
The generalized Lebesgue convergence theorem \cite{HWA} gives
$$
D_{\varepsilon}(u^{N}) \to D_{\varepsilon}(u_{\varepsilon}), \quad
\text{in } L^{n/2}(\Omega).
$$
This implies
$$
\|D_{\varepsilon}(u^{N}) - D_{\varepsilon}(u_{\varepsilon}) \|_{n/2} \to 0,
 \quad \text{as } N \to \infty.
$$
The above estimate holds  for any $t \in [0,T]$, and we can take supremum
on both sides of the above estimate to obtain
$$
\sup _{t\in [0,T]}\|D_{\varepsilon}(u^{N})
- D_{\varepsilon}(u_{\varepsilon}) \|_{n/2} \to 0, \quad \text{as }
 N \to \infty.
$$
This implies
\begin{equation}
\label{2-33}
D_{\varepsilon}(u^{N}) \to D_{\varepsilon}(u_{\varepsilon}), \quad \text{strongly in }
C (0, T; L^{n/2}(\Omega)).
\end{equation}
By $\sqrt {D_{\varepsilon}(u^{N})} \leq C (1 + |u^{N}|^{\frac{m}{2}})$, \eqref{2-22},
\eqref{2-23} and the generalized Lebesgue convergence theorem, similarly, we have
\begin{equation} \label{2-34}
\sqrt{D_{\varepsilon}(u^{N})} \to \sqrt{D_{\varepsilon}(u_{\varepsilon})},
\quad \text{strongly in } C (0, T; L^{n}(\Omega)).
\end{equation}
For any $\varphi \in L^2(0, T; L^{2^{*}}(\Omega))$, by H\"{o}lder's inequality
 we have
\begin{align*}
&\Big|\iint_{Q_{T}} \Big(\sqrt{D_{\varepsilon}(u^{N})}\nabla v^{N}\varphi
-\sqrt{D_{\varepsilon}(u_{\varepsilon})}\nabla v_{\varepsilon}\varphi\Big)
\,dx\,dt\Big|\\
&=\Big|\iint_{Q_{T}} \Big([\sqrt{D_{\varepsilon}(u^{N})}
 -\sqrt{D_{\varepsilon}(u_{\varepsilon})}]\nabla v^{N}\varphi
+\sqrt{D_{\varepsilon}(u_{\varepsilon})}[\nabla v^{N}\varphi
 -\nabla v_{\varepsilon}\varphi ]\Big) \,dx\,dt \Big|
\\
&\leq \int^T_0\|\sqrt{D_{\varepsilon}(u^{N})}
 -\sqrt{D_{\varepsilon}(u_{\varepsilon})}\|_{n}\|\nabla v^{N}\|_2
 \|\varphi\|_{2^{*}}\,dt
\\
&\quad +\Big|\iint_{Q_{T}}\sqrt{D_{\varepsilon}(u_{\varepsilon})}
 \varphi[\nabla v^{N}-\nabla v_{\varepsilon}] \,dx\,dt\Big|
\\
&\leq\sup _{t\in [0,T]}\|\sqrt{D_{\varepsilon}(u^{N})}
 -\sqrt{D_{\varepsilon}(u_{\varepsilon})}\|_{n}\|\nabla v^{N}\|_{L^2(Q_{T})}
 \|\varphi\|_{L^2(0, T; L^{2^{*}}(\Omega))}
\\
&\quad +\Big|\iint_{Q_{T}}\sqrt{D_{\varepsilon}(u_{\varepsilon})}
 \varphi[\nabla v^{N}-\nabla v_{\varepsilon}] \,dx\,dt \Big|
\\
&\equiv I + II.
\end{align*}
By \eqref{2-29} and \eqref{2-34}, $I \to 0$ as $N \to \infty$.
By H\"{o}lder's inequality and \eqref{2-34} we have
\begin{align*}
\iint_{Q_{T}}|\sqrt{D_{\varepsilon}(u_{\varepsilon})}\varphi|^2 \,dx\, dt
&\leq \int^T_0 \Big(\int_{\Omega}\big(D_{\varepsilon}(u_{\varepsilon})\big)^{n/2}
\,dx \Big)^{n/2}
\Big(\int_{\Omega}|\varphi|^{\frac {2n}{n-2}} dx \Big)^{\frac {n-2}{n}}dt
\\
&\leq \sup _{t\in [0,T]}\|\sqrt{D_{\varepsilon}
(u_{\varepsilon})}\|^2_{n}\int^T_0\|\varphi\|^2_{L^{2^{*}}(\Omega)} \,dt
\\
&\leq C \|\varphi\|^2_{L^2(0,T;L^{2^{*}}(\Omega))}.
\end{align*}
This implies
\begin{equation}\label{2-35}
\sqrt{D_{\varepsilon}(u_{\varepsilon})}\varphi \in L^2(Q_{T}).
\end{equation}
Thus $II \to 0 $ as $N \to \infty$ by \eqref{2-29}. Hence
\begin{equation} \label{2-36}
\sqrt{D_{\varepsilon}(u^{N})}\nabla v^{N} \rightharpoonup
\sqrt{D_{\varepsilon}(u_{\varepsilon})}\nabla v_{\varepsilon}, \quad
 \text{weakly in } L^2(0,T;L^{\frac {2n}{n+2}}(\Omega)).
\end{equation}
Next we consider the convergence of $D_{\varepsilon}(u^{N})\nabla v^{N}$.
By \eqref{2-17}, \eqref{2-36}
and $L^2(Q_{T}) \subset L^2(0,T;L^{\frac {2n}{n+2}}(\Omega))$,
we can extract a further
sequence, not relabeled, such that
\begin{equation}\label{2-37}
\sqrt {D_{\varepsilon}(u^{N})}\nabla v^{N} \rightharpoonup
\sqrt {D_{\varepsilon}(u_{\varepsilon})}\nabla v_{\varepsilon}, \quad
\text{weakly in }L^2(Q_{T}).
\end{equation}
By H\"{o}lder's inequality and \eqref{2-17}, we have
\begin{equation} \label{2-38}
\begin{aligned}
&\iint_{Q_{T}}\sqrt {D_{\varepsilon}(u^{N})}\nabla v^{N}
 \cdot\sqrt {D_{\varepsilon}(u_{\varepsilon})}\nabla v_{\varepsilon} \,dx\,dt
\\
&\leq \|\sqrt {D_{\varepsilon}(u^{N})}\nabla v^{N}\|_{L^2(Q_{T})}
\|\sqrt {D_{\varepsilon}(u_{\varepsilon})}\nabla v_{\varepsilon}\|_{L^2(Q_{T})}
\\
&\leq C\|\sqrt {D_{\varepsilon}(u_{\varepsilon})}\nabla v_{\varepsilon}
\|_{L^2(Q_{T})},
\end{aligned}
\end{equation}
where $C$ is independent of $\varepsilon$. Taking the limit of \eqref{2-38}
on both sides, by \eqref{2-37} we have
\begin{equation}\label{2-39}
\|\sqrt {D_{\varepsilon}(u_{\varepsilon})}\nabla v_{\varepsilon}\|_{L^2(Q_{T})}
\leq C.
\end{equation}
For any $\varphi \in L^2(0,T;L^{2^{*}}(\Omega))$, by H\"{o}lder's inequality
 we obtain
\begin{align*}
&\Big|\iint_{Q_{T}}\left( D_{\varepsilon}(u^{N})\nabla v^{N}\varphi
- D_{\varepsilon}(u_{\varepsilon})\nabla v_{\varepsilon}\varphi\right) \,dx\,dt
 \Big| \\
&\leq \Big|\iint_{Q_{T}} [\sqrt{D_{\varepsilon}(u^{N})}
 -\sqrt{D_{\varepsilon}(u_{\varepsilon})}]\sqrt{D_{\varepsilon}(u^{N})}
 \nabla v^{N}\varphi \,dx\,dt \Big| \\
&\quad + \Big|\iint_{Q_{T}} \sqrt{D_{\varepsilon}(u_{\varepsilon})}
 [\sqrt{D_{\varepsilon}(u^{N})}\nabla v^{N}\varphi
 -\sqrt{D_{\varepsilon}(u_{\varepsilon})}\nabla v_{\varepsilon}\varphi ]\,\,dx\,dt
 \Big|
\\
&\leq \int^T_0\|\sqrt{D_{\varepsilon}(u^{N})}
 -\sqrt{D_{\varepsilon}(u_{\varepsilon})}\|_{n}\|
 \sqrt{D_{\varepsilon}(u^{N})}\nabla v^{N}\|_2\|\varphi\|_{2^{*}}\,dt
\\
&\quad +\Big|\iint_{Q_{T}}\sqrt{D_{\varepsilon}(u_{\varepsilon})}
 \varphi[\sqrt{D_{\varepsilon}(u^{N})}\nabla v^{N}
 -\sqrt{D_{\varepsilon}(u_{\varepsilon})}\nabla v_{\varepsilon}] \,dx\,dt \Big|
\\
&\leq \sup_{t\in [0,T]}\|\sqrt{D_{\varepsilon}(u^{N})}
 -\sqrt{D_{\varepsilon}(u_{\varepsilon})}\|_{n}
\|\sqrt{D_{\varepsilon}(u^{N})}\nabla v^{N}\|_{L^2(Q_{T})}
 \|\varphi\|_{L^2(0, T; L^{2^*}(\Omega))}
\\
&\quad +\Big|\iint_{Q_{T}}\sqrt{D_{\varepsilon}(u_{\varepsilon})}
\varphi[\sqrt{D_{\varepsilon}(u^{N})}\nabla v^{N}
-\sqrt{D_{\varepsilon}(u_{\varepsilon})}\nabla v_{\varepsilon}] \,dx\,dt \Big|
\\
&= I + II.
\end{align*}
By \eqref{2-34} and \eqref{2-37}, $ I \to 0$ as $N \to \infty $.
By \eqref{2-35} and \eqref{2-37}, we have $II \to 0$ as $N \to \infty $. Thus
\begin{equation}
\label{2-40}
D_{\varepsilon}(u^{N})\nabla v^{N} \rightharpoonup D_{\varepsilon}(u_{\varepsilon})\nabla v_{\varepsilon}, \;\;\text{weakly in} \;L^2(0, T; L^{\frac {2n}{n+2}}(\Omega)).
\end{equation}
For any $\phi \in L^2(0, T; H^2(\Omega)\cap H^1_0(\Omega))$, we obtain $\mathscr {P}_{n}\phi= \sum^{n}_{j=1}a_J(t)\phi_J$, where $a_J(t)=\int_{\Omega}\phi\phi_Jdx$, then $\mathscr {P}_{n}\phi$ converges strongly to $\phi$ in $L^2(0, T; H^2\cap H^1_0(\Omega))$ and $a_J(t)\in L^2(0, T)$. For $\phi_J \in H^2(\Omega)$, by Sobolev embedding theorem, we obtain
$$
\|\nabla \phi_J\|_{2^{*}} \leq C\|\nabla \phi_J\|_{H^1(\Omega)} \leq C.
$$
Thus $a_J(t)\nabla\phi_J \in L^2(0, T; L^{2^{*}})$ and
\[
a_J(t)\phi_J\in L^2(0, T; H^2\cap H^1_0(\Omega))\subset
 L^2(0, T; H^{-1}(\Omega)).
\]
Taking the limit as $N \to \infty$ on both sides of \eqref{2-25}, by \eqref{2-24},
\eqref{2-40} and \eqref{2-28}, we have
\begin{equation} \label{2-41}
\begin{aligned}
&\int^T_0\langle\partial_tu_{\varepsilon}, a_J(t)\phi_J\rangle\,dt\\
&=-\int^T_0\int_{\Omega} D_{\varepsilon}(u_{\varepsilon})
 \nabla v_{\varepsilon}\cdot a_J(t)\nabla\phi_J \,dx\,dt
 -\int^T_0\int_{\Omega}v_{\varepsilon}a_J(t)\phi_J \,dx\,dt,
\end{aligned}
\end{equation}
for all $j \in N$.

Then we sum over $j= 1, 2, \dots, n$ on both sides \eqref{2-41} to get
\begin{equation} \label{2-42}
\begin{aligned}
&\int^T_0\langle\partial_tu_{\varepsilon}, \mathscr {P}_{n}\phi\rangle \,dt\\
&=-\int^T_0\int_{\Omega} D_{\varepsilon}(u_{\varepsilon})\nabla v_{\varepsilon}
 \cdot \nabla\mathscr {P}_{n}\phi \,dx\,dt
 - \int^T_0\int_{\Omega}v_{\varepsilon}\mathscr {P}_{n}\phi \,dx\,dt.
\end{aligned}
\end{equation}
Since $\mathscr {P}_{n}\phi $ converges strongly to $\phi$ in
$L^2(0, T; H^2(\Omega))$, thus as $n \to \infty$,
\begin{align*}
\int^T_0\|\nabla \mathscr {P}_{n}\phi -\nabla\phi\|^2_{2^{*}} \,dt
&\leq \int^T_0\|\nabla \mathscr {P}_{n}\phi -\nabla\phi\|^2_{H^1} \,dt\\
& \leq \int^T_0\|\mathscr {P}_{n}\phi -\phi\|^2_{H^2}\,dt \to 0.
\end{align*}
This implies that $\nabla\mathscr {P}_{n}\phi $ converges strongly to
$\nabla\phi$ in $L^2(0, T; L^{2^{*}}(\Omega))$. Thus we obtain
\begin{gather}\label{2-43}
\mathscr {P}_{n}\phi \rightharpoonup \phi, \quad \text{weakly in }
L^2(0, T; H^2(\Omega) \cap H^1_0(\Omega)),\\
\label{2-44}
\nabla\mathscr {P}_{n}\phi \rightharpoonup \nabla\phi, \quad \text{weakly in }
 L^2(0, T; L^{2^{*}}(\Omega)).
\end{gather}
By $L^2(0, T; H^1_0(\Omega)) \subset L^2(0, T; H^{-1}(\Omega))$, we take the
limit as $n \to \infty$ on both sides \eqref{2-42}, then obtain
\begin{equation} \label{2-45}
\int^T_0\langle\partial_tu_{\varepsilon}, \phi\rangle\,dt
=-\int^T_0\int_{\Omega} D_{\varepsilon}(u_{\varepsilon})\nabla v_{\varepsilon}
\cdot \nabla\phi\,\,dx\,dt
- \int^T_0\int_{\Omega}v_{\varepsilon}\phi \;\,dx\,dt.
\end{equation}
As for the initial value, by \eqref{2-9} as $N \to \infty$,
$$
u^{N}(x,0) \to u_0(x) \quad \text{in } L^2(\Omega).
$$
By \eqref{2-22}, $u_\varepsilon(x,0) = u_0(x)$ in $L^2(\Omega)$.
The proof is complete.
\end{proof}

\begin{proof}[Proof of Theorem \ref{th2.1}]
We need only to check that $u_{\varepsilon} \in L^2(0, T; H^{3}(\Omega))$,
 $v_{\varepsilon} = -\Delta u_{\varepsilon} + f(u_{\varepsilon})$ and 
$\nabla v_{\varepsilon} = -\nabla\Delta u_{\varepsilon} + F''(u_{\varepsilon})
\nabla u_{\varepsilon}$. First we consider the convergence of 
$\nabla u^{N}$ and $f(u^{N})$. By \eqref{2-21}, we have
$$
\int^T_0 \|\nabla u^{N} \|^2_2 dt \leq C.
$$
Hence we can find a subsequence of $u^{N}$, not relabeled, and 
$\upsilon \in L^2(Q_{T})$, such that
\begin{equation} \label{2-46}
\nabla u^{N} \rightharpoonup \upsilon \;\;\text{weakly in}\; L^2(Q_{T}).
\end{equation}
For any $\phi \in L^2(0, T; H^1_0(\Omega))$, by integration by parts we have
$$
\lim _{N \to  \infty}\int^T_0\int_{\Omega}\nabla u^{N}\phi \,dx\,dt
=\lim _{N \to  \infty}\int^T_0\int_{\Omega}u^{N}\nabla \phi \,dx\,dt.
$$
By \eqref{2-21}, \eqref{2-46} and 
$\nabla \phi\in L^2(Q_{T}) \subset L^1(0,T;H^{-1}(\Omega))$ we have
$$
\int^T_0\int_{\Omega}\upsilon\phi \,dx\,dt
=\int^T_0\int_{\Omega}u_{\varepsilon}\nabla \phi \,dx\,dt
=\int^T_0\int_{\Omega}\nabla u_{\varepsilon}\phi \,dx\,dt.
$$
Hence $\upsilon = \nabla u_{\varepsilon} $ almost all in $\Omega\times[0, T]$ and
\begin{equation} \label{2-47}
\nabla u^{N} \rightharpoonup \nabla u_{\varepsilon} \quad \text{weakly in }
 L^2(Q_{T}).
\end{equation}
By $|F'(u^{N})|\leq C (1 + |u^{N}|^{r})$, \eqref{2-22}, \eqref{2-23} 
and the general dominated convergence theorem, similarly, we have
\begin{equation} \label{2-48}
F'(u^{N}) \to F'(u_{\varepsilon}) \quad \text{strongly in } C (0, T; L^{q}(\Omega)),
\end{equation}
for $1 \leq q < \infty$ if $n = 1, 2$ and $2 \leq q < \frac {2n}{r(n-2)}$ if 
$n \geq 3$.

By the growth condition \eqref{1-7} and the Sobolev embedding theorem, we obtain
\begin{align*}
\|f(u^{N})\|^2_{L^2(\Omega)} &= \int_{\Omega}(F'(u^{N}))^2\,dx \\
&\leq C\int_{\Omega}(|u^{N}|^{r}+1)^2\,dx\\
&\leq 2C\int_{\Omega}|u^{N}|^{2r}\,dx+2C|\Omega|\\
&\leq C \|u^{N}\|^{2r}_{H^1(\Omega)}+C.
\end{align*}
Thus there exists a $w \in L^{\infty}(0, T; L^2(\Omega))$ such that
$$
F'(u^{N}) \rightharpoonup w \quad \text{weakly-* in } L^{\infty}(0, T; L^2(\Omega)).
$$
This implies
\begin{equation} \label{2-49}
\lim _{N \to \infty}\int^T_0\int_{\Omega}F'(u^{N})g \,dx\,dt 
= \int^T_0\int_{\Omega} wg \,dx\,dt,
\end{equation}
for any $g \in L^1(0, T; L^2(\Omega))$.

By H\"{o}lder's inequality, \eqref{2-48} and \eqref{2-49}, we have as 
$N \to \infty$
\begin{align*}
&\Big|\iint_{Q_{T}} \big(F'(u_{\varepsilon})-w \big)g \,dx\,dt\Big|\\
&\leq \iint_{Q_{T}} |F'(u_{\varepsilon})-F'(u^{N})||g| \,dx\,dt 
+\Big|\iint_{Q_{T}}[F'(u^{N})-w]g \,dx\,dt\Big|\\
&\leq \int^T_0\|F'(u_{\varepsilon})-F'(u^{N})\|_2\|g\|_2 dt
 + \Big|\iint_{Q_{T}}[F'(u^{N})-w]g \,dx\,dt \Big|\leq 0,
\end{align*}
for any $g \in L^1(0, T; L^2(\Omega))$. Hence
 $F'(u_{\varepsilon}) = w$ a.e. in $Q_{T}$ and
\begin{equation} \label{2-50}
F'(u^{N}) \rightharpoonup F'(u_{\varepsilon}) \quad \text{weak-* in } 
L^{\infty}(0, T; L^2(\Omega)).
\end{equation}
Multiplying \eqref{2-8} by $a_J(t)$ and integrating \eqref{2-8} over
$t \in [0, T]$, we obtain
\begin{equation} \label{2-51}
\begin{aligned}
&\int^T_0\int_{\Omega} v^{N}a_J(t)\phi_J \,dx\,dt\\
&= \int^T_0\int_{\Omega}\left(\nabla u^{N}\cdot a_J(t)\nabla\phi_J
+ F'(u^{N})a_J(t)\phi_J\right) \,dx\,dt.
\end{aligned}
\end{equation}
For any $\phi \in L^2(0, T; H^1_0(\Omega))$, we obtain 
$\mathscr {P}_{n}\phi= \sum^{n}_{j=1}a_J(t)\phi_J$, where
$a_J(t)\in L^2(0, T)$. Thus $a_J(t)\phi_J \in L^2(0,T;H^1_0(\Omega))$
and $a_J(t)\nabla\phi_J \in L^2(Q_{T})$. By \eqref{2-28}, \eqref{2-47}
and \eqref{2-50}, we take the limit as $N \to \infty$ on both sides of
 \eqref{2-51} to get
\begin{equation} \label{2-52}
\int^T_0\int_{\Omega}v_{\varepsilon}a_J(t)\phi_J \,dx\,dt
= \int^T_0\int_{\Omega}\left(\nabla u_{\varepsilon}a_J(t)\nabla\phi_J
+ F'(u_{\varepsilon})a_J(t)\phi_J\right) \,dx\,dt,
\end{equation}
for all $j \in N$.

Then we sum over $j= 1, \dots, n$ on both sides \eqref{2-52}, and obtain
\begin{equation} \label{2-53}
\int^T_0\int_{\Omega}v_{\varepsilon}\mathscr {P}_{n}\phi \,dx\,dt 
=\int^T_0\int_{\Omega} \left(\nabla u_{\varepsilon}\cdot \nabla\mathscr {P}_{n}\phi
 + F'(u_{\varepsilon})\mathscr {P}_{n}\phi\right) \,dx\,dt.
\end{equation}
Since $\mathscr {P}_{n}\phi $ converges strongly to $\phi$ in 
$L^2(0, T; H^1_0(\Omega))$, we have as $n \to \infty$
$$
\int^T_0\|\nabla \mathscr {P}_{n}\phi -\nabla\phi\|^2_2 \,dt
\leq \int^T_0\| \mathscr {P}_{n}\phi -\phi\|^2_{H^1_0} \,dt \to 0.
$$
This implies that $\nabla\mathscr {P}_{n}\phi $ converges strongly to 
$\nabla\phi$ in $L^2(Q_{T})$. Thus we obtain
\begin{gather} \label{2-54}
\mathscr {P}_{n}\phi \rightharpoonup \phi \quad \text{weakly in }
  L^2(0, T; H^1_0(\Omega))\\
\label{2-55}
\nabla\mathscr {P}_{n}\phi \rightharpoonup \nabla\phi \quad
\text{weakly in } L^2(Q_{T}).
\end{gather}
By $L^2(0, T; H^1_0(\Omega)) \subset L^2(0, T; H^{-1}(\Omega))$ and 
$L^{\infty}(0, T; L^2(\Omega)) \subset L^2(0, T; H^{-1}(\Omega))$, 
we take the limit as $n \to \infty$ on both sides \eqref{2-53}, and we obtain
$$
\iint_{Q_{T}}v_{\varepsilon}\phi \,dx\, dt 
=\iint_{Q_{T}} \left(\nabla u_{\varepsilon}\cdot \nabla\phi 
+ F'(u_{\varepsilon})\phi\right)\,dx\,dt.
$$
Since $F'(u_{\varepsilon}) \in L^{\infty}(0, T; L^2(\Omega))$ and 
$v_{\varepsilon} \in L^2(0, T; H^1_0(\Omega))$, it follows
from regularity theory \cite {LCE} that $u_{\varepsilon} \in L^2(0, T; H^2(\Omega))$.
 Hence
\begin{equation} \label{2-56}
v_{\varepsilon} = -\Delta u_{\varepsilon} + F'(u_{\varepsilon}) \quad 
\text{almost everywhere in}\;Q_{T}.
\end{equation}
Next we show $F'(u_{\varepsilon}) \in L^2(0, T; H^1(\Omega))$. 
By H\"{o}lder's inequality, the Sobolev embedding theorem and \eqref{1-8}, we have
\begin{align*}
\int^T_0\int_{\Omega}|\nabla F'(u_{\varepsilon})|^2 \,dx\,dt 
&= \int^T_0\int_{\Omega}|F''(u_{\varepsilon})|^2|\nabla u_{\varepsilon}|^2 \,dx\,dt\\
&\leq \int^T_0\big(\int_{\Omega}| F''(u_{\varepsilon})|^{2\times \frac {n}{2}} \,dx 
\big)^{2/n}\big(\int_{\Omega}|\nabla u_{\varepsilon}|
 ^{2\times \frac {n}{n-2}} \,dx \big)^{\frac {n-2}{n}}\,dt
\\
&\leq C \int^T_0\big(\int_{\Omega}(1+ |u_{\varepsilon}|^{r-1})^{n}\,dx 
\big)^{2/n}\|\nabla u_{\varepsilon}\|^2_{\frac {2n}{n-2}}\,dt
\\
&\leq C \int^T_0\big( 1+\int_{\Omega} |u_{\varepsilon}|^{(r-1)n} \,dx 
\big)^{2/n}\|\nabla u_{\varepsilon}\|^2_{H^1(\Omega)}\,dt
\\
&\leq C\int^T_0\big( 1+\|u_{\varepsilon}\|
^{\frac {4}{n-2}}_{\frac {2n}{n-2}}\big)\|u_{\varepsilon}\|^2_{H^2(\Omega)}\,dt
\\
&\leq C\big( 1+\|u_{\varepsilon}\|^{\frac {4}{n-2}}_{L^{\infty}(0, T; H^1(\Omega))}
 \big)\int^T_0\|u_{\varepsilon}\|^2_{H^2(\Omega)}\,dt
\\
&\leq C\big( 1+\|u_{\varepsilon}\|^{\frac {4}{n-2}}_{L^{\infty}
(0, T; H^1(\Omega))}\big)\|u_{\varepsilon}\|^2_{L^2(0, T; H^2(\Omega))} \leq C.
\end{align*}
Thus $\nabla F'(u_{\varepsilon}) \in L^2(Q_{T})$ and 
$F'(u_{\varepsilon}) \in L^2(0, T; H^1(\Omega))$. Combined with 
$v_{\varepsilon} \in L^2(0, T; H^1_0(\Omega))$, by \eqref{2-56} and 
regularity theory we have $u_{\varepsilon} \in L^2(0, T; H^{3}(\Omega))$ and
\begin{equation}\label{2-57}
\nabla v_{\varepsilon} = -\nabla\Delta u_{\varepsilon} 
+ F''(u_{\varepsilon})\nabla u_{\varepsilon}, \quad \text{almost everywhere in } Q_{T}.
\end{equation}
By \eqref{2-45}, \eqref{2-56} and \eqref{2-57}, we obtain
\begin{equation} \label{2-58}
\begin{aligned}
&\int^T_0\langle\partial_tu_{\varepsilon}, \phi\rangle \,dt
+ \int^T_0\int_{\Omega}(-\Delta u_{\varepsilon} + F'(u_{\varepsilon}))\phi \,dx\,dt
\\
&=-\int^T_0\int_{\Omega} D_{\varepsilon}(u_{\varepsilon})
(-\nabla\Delta u_{\varepsilon}
+ F''(u_{\varepsilon})\nabla u_{\varepsilon})\cdot \nabla\phi \,dx\,dt,
\end{aligned}
\end{equation}
for all $\phi \in L^2(0, T; H^2(\Omega)\cap H^1_0(\Omega))$.

Last we show that a weak solution $u_{\varepsilon}$ to \eqref{2-2} satisfies 
energy inequality \eqref{2-4}. Replacing $t$ by $\tau$ in \eqref{2-14} and 
integrating over $\tau\in[0,T]$, we have
\begin{equation} \label{2-59}
\begin{aligned}
&E(u^{N}(x,t)) +\int^{t}_0\int_{\Omega}D_{\varepsilon}(u^{N}(x,\tau))
 |\nabla v^{N}(x,\tau)|^2 \,dx\,d\tau \\
&+ \int^{t}_0\int_{\Omega}|v^{N}(x,\tau)|^2\,dx\,d\tau
= E(u^{N}(x,0)).
\end{aligned}
\end{equation}
Next, we pass to the limit in \eqref{2-59}. First, by mean value theorem
and \eqref{1-7} we have
\begin{equation} \label{2-60}
\begin{aligned}
&\big|\int_{\Omega}\left(F(u^{N}(t))-F(u_{\varepsilon}(t))\right)dx \big|\\
&\leq \int_{\Omega}|F'(\xi)||u^{N}(t)-u_{\varepsilon}(t)| \,dx\\
&\leq \int_{\Omega} C (|u^{N}(t)|^{r}
+ |u_{\varepsilon}(t)|^{r} + 1)|u^{N}(t)-u_{\varepsilon}(t)|\,dx,
\end{aligned}
\end{equation}
for $1 \leq r < \infty $ if $n= 1, 2 $ and $1 \leq r \leq \frac {n}{n-2}$ if
$n \geq 3$, $\xi = \lambda u^{N}(t) + (1-\lambda)u_{\varepsilon}(t)$ for some
$\lambda \in (0,1)$.
By H\"{o}lder's inequality, we have
\begin{equation} \label{2-61}
\int_{\Omega} |u^{N}(t)|^{r}|u^{N}(t)-u_{\varepsilon}(t)| \,dx
\leq \|u^{N}(t)-u_{\varepsilon}(t)\|_2\|u^{N}(t)\|^{r}_{2r}.
\end{equation}
Since the Sobolev embedding theorem says that
$H^1_0(\Omega) \hookrightarrow L^{p}(\Omega)$ for $1 \leq p \leq 2^{*}$
and the embedding is compact if $1 \leq p < 2^{*}$, by \eqref{2-21},
then for a subsequence, not relabeled, we have $u^{N}\to u_{\varepsilon} $
strongly in $L^{\infty}(0, T; L^2(\Omega))$ and $u^{N}$ is bounded in
$L^{\infty}(0, T; L^{2r}(\Omega))$. Hence, it follows from \eqref{2-61} that
\begin{equation} \label{2-62}
\int_{\Omega} |u^{N}(t)|^{r}|u^{N}(t)-u_{\varepsilon}(t)|\,dx \to 0,
\end{equation}
as $N \to \infty$, for almost all $t \in [0, T]$.

Similarly, we can prove that
\begin{equation} \label{2-63}
\int_{\Omega} (|u_{\varepsilon}(t)|^{r} + 1)|u^{N}(t)-u_{\varepsilon}(t)| \,dx 
\to 0,
\end{equation}
as $N \to \infty$, for almost all $t \in [0, T]$, by \eqref{2-60}, \eqref{2-62} 
and \eqref{2-63}, we have
\begin{equation} \label{2-64}
\lim _{N \to \infty}\int_{\Omega}F(u^{N}(t)) \,dx 
= \int_{\Omega}F(u_{\varepsilon}(t))\,dx.
\end{equation}
Since $u^{N}(x,0) \to u_0(x)$ strongly in $L^2(\Omega)$, we obtain
\begin{equation} \label{2-65}
\lim _{N \to \infty}\int_{\Omega}F(u^{N}(0))\,dx = \int_{\Omega}F(u_0(x))\,dx.
\end{equation}
By \eqref{2-47}, \eqref{2-64}, \eqref{2-37}, \eqref{2-29}, \eqref{2-59} 
and the weak lower semicontinuity of the $L^{p}$ norms \cite{LC}. Then
\begin{equation} \label{2-66}
\begin{aligned}
&\int_{\Omega} \Big(\frac{1}{2}|\nabla u_{\varepsilon}(x,t)|^2
+ F(u_{\varepsilon}(x,t))\Big) dx\\
&+ \int^{t}_0\int_{\Omega}\left(D_{\varepsilon}(u_{\varepsilon}(x,\tau))
|\nabla v_{\varepsilon}(x,\tau)|^2+ |v_{\varepsilon}(x,\tau)|^2\right)\,dx\,d\tau
\\
&\leq \lim_{N \uparrow \infty}\text{inf}\int_{\Omega}
\left( \frac{1}{2}|\nabla u^{N}(x,t)|^2 + F(u^{N}(x,t))\right) dx
\\
&\quad +\lim_{N \uparrow \infty}\text{inf}\iint_{Q_t}
 \left(D_{\varepsilon}(u^{N}(x,\tau))|\nabla v^{N}(x,\tau)|^2 + |v^{N}(x, \tau)|^2\right) \,dx\,d\tau
\\
&= \lim_{N \uparrow \infty}\text{inf}\; E(u^{N}(x,0)).
\end{aligned}
\end{equation}
Since $u^{N}(x,0) \to u_0(x)$ strongly in $H^1(\Omega)$, by \eqref{2-65} we have
\begin{equation} \label{2-67}
\lim _{N \to \infty} E(u^{N}(x,0)) = \int_{\Omega}
\Big( \frac{1}{2}|\nabla u_0(x)|^2 + F(u_0(x))\Big)dx.
\end{equation}
Combining \eqref{2-66} with \eqref{2-67} gives the energy inequality \eqref{2-4}.
The proof is complete.
\end{proof}

\section{Degenerate mobility}

This section is devoted to the existence of weak solutions to the equations 
\eqref{1-10}. Here we consider the limit of approximate solutions 
$u_{\varepsilon_i}$ defined in section $2$. The limiting value $u$ does
exist and solves the degenerate Allen-Cahn/Cahn-Hilliard equation 
in the weak sense.

\begin{theorem}\label{th3.1}
Suppose $u_0\in H^1(\Omega)$, under assumptions \eqref{1-2} and 
\eqref{1-6}-\eqref{1-8}, for any $T > 0$, problem \eqref{1-10} has 
a weak solution $u: Q_T\to R$ satisfying
\begin{enumerate}
\item $u \in {L^{\infty}(0, T; H^1_0(\Omega))\cap C([0, T]; L^{p}(\Omega))
\cap L^2(0, T; H^2(\Omega))}$, where
$1\leq p < \infty$ if $n= 1, 2$ and $2\leq p <\frac{2n}{n-2} $ if $n\geq 3$,

\item $\partial_t u \in {L^2(0, T; (H^2(\Omega))')}$,

\item $u(x,0)= u_0(x)$ for all $x\in\Omega$,
\end{enumerate}
which satisfies  \eqref{1-10} in the following weak sense:
\begin{enumerate}
\item Define $P$ as the set where $D(u)$ is non-degenerate, that is
$$
P:=\{(x,t)\in Q_T:|u|\neq 0\}.
$$
There exists a set $A \subset Q_T$ with $|Q_T\setminus A|=0$ and a function 
$\zeta:Q_T\to R^{n}$
satisfying $\chi_{A\cap P} D(u)\zeta \in {L^2(0, T; L^{\frac {2n}{n+2}}(\Omega))}$, 
such that
\begin{equation} \label{3-1}
\begin{aligned}
&\int^T_0\langle \partial_t u, \phi\rangle \,dt\\
&=-\int^T_0\int_{A\cap P} D(u)\zeta\cdot\nabla\phi \,dx\,dt
- \int^T_0\int_\Omega(-\Delta u+ f(u))\phi \,dx\,dt
\end{aligned}
\end{equation}
for all test functions $\phi\in L^2(0, T; H^2(\Omega)\cap H^1_0(\Omega))$.

\item For each $j \in N$, there exists 
$E_J:=\{(x, t)\in Q_{T}; u_i\to u  \text{ uniformly},\; |u|> \delta_J
\text{ for } \delta_J>0\} =T_J\times S_J$ such that
\begin{gather*}
u\in L^2(T_J; H^{3}(S_J)),\\
\zeta = -\nabla\Delta u + F''(u)\nabla u, \quad \text{in } \;E_J.
\end{gather*}
\end{enumerate}
In addition, $u$ satisfies the  energy inequality
\begin{equation} \label{3-2}
\begin{aligned}
&E(u)+\iint_{Q_t\cap A\cap P} D(u(x,\tau))|\zeta(x,\tau)|^2\,dx\,d\tau\\
&+\iint_{Q_t}|-\Delta u +f(u)|^2\,dx\,d\tau \leq E(u_0),
\end{aligned}
\end{equation}
for all $t > 0$.
\end{theorem}

\begin{proof} 
We consider a sequence of positive numbers $\varepsilon_i$ monotonically
decreasing to $0$ as $i \to \infty$. Fix $u_0\in H^1(\Omega)$, for any 
fixed $\varepsilon_i$, here, for the sake of simplicity, we write
$u_i := u_{\varepsilon_i}$ and
$D_i(u_i) := D_{\varepsilon _i}(u_{\varepsilon _i})$.
By Theorem \ref{th2.1}, there exists a function $u_i$ such that
\begin{enumerate}
\item $u_i\in {L^{\infty}(0, T; H^1_0(\Omega))\cap C([0, T]; L^{p}(\Omega))
\cap L^2(0, T; H^{3}(\Omega))}$, where $1\leq p < \infty$ if $n= 1, 2$ and 
$2\leq p <\frac{2n}{n-2} $ if $n\geq 3$,

\item $\partial_t u_i\in {L^2(0, T; (H^2(\Omega))')}$,
\begin{equation} \label{3-3}
\int^T_0\langle \partial_t u_i, \phi\rangle\,dt
=-\int^T_0\int_\Omega D_i(u_i)\nabla v_i \cdot\nabla\phi \,dx\,dt
-\int^T_0\int_\Omega v_i\phi \,dx\,dt
\end{equation}
for all test functions $\phi\in L^2(0, T; H^2(\Omega)\cap H^1_0(\Omega))$, where
\begin{equation} \label{3-4}
v_i=-\Delta u_i + f(u_i), \text{almost everywhere in } Q_T.
\end{equation}
\end{enumerate}
By the arguments in the proof of Theorem \ref{th2.1}, the bounds on the 
right hand side of \eqref{2-16}, \eqref{2-20}, \eqref{2-39} and \eqref{2-32} 
depend only on the growth conditions of the mobility and potential, so there
 exists a constant $C > 0$ independent of $\varepsilon_i$ such that
\begin{gather} \label{3-5}
\|u_i\|_{L^{\infty}(0, T; H^1_0(\Omega))} \leq C,\\
\label{3-6}
\|\partial_tu_i\|_{L^2(0 ,T; (H^2(\Omega))')} \leq C,\\
\label{3-7}
\|\sqrt{D_i(u_i)}\nabla v_i\|_{L^2(Q_{T})} \leq C,\\
\label{3-8}
\|v_i\|_{L^2(Q_{T})} \leq C.
\end{gather}
Similar to the proof of Theorem \ref{th2.1}, the above boundedness of 
$\{u_i\}$ and $\{\partial_tu_i\}$ enable us to find a subsequence,
not relabeled, and $u \in L^{\infty}(0, T; H^1_0(\Omega))$ such that as 
$i \to \infty$,
\begin{gather}
\label{3-9}
u_i \rightharpoonup u, \quad \text{weak-* in } L^{\infty}(0, T; H^1_0(\Omega)),\\
\label{3-10}
u_i \to u, \quad \text{strongly in } C(0, T; L^{p}(\Omega)), \\
\label{3-11}
u_i \to u, \quad \text{strongly in} L^2(0, T; L^{p}(\Omega))
\text{ and almost all in } Q_{T}, \\
\label{3-12}
\partial_tu_i \rightharpoonup \partial_tu, \quad \text{weakly in }
 L^2(0, T; (H^2(\Omega))'),
\end{gather}
where $1 \leq p < \infty$ if $n = 1, 2$ and $2 \leq p < \frac {2n}{n-2}$ if 
$n \geq 3$.

By \eqref{3-7} and \eqref{3-8}, there exists $\xi, \eta \in L^2(Q_{T})$ such that
\begin{gather} \label{3-13}
\sqrt{D_i(u_i)} \nabla v_i \rightharpoonup \xi, \quad \text{weakly in }
 L^2(Q_{T}),\\
\label{3-14}
v_i \rightharpoonup \eta, \quad \text{weakly in } L^2(Q_{T}).
\end{gather}
Next we show the convergence of $D_i(u_i)\nabla v_i$ and
$\eta = - \Delta u + f(u) \;\text{a.e.}\;Q_{T}$.
Similar to having \eqref{2-33} and \eqref{2-34}, by the uniform convergence 
of $D_i \to D$, we obtain
\begin{gather} \label{3-15}
D_i(u_i) \to D(u), \quad \text{strongly in } C (0, T; L^{n/2}(\Omega)),\\
\label{3-16}
\sqrt{D_i(u_i)} \to \sqrt{D(u)}, \quad \text{strongly in } C (0, T; L^{n}(\Omega)).
\end{gather}

For any $\varphi \in L^2(0, T; L^{2^{*}}(\Omega))$, by H\"{o}lder's inequality,
 we have
\begin{align*}
&\Big|\iint_{Q_{T}} \left(D_i(u_i)\nabla v_i\varphi 
 - \sqrt{D(u)}\xi\varphi\right) \,dx\,dt\Big|\\
&\leq\Big|\iint_{Q_{T}} [\sqrt{D_i(u_i)}
 -\sqrt{D(u)}]\sqrt{D_i(u_i)}\nabla v_i\varphi \,dx\,dt \Big|\\
&\quad+ \Big|\iint_{Q_{T}}\sqrt{D(u)}[\sqrt{D_i(u_i)}\nabla v_i\varphi-\xi\varphi ] 
 \,dx\,dt \Big| \\
&\leq\int^T_0\|\sqrt{D_i(u_i)}-\sqrt{D(u)}\|_{n}\|\sqrt{D_i(u_i)}\nabla v_i\|_2
 \|\varphi\|_{2^*}\,dt \\
&\quad +\Big|\iint_{Q_{T}}\sqrt{D(u)}\varphi[\sqrt{D_i(u_i)}\nabla v_i-\xi]
  \,dx\,dt\Big| \\
&\leq\sup _{t\in [0, T]}\|\sqrt{D_i(u_i)}
 -\sqrt{D(u)}\|_{n}\|\sqrt{D_i(u_i)}\nabla v_i\|_{L^2(Q_{T})}
 \|\varphi\|_{L^2(0, T; L^{2^{*}}(\Omega))} \\
&\quad +\Big|\iint_{Q_{T}}\sqrt{D(u)}\varphi[\sqrt{D_i(u_i)}\nabla v_i-\xi]\,dx\,dt
 \Big|\\
&=: I + II.
\end{align*}
By \eqref{3-16} and \eqref{3-7}, $I \to 0 \;\text{as}\; N \to \infty $. 
By H\"{o}lder's inequality and the boundedness of 
$D(u)$ in $C (0, T; L^{n/2}(\Omega))$ we have
\begin{align*}
\iint_{Q_{T}}|\sqrt{D(u)}\varphi|^2 \,dx\, dt 
&\leq \int^T_0 \Big(\int_{\Omega}\big(D(u)\big)^{n/2}\,dx \Big)^{n/2}
 \Big(\int_{\Omega}|\varphi|^{2^{*}} \,dx \Big)^{\frac {n-2}{n}}\,dt\\
&\leq \sup _{t\in [0, T]}\|D(u)\|_{n/2}\int^T_0\|\varphi\|^2_{L^{2^{*}}(\Omega)} 
\,dt\\
&\leq C \|\varphi\|^2_{L^2(0, T; L^{2^{*}}(\Omega))}\,.
\end{align*}
This implies
\begin{equation}\label{3-17}
\sqrt{D(u)}\varphi \in L^2(Q_{T}).
\end{equation}
By \eqref{3-13}, thus $II \to 0 $ as $N \to \infty$, this implies
\begin{equation} \label{3-18}
D_i(u_i)\nabla v_i \rightharpoonup \sqrt{D(u)}\xi \quad
\text{weakly in } L^2(0, T; L^{\frac {2n}{n+2}}(\Omega)).
\end{equation}
By \eqref{3-4}, for any $\phi \in L^2(0, T; H^1_0(\Omega))\subset L^2(Q_{T})$ 
we have
\begin{equation} \label{3-19}
\begin{aligned}
\iint_{Q_{T}}v_i\phi \,dx\,dt
&= -\iint_{Q_{T}} \Delta u_i\phi \,dx\,dt + \iint_{Q_{T}}f(u_i)\phi \,dx\,dt\\
& = \iint_{Q_{T}}\nabla u_i\nabla\phi \,dx\,dt
+ \iint_{Q_{T}}f(u_i)\phi \,dx\,dt.
\end{aligned}
\end{equation}
Recalling that the convergence of $\nabla u_i$ and $f(u_i)$ are similar to get
\eqref{2-47} and \eqref{2-50}, we have
\begin{gather} \label{3-20}
\nabla u_i \rightharpoonup \nabla u, \quad \text{weak-* in }
 L^{\infty}(0, T; L^2(\Omega)),\\
\label{3-21}
f(u_i) \rightharpoonup f(u), \quad \text{weak-* in } L^{\infty}(0, T; L^2(\Omega)).
\end{gather}
By \eqref{3-20}, \eqref{3-21} and
$L^2(0, T; H^1_0(\Omega))\subset L^1(0, T; L^2(\Omega))$,
taking the limits of \eqref{3-19} on both sides, we have
$$
\iint_{Q_{T}}\eta\phi \,dx\,dt
 = \iint_{Q_{T}}\nabla u\nabla\phi \,dx\,dt
+ \iint_{Q_{T}}f(u)\phi \,dx\,dt.
$$
Since $f(u)\in L^{\infty}(0, T; L^2(\Omega))$ and
$\eta\in L^2(Q_{T})$, by regularity theory we see that
$u\in L^2(0, T; H^2(\Omega))$ and
\begin{equation} \label{3-22}
\eta = - \Delta u + f(u), \quad \text{almost everywhere in } Q_{T}.
\end{equation}
By \eqref{3-12}, \eqref{3-18} and \eqref{3-22}, taking the limits of \eqref{3-3},
 we have
\begin{equation} \label{3-23}
\int^T_0\langle \partial_tu, \phi\rangle\,dt
= -\int^T_0\int_{\Omega}\sqrt{D(u)}\xi\nabla\phi \,dx\,dt
- \int^T_0\int_{\Omega}[- \Delta u + f(u)]\phi \,dx\,dt.
\end{equation}
As for the initial value, since $u_i(x,0) = u_0(x) $ in $L^2(\Omega)$,
 by \eqref{3-10} we have $u(x,0) = u_0(x)$.

Now we consider the weak convergence of $\nabla v_i$. Choose a sequence 
of positive numbers $\delta_J$ that monotonically decreases to 0 as
$j \to \infty$. By \eqref{3-11} and Egorov's theorem, for every $\delta_J > 0 $,
there exists a subset $B_J\subset Q_{T}$ with
$|Q_{T} \backslash B_J| < \delta_J$ such that
$$
u_i \to u, \quad \text{uniformly in } B_J.
$$
Define $A_1= B_1, A_2= B_1 \cup B_2, \dots, A_J= B_1 \cup B_2 \cup \dots
\cup B_J$. Then
\begin{equation} \label{3-24}
A_1\subset A_2\subset \dots \subset A_J\subset A_{j+1} \subset \dots
\subset Q_{T}.
\end{equation}
Thus the limit of $\{A_J\}$ exists, then we have
$\lim_{j \to \infty}A_J = \cup^{\infty}_{j=1}A_J := A $ and
$|Q_{T} \backslash A| = 0$.

Define $P_J:=\{(x,t)\in Q_{T}; |u| > \delta _J\}$. Then
\begin{equation}\label{3-25}
P_1 \subset P_2\subset \dots \subset P_J\subset P_{j+1}
\subset \dots \subset Q_{T}.
\end{equation}
Thus the limit of $\{P_J\}$ exists, then we have
$ \lim_{j \to \infty}P_J = \cup^{\infty}_{j=1}P_J:= P$.
For each $j$, we define
\begin{gather*}
E_J := A_J \cap P_J, \quad\text{where $|u| > \delta_J$ and $u_i \to u$
uniformly},\\
G_J := A_J \backslash P_J, \quad  \text{where $|u| \leq \delta_j$ and
$u_i \to u$ uniformly.} 
\end{gather*}
Thus we obtain $A_J= E_J \cup G_J$.
By \eqref{3-24} and \eqref{3-25}, we have
$$
E_1 \subset E_2\subset \dots \subset E_J\subset E_{j+1}
\subset \dots \subset Q_{T}.
$$
Thus the limit of $\{E_J\}$ exists, then we have 
$ \lim_{j \to \infty}E_J = \cup^{\infty}_{j=1}E_J = A \cap P:= E$.

For any $\psi \in L^2(0, T; L^{2^{*}}(\Omega))$,
\begin{equation} \label{3-26}
\begin{aligned}
&\iint_{Q_{T}}D_i(u_i)\nabla v_i\psi \,dx\,dt\\
&= \iint_{Q_{T}\backslash A_J}D_i(u_i)\nabla v_i\psi \,dx\, dt
+\iint_{G_J}D_i(u_i)\nabla v_i\psi \,dx\,dt\\
&\quad  +\iint_{E_J}D_i(u_i)\nabla v_i\psi \,dx\,dt.
\end{aligned}
\end{equation}
As $i \to \infty$, by \eqref{3-18} we obtain
\begin{gather}\label{3-27}
\lim_{i \to \infty}\iint_{Q_{T}}D_i(u_i)\nabla v_i\psi \,dx\,dt
=\iint_{Q_{T}}\sqrt{D(u)}\xi\psi \,dx\,dt, \\
\label{3-28}
\lim_{i \to \infty}\iint_{Q_{T}\backslash A_J}D_i(u_i)\nabla v_i\psi \,dx\,dt
 =\iint_{Q_{T}\backslash A_J}\sqrt{D(u)}\xi\psi \,dx\,dt.
\end{gather}
By $|Q_{T}\backslash A|=0$, taking the limit of \eqref{3-28} as $j\to \infty$,
we have
\begin{equation} \label{3-29}
\lim_{j \to \infty}\lim_{i \to \infty}\iint_{Q_{T}\backslash A_J}D_i(u_i)
\nabla v_i\psi \,dx\,dt= 0.
\end{equation}
To analyze the second and third terms of \eqref{3-26}, we write
$u_{j-1,i}: = u_i$ and $v_{j-1,i}: = v_i$ in $A_J$, then we have
$$
u_{j-1,i} \to u, \quad \text{uniformly in $A_J$ for all }j\in N.
$$
This implies that there exists an index $N_J \in N^{+}$ such that for all
$i \geq N_J$,
$$
|u_{j-1,i} - u| < \frac{\delta_J}{2}.
$$
We can easily get the following result:
\begin{equation}  \label{3-30}
\begin{gathered}
|u_{j-1, i}|\geq \frac{\delta_J}{2},\quad\text{if } (x,t) \in E_J,\\
|u_{j-1, i}|\leq 2\delta_J,\quad \text{if } (x,t) \in G_J.
\end{gathered}
\end{equation}
Considering the limit of the second term of \eqref{3-26}, by H\"{o}lder's
inequality and \eqref{3-7} we have
\begin{equation} \label{3-31}
\begin{aligned}
&\Big|\iint_{G_J}D_{j-1,i}(u_{j-1,i})\nabla v_{j-1,i}\psi\,dx\,dt\Big|\\
&\leq \sup_{(x, t)\in G_J} \sqrt{D_{j-1,i}(u_{j-1,i})}
\iint_{Q_{T}}|\sqrt{D_{j-1,i}(u_{j-1,i})}\nabla v_{j-1,i}||\psi| \,dx\,dt \\
&\leq \sup_{(x, t) \in G_J}\sqrt{D_{j-1,i}(u_{j-1,i})} \|
 \sqrt{D_{j-1,i}(u_{j-1,i})}\nabla v_{j-1,i}\|_{L^2(Q_{T})}\|\psi\|_{L^2(Q_{T})}\\
&\leq C\sup_{(x, t) \in G_J}\sqrt{D_{j-1,i}(u_{j-1,i})}
|\Omega|^{1/n}\|\psi\|_{L^2(0, T; L^{2^{*}}(\Omega))}\\
&\leq C \max\{(2\delta_J)^{m/2},\varepsilon^{m/2}_{j-1,i}\}.
\end{aligned}
\end{equation}
Taking the limits of \eqref{3-31} as $i \to \infty$ and $j \to \infty$, we have
\begin{equation} \label{3-32}
\begin{aligned}
&\lim_{j \to \infty}\lim_{i \to \infty}\big|\iint_{G_J}D_{j-1,i}(u_{j-1,i})
\nabla v_{j-1,i}\psi \,dx\,dt\big|\\
&\leq C \lim_{j \to \infty}\lim_{i \to \infty}\max \{(2\delta_J)^{m/2},
\varepsilon^{m/2}_{j-1,i}\} = 0.
\end{aligned}
\end{equation}
By \eqref{3-7} and \eqref{3-30}, we obtain
\begin{align*}
(\frac {\delta_J}{2})^m\iint_{D_J}|\nabla v_{j-1,i}|^2 \,dx\,dt
 &\leq \iint_{D_J}D_{j-1,i}(u_{j-1,i})|\nabla v_{j-1,i}|^2\,dx \,dt
\\
&\leq \iint_{Q_{T}}D_{j-1,i}(u_{j-1,i})|\nabla v_{j-1,i}|^2 \,dx\,dt \;\leq C.
\end{align*}
This implies
$$
\iint_{D_J}|\nabla v_{j-1,i}|^2 \,dx\,dt \;\leq C(\delta_J)^{-m}.
$$
So $\nabla v_{j-1,i}$ is bounded in $L^2(E_J)$, thus there exists a
subsequence, labeled as $\{\nabla v_{j,i}\}$, and $\zeta_J \in L^2(E_J)$ such that
\begin{equation}
\label{3-33}
\nabla v_{j,i} \rightharpoonup \zeta_J, \quad \text{weakly in } L^2(E_J).
\end{equation}
By $E_{j-1}\subset E_J$, for any $g \in L^2(E_J)$, we have $g \in L^2(E_{j-1})$
and $\nabla v_{j-1,i} = \nabla v{_{j, i}}\; \text{in } E_{j-1}$.
By \eqref{3-33} we have
\begin{align*}
\lim_{i \to \infty}\iint_{E_{j-1}}\nabla v_{j,i}g\,dx\,dt
&=\lim _{i \to \infty}\iint_{E_{j-1}}\nabla v_{j-1,i}g \,dx\,dt\\
&=\iint_{E_{j-1}}\zeta_Jg \,dx\,dt = \iint_{E_{j-1}}\zeta_{j-1}g \,dx\,dt.
\end{align*}
Thus $\zeta_J = \zeta_{j-1}$ almost everywhere in $E_{j-1}$. we define
$$
\omega_J:=  \begin{cases}
 \zeta_J, & \text{if }\ (x,t) \in E_J,\\
 0, & \text{if }\ (x,t) \in E \backslash E_J.
 \end{cases}
$$
So for almost every $(x, t) \in E$, there exists a limit of $\omega_J(x, t)$
as $j\to \infty$. We write
$$
\zeta(x,t) = \lim _{j\to \infty}\omega_J(x,t),\quad \text{almost everywhere in } E.
$$
Clearly $\zeta(x,t) = \zeta_J(x,t)$ for almost all $(x, t)\in E_J$ for all $j$.
 Using a standard diagonal argument, we can extract a subsequence such that
\begin{equation}
\label{3-34}
\nabla v_{k, N_{k}} \rightharpoonup \zeta, \quad \text{weakly in }
L^2(E_J) \text{ for all } j.
\end{equation}
For any $\varphi \in L^2(0, T; L^{2^{*}}(\Omega))$, by H\"{o}lder's inequality
we have
\begin{align*}
&\Big|\iint_{Q_{T}}\Big( \chi_{E_J}\sqrt{D_{k, N_{k}}(u_{k, N_{k}})}
 \nabla v_{k, N_{k}}\varphi -\chi_{E_J}\sqrt{D(u)}\zeta\varphi\Big) \,dx\,dt\Big|\\
&\leq\Big|\iint_{Q_{T}} \chi_{E_J}[\sqrt{D_{k, N_{k}}(u_{k, N_{k}})}
 -\sqrt{D(u)}]\nabla v_{k, N_{k}}\varphi \,dx\,dt\Big|\\
&\quad + \Big|\iint_{Q_{T}}\chi_{E_J} \sqrt{D(u)}[\nabla v_{k, N_{k}}\varphi
 -\zeta \varphi ] \,dx\,dt\Big|\\
&\leq \sup_{t\in [0, T]}\|\sqrt{D_{k, N_{k}}(u_{k, N_{k}})}
 -\sqrt{D(u)}\|_{n}\int^T_0\| \chi_{E_J}\nabla v_{k, N_{k}}\|_2
 \|\varphi\|_{\frac {2n}{n-2}}\,dt\\
&\quad +\Big|\iint_{E_J}\sqrt{D(u)}\varphi[\nabla v_{k, N_{k}}
 -\zeta] \,dx\,dt \Big|\\
&\leq\sup_{t\in [0, T]}\|\sqrt{D_{k, N_{k}}(u_{k, N_{k}})}
 -\sqrt{D(u)}\|_{n}\|\nabla v_{k, N_{k}}\|_{L^2(E_J)}
 \|\varphi\|_{L^2(0, T; L^{2^{*}}(\Omega))}\\
&\quad +\Big|\iint_{E_J}\sqrt{D(u)}\varphi[\nabla v_{k, N_{k}}-\zeta]\,\,dx\,dt
 \Big|\\
&=: I + II.
\end{align*}
By \eqref{3-16} and \eqref{3-34}, $I \to 0$ as $N \to \infty$.
By \eqref{3-17} and \eqref{3-34}, we have $II \to 0 $ as $N \to \infty$. Thus
$$
\chi_{E_J}\sqrt{D_{k, N_{k}}(u_{k, N_{k}})}\nabla v_{k, N_{k}}
\rightharpoonup \chi_{E_J}\sqrt{D(u)}\zeta, \quad\text{weakly in }
 L^2(0, T; L^{\frac {2n}{n+2}}(\Omega)),
$$
for all $j$.

From $L^2 \subset L^{\frac {2n}{n+2}}$ and \eqref{3-13}, we see  that 
$\xi = \sqrt{D(u)}\zeta$ in every $E_J$ and
\begin{equation}\label{3-35}
\xi = \sqrt{D(u)}\zeta \quad \text{in } E.
\end{equation}
Consequently, by \eqref{3-18},
$$
\chi_{E}D_{k,N_{k}}(u_{k,N_{k}})\nabla v_{k,N_{k}}
 \rightharpoonup \chi_{E}D(u)\zeta, \quad \text{weakly in }
 L^2(0, T; L^{\frac {2n}{n+2}}(\Omega)).
$$
Thus by Taking the limits of third term of \eqref{3-26}, we have
\begin{equation} \label{3-36}
\begin{aligned}
&\lim_{j \to \infty}\lim_{k \to \infty}
\iint_{E_J}D_{k,N_{k}}(u_{k,N_{k}})\nabla v _{k,N_{k}}\psi \,dx\,dt\\
& =\lim_{j \to \infty}\iint_{E_J}D(u)\zeta \psi \,dx\,dt
= \iint_{E}D(u)\zeta \psi \,dx\,dt.
\end{aligned}
\end{equation}
By \eqref{3-27}, \eqref{3-29}, \eqref{3-32} and \eqref{3-36}, we have
\begin{equation}\label{3-37}
\iint_{Q_{T}}\sqrt {D(u)}\xi\psi \,dx\,dt = \iint_{E}D(u)\zeta \psi \,dx\, dt.
\end{equation}
By \eqref{3-23} and \eqref{3-37}, we find that $u$ and $\zeta$
solve equation \eqref{1-10} in the following weak sense
\begin{equation} \label{3-38}
\int^T_0\langle \partial_tu, \phi\rangle \,dt
= -\iint_{E}D(u)\zeta\nabla\phi\,dx\,dt
- \int^T_0\int_{\Omega}[- \Delta u + f(u)]\phi \,dx\,dt,
\end{equation}
for all $\phi \in L^2(0,T; H^2(\Omega)\cap H^1_0(\Omega))$.

From \eqref{3-14} and \eqref{3-34}, we notice that $v_i$ is bounded in
 $L^2(T_J; H^1(S_J))$, where $E_J=T_J\times S_J$. So
we can extract a further sequence, not relabeled, and
\begin{equation} \label{3-39}
\begin{gathered}
v \in L^2(T_J; H^1(S_J)),\\
v_i\rightharpoonup v \quad \text{weakly in } L^2(T_J; H^1(S_J)).
\end{gathered}
\end{equation}
Similar to show  $F'(u_{\varepsilon}) \in L^2(0, T; H^1(\Omega))$ and
\eqref{3-22}.
Hence, we have $F'(u) \in L^2(0, T; H^1(\Omega))$ and
$v= -\Delta u + f(u)$, a.e. in $E_J$, By $v \in L^2(T_J; H^1(S_J))$ we have
$u \in L^2(T_J; H^{3}(S_J))$ and
\begin{equation} \label{3-40}
\nabla v= - \nabla\Delta u + F''(u)\nabla u, \quad \text{almost everywhere in}\;\;E_J.
\end{equation}
Obviously we have
$\eta= v$, $\zeta= \nabla v$, a. e. in$E_J$.
So we obtain the desired relation between $\zeta$ and $u$:
$$
\zeta = -\nabla\Delta u + F''(u)\nabla u, \quad \text{in } E_J.
$$
Finally, we show that a weak solution $u$ to \eqref{1-10} satisfies energy
inequality \eqref{3-2}. By \eqref{2-4} we have
\begin{equation} \label{3-41}
\begin{aligned}
&\int_\Omega\Big(\frac{1}{2}|\nabla u_{k, N_{k}}(x,t)|^2
+ F(u_{k, N_{k}}(x,t))\Big)dx\\
&+ \iint_{Q_t\cap E}D_{k, N_{k}}(u_{k, N_{k}}(x,\tau))
 |\nabla v_{k, N_{k}}(x,\tau)|^2 \,dx\,d\tau \\
& + \iint_{Q_t}|v_{k, N_{k}}(x,\tau)|^2 \,dx\, d\tau\\
&\leq \int_\Omega\Big( \frac{1}{2}|\nabla u_0|^2 + F(u_0)\Big) dx.
\end{aligned}
\end{equation}
By having \eqref{2-47} and \eqref{2-66}, similarly we have
\begin{gather} \label{3-42}
\nabla u_{k, N_{k}}\rightharpoonup \nabla u, \quad \text{weakly in } L^2(Q_{T}),\\
\label{3-43}
\lim _{N \to \infty}\int_{\Omega}F(u_{k, N_{k}}(t))\,dx = \int_{\Omega}F(u(t))\,dx.
\end{gather}
By \eqref{3-42}, \eqref{3-43}, \eqref{3-13}, \eqref{3-35}, \eqref{3-14},
\eqref{3-22}, \eqref{3-41} and the weak lower semicontinuity of the $L^{p}$ norms.
Then
\begin{align*}
&\int_\Omega\left(\frac{1}{2}|\nabla u(x,t)|^2 + F(u(x,t))\right) dx
+ \iint_{Q_t\cap E}D(u(x,\tau))|\zeta(x,\tau)|^2 dx d\tau \\
&\quad+ \iint_{Q_t}|-\Delta u +f(u)|^2\,dx\,d\tau
\\
&\leq \lim_{N \uparrow \infty}\inf\int_{\Omega}
\Big(\frac{1}{2}|\nabla u_{k, N_{k}}(x,t)|^2 + F(u_{k, N_{k}}(x,t))\Big) dx\\
&\quad+ \lim_{N \uparrow \infty}\inf\iint_{Q_t\cap E}D_{k, N_{k}}(u_{k, N_{k}}(x,\tau))|\nabla v_{k, N_{k}}(x,\tau)|^2 \,dx\,d\tau
\\
&\quad +\lim_{N \uparrow \infty}\inf\iint_{Q_t}|v_{k, N_{k}}(x,\tau)|^2 dx d\tau
\\
&\leq \int_\Omega\Big( \frac12|\nabla u_0|^2 + F(u_0)\Big)dx.
\end{align*}
This gives the energy inequality \eqref{3-2}. The proof is complete.
\end{proof}

\subsection*{Acknowledgments}
The authors would like to thanks the anonymous referees for the  valuable suggestions
 for the revision and improvement of this article.

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\end{document}