\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2004(2004), No. 126, pp. 1--24.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or
http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2004 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE-2004/126\hfil A generalized solution]
{A generalized solution to a Cahn-Hilliard/Allen-Cahn system}

\author[J. L. Boldrini, P. N. da Silva\hfil EJDE-2004/126\hfilneg]
{Jos\'e Luiz Boldrini, Patr\'{\i}cia Nunes da Silva}
% in alphabetical order

\address{Jos\'e Luiz Boldrini \hfill\break
DM-IMECC-UNICAMP\\
CP 6065 \\
13083-970 Campinas, SP, Brazil}
\email{boldrini@ime.unicamp.br}

\address{Patr\'{\i}cia Nunes da Silva \hfill\break
DM-IME-UERJ\\
Rio de Janeiro, RJ, Brazil}
\email{nunes@ime.uerj.br}

\date{}
\thanks{Submitted June 10, 2004. Published October 25, 2004.}
\thanks{P.N.d.S. was supported by grant 98/15946-5 from FAPESP, Brazil}

\subjclass[2000]{47J35, 35K57, 35Q99}
\keywords{Cahn-Hilliard and Allen-Cahn equations; Ostwald ripening;
\hfill\break\indent
phase transitions}

\begin{abstract}
 We study a system consisting of a Cahn-Hilliard and
 several Allen-Cahn type equations. This system was proposed by Fan,
 L.-Q. Chen, S. Chen and Voorhees for modelling Ostwald ripening
 in two-phase system. We prove the existence of a generalized
 solution whose concentration component is in $L^{\infty}$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corolary}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}

\section{Introduction}

Ostwald ripening is a phenomenon observed in a wide variety of
two-phase systems
in which there is coarsening of one phase dispersed in the matrix of
another.
Because its practical importance, this process has been extensively
studied in
several degrees of generality.
In particular for Ostwald ripening of anisotropic crystals,
Fan et al. \cite{chen998} presented a model taking in consideration
both the
evolution of the compositional field and of the
crystallographic orientations.
In the work of Fan et al. \cite{chen998}, there are also numerical
experiments used to validate the model,
but there is no rigorous mathematical analysis of the model.
Our objective in this paper is to do such mathematical analysis.

By defining orientation and composition field variables, the kinetics
of coupled grain growth can be described by their spatial and temporal
evolution, which is related with the total free energy of the system.
The microstructural evolution of Ostwald ripening can be described by
the Cahn-Hilliard/Allen-Cahn system
%
\begin{equation}
\label{tpgep}
\begin{gathered}
\partial_{t} c=\nabla \cdot[
D\nabla (
\partial_{c} \mathcal{F}-\kappa_{c} \Delta c
)],\quad (x,t)\in \Omega_{T}\\
\partial_{t} \theta_{i}=-L_{i}(
\partial_{\theta_{i}} \mathcal{F}-\kappa_{i}\Delta \theta_{i}
), \quad (x,t)\in \Omega_{T}\\
\partial_{{\bf n}}c=
\partial_{{\bf n}}( \partial_{c} \mathcal{F}-\kappa_{c} \Delta c)=
\partial_{{\bf n}}\theta_{i}=0, \quad (x,t)\in S_{T}\\
c(x,0)= c_{0}(x),\quad \theta_{i}(x,0)=\theta_{i0}(x), \quad x\in\Omega
\end{gathered}
\end{equation}
for $i=1,\dots, p$.
Here, $\Omega$ is the physical region where the Ostwald process is
occurring;
$\Omega_{T}=\Omega\times(0,T)$; $S_{T}=\partial\Omega\times(0,T)$;
$0 < T < +\infty$; ${\bf n}$ denotes the unitary exterior normal
vector and $\partial_{{\bf n}}$ is the exterior normal derivative at
the boundary; $c(x,t)$ is the compositional field
(fraction of the soluto with respect to the mixture) which takes one
value within the matrix phase, another value
within a second phase grain and $c(x,t)$ varies
between these values at the interfacial region between the
matrix phase and a second phase grain; $\theta_i(x,t)$, for $i=1,
\dots, p$, are the crystallographic orientations fields; $D$,
$\lambda_c$, $L_{i}$, $\lambda_i$ are positive constants related to
the material properties. The function
$\mathcal{F}=\mathcal{F}(c,\theta_{1},\dots,\theta_{p})$ is the local
free energy density which is given by
%
\begin{equation} \label{energia}
\begin{aligned}
&\mathcal{F}(c,\theta_{1},\dots,\theta_{p})\\
&=-\frac{A}{2}(c-c_{m})^{2}+\frac{B}{4}(c-c_{m})^{4}
+\frac{D_{\alpha}}{4}(c-c_{\alpha})^{4}\\
&\quad +\frac{D_{\beta}}{4}(c-c_{\beta})^{4}
+\sum_{i=1}^{p}[-\frac{\gamma}{2}(c-c_{\alpha})^{2}\theta_{i}^{2}
+\frac{\delta}{4}
\theta_{i}^{4}]+\sum_{i=1}^{p}\sum_{i\not=j=1}^{p}\frac{\varepsilon_{ij}}{2}
\theta_{i}^{2}\theta_{j}^{2}.
\end{aligned}
\end{equation}
%
where $c_{\alpha}$ and $c_{\beta}$ are the solubilities in the matrix
phase and the second phase respectively, and
$c_{m}=(c_{\alpha}+c_{\beta})/2$. The positive coefficients $A$, $B$,
$D_{\alpha}$, $D_{\beta}$, $\gamma$, $\delta$ and $\varepsilon_{ij}$
are phenomenological parameters.

In this paper we obtain a $(p+2)$-tuple which satisfies a
variational inequality related to Problem~(\ref{tpgep}) and also
satisfies the physical requirement that the concentration
takes values in the closed interval $[0,1]$.

Our approach to the problem
is to analyze a three-parameter family of suitable systems which
contain a logarithmic perturbation term and approximate the
model presented by Fan et al. \cite{chen998}. In this
analysis, we show that
the approximate solutions converge to a generalized
solution of the original
continuous model and this, in particular,
will furnish a rigorous proof of the existence of
generalized solutions (see the statement of Theorem~\ref{tpt1}).
Our approach is similar to that
used by Passo et al.~\cite{passo1299} for an Cahn-Hilliard/Allen-Cahn
system with degenerate mobility.

\section{Existence of Solutions}

Including the physical restriction on the
concentration, Problem~(\ref{tpgep}) is stated as
follows:
%
\begin{equation} \label{tpgepmv}
\begin{gathered}
\partial_{t} c=
\nabla [D\nabla (
\partial_{c} \mathcal{F}
-\kappa_{c} \Delta c
)], \quad (x,t)\in \Omega_{T}\\
\partial_{t} \theta_{i}=
-L_{i}(
\partial_{\theta_{i}} \mathcal{F}
-\kappa_{i} \Delta \theta_{i}
), \quad (x,t)\in \Omega_{T}\\
\partial_{{\bf n}}c=
\partial_{{\bf n}}(\partial_{c} \mathcal{F}
-\kappa_{c} \Delta c)=
\partial_{{\bf n}} \theta_{i}=0, \quad (x,t)\in S_{T}\\
c(x,0)= c_{0}(x),\quad \theta_{i}(x,0)=\theta_{i0}(x),
\quad x\in \Omega\\
0\leq c \leq 1 \quad (x,t)\in \Omega_{T}
\end{gathered}
\end{equation}
for $i=1,\dots, p$.

Throughout this paper, standard notation
will be used for the several required functional spaces. We denote by
$\overline{f}$ the mean value of $f$ in $\Omega$
of a given $f\in L^{1}(\Omega)$. The duality
pairing between $H^{1}(\Omega)$ and its dual is denoted
by $\langle\cdot,\cdot\rangle$
and $(\cdot,\cdot)$ denotes the inner product in
$L^{2}(\Omega)$.
We will prove the following:

\begin{theorem} \label{tpt1}
Let $T>0$ and $\Omega\subset R^d$, $1 \leq d \leq 3$ be a
bounded domain with $C^3$-- boundary.
For all $c_{0},\theta_{i0}, i=1,\dots,p$, satisfying
$c_{0}, \theta_{i0}\in H^{1}(\Omega)$, for $i=1,\dots,p$,
$0\leq c_{0} \leq 1$,
there exists a unique
$(p+2)$-tuple $(c,w-\overline{w},\theta_{1},\dots,\theta_{p})$ such 
that,
for $i=1,\dots,p$,
\begin{itemize}
\item[(a)] $c, \; \theta_{i} \in
L^{\infty}(0,T,H^{1}(\Omega))\cap
L^{2}(0,T,H^{2}(\Omega))$,

\item[(b)] $ w\in L^{2}(0,T,H^{1}(\Omega))$

\item[(c)] $\partial_{t}c\in L^{2}(0,T,[H^{1}(\Omega)]')$,
$\partial_{t}\theta_{i}\in L^{2}(\Omega_{T})$

\item[(d)] $0\leq c \leq 1$ a.e. in $\Omega_{T}$

\item[(e)] $c(x,0)=c_{0}(x)$, $\theta_{i}(x,0)=\theta_{i0}(x)$

\item[(f)] $ \partial_{c}
\mathcal{F}(c,\theta_{1},\dots,\theta_{p})$,
$\partial_{\theta_{i}} \mathcal{F}(c,\theta_{1},\dots,\theta_{p}) \in
L^{2}(\Omega_{T})$

\item[(g)]  ${\partial_{{\bf n}}c}_{|_{S_{T}}}=
{\partial_{{\bf n}}\theta_{i}}_{|_{S_{T}}}=0$ in $L^{2}(S_{T})$

\item[(h)] $(c,w,\theta_{1},\dots,\theta_{p})$ satisfies
\begin{equation}
\int_{0}^{T}\langle \partial_{t} c ,\phi\rangle dt=
-\iint_{\Omega_{T}}\nabla w\nabla \phi,\quad \forall \phi \in
L^{2}(0,T,H^{1}(\Omega)), \label{tpge1.5}
\end{equation}
%
\begin{equation}
\int_{0}^{T}\xi(t)\left\{\kappa_{c}D (\nabla c,\nabla \phi-\nabla c)
-(w-D\partial_{c}\mathcal{F}(c,\theta_{1},\dots,\theta_{p}),
\phi-c)\right\}dt\geq 0,
\label{tpge1.7}
\end{equation}
%
for all $\xi\in C[0,T), \xi \geq 0,\;
\forall \phi\in K=\{\eta \in H^{1}(\Omega),\; 0\leq \eta\leq 1,\,
\overline{\eta}=\overline{c_{0}}\}$, and
%
\begin{equation}
\iint_{\Omega_{T}}\partial_{t} \theta_{i} \psi_{i}=
-\iint_{\Omega_{T}}
L(\partial_{\theta_{i}}\mathcal{F}(c,\theta_{1},\dots,\theta_{p})-
\kappa_{i} \Delta \theta_{i})\psi_{i}, \label{tpge1.6}
\end{equation}
%
for all $\psi_{i} \in L^{2}(\Omega_{T}), i=1,\dots,p$,
where $\mathcal{F}$ is given by (\ref{energia}).
\end{itemize}
\end{theorem}
%
\begin{remark} \rm
The inequality obtained (\ref{tpge1.7}) is similar to one
obtained by Elliott and Luckhaus~\cite{elliott88791} in the case of
the deep quench limit problem for a system of nonlinear diffusion
equations.
\end{remark}

\begin{remark} \rm
We observe that (\ref{tpge1.7}) comes from the fact that
classically $w$ is expected to be equal to $D(
\partial_{c} \mathcal{F}-\kappa_{c} \Delta c
)$ up to a function of time.
\end{remark}

\begin{remark} \rm
The solution presented in Theorem~\ref{tpt1} is a generalized
solution of (\ref{tpgepmv}). In fact, as will be shown at
the end of this article, (\ref{tpge1.7}) holds as an
equality in the region where $0<c(x,t)<1$ for almost all times.
\end{remark}

We start by proving the uniqueness referred to in Theorem~\ref{tpt1}.

\begin{lemma} \label{unicidade}
Consider a solution of (\ref{tpge1.5})--(\ref{tpge1.6}) as
in Theorem~\ref{tpt1}. Under the hypotheses (a)--(e) and
(h) of Theorem~\ref{tpt1}, the components
$c$, $\theta_{1}$, $\dots$, $\theta_{p}$  are uniquely determined;
the component $w$ is uniquely determined up to a function of time.
\end{lemma}

\begin{proof} We argue as Elliott and Luckhaus~\cite{elliott88791}.
We introduce the Green's operator $G$:
given $f\in [H^{1}(\Omega)]_{null}'=\{f\in
[H^{1}(\Omega)]',\;\langle f,1\rangle =0\}$, we define $Gf \in
H^{1}(\Omega)$ as the unique solution of
%
\begin{equation*}
\int_{\Omega}\nabla Gf\nabla\psi=\langle f,\psi\rangle,\quad \forall
\psi \in H^{1}(\Omega) \quad \mbox{and} \quad
\int_{\Omega}Gf =0.
\end{equation*}
%
Let $z^{c}=c_{1}-c_{2}$, $z^{w}=w_{1}-w_{2}$ and
$z^{\theta_{i}}=\theta_{i1}-\theta_{i2},\;i=1,\dots,p$ be
the differences of two pair of solutions to
(\ref{tpge1.5})--(\ref{tpge1.6}) as in Theorem~\ref{tpt1}.
Since equation~(\ref{tpge1.5})
implies that the mean value of the composition field in $\Omega$
is conserved, we have that $(z^{c},1)=0$ and
we find from (\ref{tpge1.5}) that
%
\[
-Gz_{t}^{c}=\overline{z^{w}}.
\]
The definition of the Green operator and the fact that
$(z^{c},1)=0$  give
%
\[
-(\nabla Gz_{t}^{c},\nabla Gz^{c})=-(Gz_{t}^{c},z^{c})
=(\overline{z^{w}},z^{c})=(z^{w},z^{c}).
\]
%
Since $c_{1},\,c_{2}\in K=\{\eta \in H^{1}(\Omega),\; 0\leq \eta\leq 
1,\,
\overline{\eta}=\overline{c_{0}}\}$, we find from (\ref{tpge1.7}) that
%
\[
-\kappa_{c}D|\nabla z^{c}|^{2}+(z^{w},z^{c})
-D(\partial_{c}\mathcal{F}(c_{1},\theta_{11},\dots,\theta_{p1})
-\partial_{c}\mathcal{F}(c_{2},\theta_{12},\dots,\theta_{p2}),z^{c})
\geq 0.
\]
%
Thus, we have
\begin{align*}
&\frac{1}{2}\frac{d}{dt}|\nabla G z^{c}|+\kappa_{c}D|\nabla 
z^{c}|^{2}\\
&\leq -D(\partial_{c}\mathcal{F}(c_{1},\theta_{11},\dots,\theta_{p1})
-\partial_{c}\mathcal{F}(c_{2},\theta_{12},\dots,\theta_{p2})
-\kappa_{c}\Delta z^{c},z^{c}).
\end{align*}
%
We find from (\ref{tpge1.6}) that
\begin{align*}
&\frac{D}{2L_{i}}\frac{d}{dt}|z^{\theta_{i}}|^{2}
+D\lambda_{i}|\nabla  z^{\theta_{i}}|^{2}\\
&+D\bigl(\partial_{\theta_{i}}\mathcal{F}
(c_{1},\theta_{11},\dots,\theta_{p1})
-\partial_{\theta_{i}}\mathcal{F}(c_{2},\theta_{12},\dots,\theta_{p2}),
z^{\theta_{i}}\bigr)=0.
\end{align*}
%
By adding the above equations, using the convexity of
$[\mathcal{F}+H](c,\theta_{1},\dots,\theta_{p})$,
%
\[
 [\mathcal{F}+H](c,\theta_{1},\dots,\theta_{p})=
\frac{D_{\alpha}}{4}(c-c_{\alpha})^{4}
+\frac{D_{\beta}}{4}(c-c_{\beta})^{4}
+\frac{\delta}{4}\sum_{i=1}^{p}\theta_{i}^{4}
\]
where
%
\[
H(c,\theta_{1},\dots,\theta_{p})=
\frac{A}{2}(c-c_{m})^{2}
+ \frac{\gamma}{2}\sum_{i=1}^{p}c^{2}\theta_{i}^{2}
-\sum_{i=1}^{p}\sum_{i\not=j=1}^{p}\frac{\varepsilon_{ij}}{2}
\theta_{i}^{2}\theta_{j}^{2},
\]
%
we obtain
%
\begin{equation} \label{aux1}
\begin{split}
&\frac{1}{2}\frac{d}{dt}|\nabla Gz^{c}|^{2}
+\kappa_{c}D|\nabla z^{c}|^{2}
+\sum_{i=1}^{p}[\frac{D}{2L_{i}}\frac{d}{dt}|z^{\theta_{i}}|^{2}
+D\lambda_{i}|\nabla  z^{\theta_{i}}|^{2}]\\
&\leq \bigl(\nabla (H(c_{1},\theta_{11},\dots,\theta_{p1})
-H(c_{2},\theta_{12},\dots,\theta_{p2}))
\cdot (z^{c},z^{\theta_{1}},\dots,z^{\theta_{p}}),1\bigr)
\end{split}
\end{equation}
%
To estimate the right-hand side of the above
inequality, we use the regularity of $c_{i}$ and $\theta_{ik}$. Then
\begin{align*}
&(\nabla(\theta_{i1}^{2}\theta_{j1}^{2}-\theta_{i2}^{2}\theta_{j2}^{2})
\cdot(z^{\theta_{i}},z^{\theta_{j}}),1)\\
&=2((\theta_{i1}\theta_{j1}^{2}-\theta_{i2}\theta_{j2}^{2},
\theta_{i1}^{2}\theta_{j1}-\theta_{i2}^{2}\theta_{j2})
\cdot(z^{\theta_{i}},z^{\theta_{j}}),1)\\
&=2((z^{\theta_{i}}\theta_{j1}^{2}
+\theta_{i2}(\theta_{j1}^2-\theta_{j2}^{2}),
\theta_{i1}^{2}\theta_{j1}-\theta_{i2}^{2}\theta_{j2})
\cdot(z^{\theta_{i}},z^{\theta_{j}}),1)\\
&=2((
z^{\theta_{i}}\theta_{j1}^{2}
+\theta_{i2}(\theta_{j1}+\theta_{j2})z^{\theta_{j}},
z^{\theta_{j}}\theta_{i1}^{2}
+\theta_{j2}(\theta_{i1}+\theta_{i2})z^{\theta_{i}})
\cdot(z^{\theta_{i}},z^{\theta_{j}}),1)\\
&\leq
C[|z^{\theta_{i}}|^{2}+|z^{\theta_{j}}|^{2}]
+\frac{D\lambda_{i}}{8(p-1)}|\nabla  z^{\theta_{i}}|^{2}
+\frac{D\lambda_{j}}{8(p-1)}|\nabla  z^{\theta_{j}}|^{2}
\end{align*}
%
and
%
\begin{align*}
\gamma(\nabla(c_{1}^{2}\theta_{i1}^{2}-c_{2}^{2}\theta_{i2}^{2})
\cdot(z^{c},z^{\theta_{i}}),1)
&\leq
C[|z^{c}|^{2}+|z^{\theta_{i}}|^{2}]
+\frac{\kappa_{c}D}{2p}|\nabla  z^{c}|^{2}
+\frac{D\lambda_{i}}{4}|\nabla  z^{\theta_{i}}|^{2}.
\end{align*}
%
The above inequalities and (\ref{aux1}) imply
%
\[
\begin{split}
&\frac{1}{2}\frac{d}{dt}|\nabla Gz^{c}|^{2}
+\frac{\kappa_{c}D}{2}|\nabla z^{c}|^{2}
+\sum_{i=1}^{p}\big[\frac{D}{2L_{i}}\frac{d}{dt}|z^{\theta_{i}}|^{2}
+\frac{D\lambda_{i}}{2}|\nabla  z^{\theta_{i}}|^{2}\big]\\
&\leq 
C\big[\|z^{c}\|_{L^{2}(\Omega)}^{2}
+\sum_{i=1}^{p}\|z^{\theta_{i}}\|_{L^{2}(\Omega)}^{2}\big].
\end{split}
\]
%
From the definition of the Green operator,
$|z^{c}|^{2}=(\nabla  G z^{c},\nabla  z^{c})$. Using the
H\"older inequality, we rewrite the above inequality as
%
\[
\begin{split}
&\frac{1}{2}\frac{d}{dt}|\nabla  G z^{c}|^{2}
+\frac{\kappa_{c}D}{4}|\nabla z^{c}|^{2}
+\sum_{i=1}^{p}\big[\frac{D}{2L_{i}}\frac{d}{dt}|z^{\theta_{i}}|^{2}
+\frac{D\lambda_{i}}{2}|\nabla  z^{\theta_{i}}|^{2}\big]\\
&\leq C\big[\|\nabla G z^{c}\|_{L^{2}(\Omega)}^{2}
+\sum_{i=1}^{p}\|z^{\theta_{i}}\|_{L^{2}(\Omega)}^{2}\big].
\end{split}
\]
%
A standard Gronwall argument then yields
%
\[
\nabla Gz^{c}=0
\quad \mbox{and} \quad
z^{\theta_{i}}=0,\quad i=1,\dots,p,
\]
%
since
%
\[
Gz^{c}(0)=0\quad \mbox{and} \quad
z^{\theta_{i}}(0)=0,\quad i=1,\dots,p.
\]
%
The uniqueness is proved since $|z^{c}|^{2}=(\nabla
\mathcal{G}z^{c},\nabla  z^{c})=0$.
\end{proof}

As a corollary of this, we have the following:

\begin{lemma} \label{unicidade1}
Under the conditions of Theorem~\ref{tpt1}, when
either $c_0\equiv 0$ or $c_{0}\equiv 1$ almost everywhere on $\Omega$,
there is a solution of \eqref{tpge1.5}--\eqref{tpge1.6}.
\end{lemma}

\begin{proof} In such cases, since $0\leq
c_{0}(x)\leq 1$, we have in fact that
either $c_{0}(x)=0$ or $c_{0}(x)=1$.
Now, take $c$
identically zero or one, respectively. Then, equation (\ref{tpge1.5}) 
is
trivially satisfied and will imply that $w$ is a constant. Otherwise,
(\ref{tpge1.7}) is also trivially satisfied and to obtain a solution
of the Problem~(\ref{tpge1.5})--(\ref{tpge1.6}), we just have to solve
the nonlinear parabolic  system (\ref{tpge1.6}). But
this system can be solved
rather easily by standard methods, like Galerkin
method, for instance, since
the nonlinearities have the right sign and thus furnish suitable 
estimates.
\end{proof}

By the above lemma,
we have uniqueness of $c$ in the cases where either
$c_0\equiv 0$ or $c_{0}\equiv 1$ almost everywhere on
$\Omega$.
Thus, to prove Theorem~\ref{tpt1}, it just remain to
deal with the cases where the mean value of the
initial condition $c_{0}$ is strictly between to zero and one.
Thus,
in the following we assume that
%
\begin{equation} \label{tpge1.4}
\begin{split}
 c_{0}, \;\theta_{i0}\in H^{1}(\Omega),\quad i=1,\dots,p,\\
0\leq c_{0} \leq 1,\quad \overline{c_{0}}\in (0,1),
\end{split}
\end{equation}

To obtain the result in Theorem~\ref{tpt1}, we approximate system
(\ref{tpgepmv}) by a three-parameter family of suitable systems which
contain a logarithmic perturbation term and then pass to the limit.
In Section~\ref{sistap}, we use the results of Passo et 
al.~\cite{passo1299} to construct such
perturbed systems and together with some ideas presented by Copetti and
Elliott~\cite{copetti6392} and by Elliott and
Luckhaus~\cite{elliott88791},
we take the limit in these systems in the last three sections.

For sake of simplicity of exposition, without loosing generality,
we develop the proof for the case of dimension
one and for only one orientation field variable, that is,
when $\Omega$ is a bounded open interval and
$p$ is equal to one, and thus we have just one orientation field that
we denote $\theta$. In this case, the local free energy density is
reduced to
%
\begin{equation}\label{energia1}
\begin{split}
\mathcal{F}(c,\theta)
&=-\frac{A}{2}(c-c_{m})^{2}+\frac{B}{4}(c-c_{m})^{4}
+\frac{D_{\alpha}}{4}(c-c_{\alpha})^{4} +
\frac{D_{\beta}}{4}(c-c_{\beta})^{4}\\
&\quad
+\frac{\delta}{4}\theta^{4}
-\frac{\gamma}{2}(c-c_{\alpha})^{2}\theta^{2}.
\end{split}
\end{equation}
%
We remark that, even though the cross terms of
(\ref{energia}) involving the orientation field
variables are absent in above expression,
their presences when $p$ is greater than one will not
bring any difficulty for extending the result,
as we will point out at the end of the paper.

\section{Perturbed Systems}\label{sistap}
%
In this section we construct a three-parameter family of perturbed
systems. The auxiliary parameter $M$ controls a
truncation of the local free
energy $\mathcal{F}$ which will permit the application of an existence
result of Passo et al.~\cite{passo1299}. The parameters
$\sigma$ and $\varepsilon$ are associated to
the logarithmic term, their introduction will enable us to guarantee
that the composition field variable $c$ takes values in
the closure of the set $I=(0,1)$.

For each positive constants $\sigma$,
$M$ and $\varepsilon \in (0,1)$, we define the perturbed local free
energy density as follows:
%
\begin{equation} \label{energia2}
\mathcal{F}_{\sigma \varepsilon M}(c,\theta)=
f(c)+g_{M}(\theta)+h_{M}(c,\theta)
+\varepsilon
[F_{\sigma}(c)+ F_{\sigma}(1-c)].
\end{equation}
%
where the first three terms give a truncation of the original
$\mathcal{F}(c,\theta)$ given in (\ref{energia1}), and the last term
is a logarithmic perturbation. To obtain a truncation of the local
free energy density, we introduce
bounded functions whose summation coincides with
$\mathcal{F}$ for $(c,\theta) \in [0,1]\times [-M,M]$. Let
$f, g_{M}$ and $h_{M}$ be such that
%
\begin{gather*}
f(c)=-\frac{A}{2}(c-c_{m})^{2}
+\frac{B}{4}(c-c_{m})^{4}
+\frac{D_{\alpha}}{4}(c-c_{\alpha})^{4}+
\frac{D_{\beta}}{4}(c-c_{\beta})^{4},\quad
0\leq c \leq 1,\\
g_{M}(\theta)=\frac{\delta}{4}h_{2}^{2}(M;\theta)
\quad \mbox{and} \quad
h_{M}(c,\theta)=h_{1}(c)h_{2}(M;\theta)
\end{gather*}
%
with
%
\begin{gather*}
h_{1}(c)=-\frac{\gamma}{2}(c-c_{\alpha})^{2},\quad 0\leq c \leq 1,\\
h_{2}(M;\theta)= \theta^{2},\quad -M\leq \theta \leq M.
\end{gather*}
%
Outside the intervals $[0,1]$ and $[-M,M]$, we extend the above
functions to satisfy
%
\begin{gather}
\label{limitacoes1}
\|f\|_{C^{2}({\bf R})}\leq U_{0},\quad
\|g_{M}\|_{C^{2}({\bf R})}\leq V_{0}(M),\\
\|h_{1}\|_{C^{2}({\bf R})}\leq W_{0}, \quad
\|h_{M}\|_{C^{2}({\bf R}^{2})}\leq Z_{0}(M),
\label{limitacoes2}\\
|h_{2}(M;\theta)|\leq K\theta^{2},\quad
|h_{2}'(M;\theta)|\leq K|\theta|,\quad \forall M>0,
|\;\forall\theta\in {\bf R},
\label{limitacoes3}
\end{gather}
%
where $U_{0}, W_{0},K>0$ are constants and, for each $M$, $V_{0}(M)$
and $Z_{0}(M)$ are also constants.

We took the logarithmic term
$\varepsilon[F_{\sigma}(c)-F_{\sigma}(1-c)]$
as in Passo et al.~\cite{passo1299}. Let us denote
%
\[
F(s)=s\ln s.
\]
%
\label{fsigma}
For $\sigma \in (0,1/e)$, we choose $F_{\sigma}'(s)$ such that
%
\[
F_{\sigma}'(s)=
\begin{cases}
\frac{\sigma}{2\sigma-s}+\ln \sigma, & \mbox{if $s<\sigma$},\\
 \ln s +1, & \mbox{if $\sigma \leq s \leq 1-\sigma$},\\
 f_{\sigma}(s), & \mbox{if $1-\sigma < s<2$},\\
 1, & \mbox{if $s\geq 2$},
\end{cases}
\]
%
where $f_{\sigma}\in C^{1}([1-\sigma,2])$ is chosen having
the following properties:
%
\begin{gather*}
f_{\sigma}\leq F', \quad f_{\sigma}'\geq 0,\\
f_{\sigma}(1-\sigma)= F'(1-\sigma), \quad f_{\sigma}(2)=1,\\
f_{\sigma}'(1-\sigma)= F''(1-\sigma), \quad f_{\sigma}'(2)=0.
\end{gather*}
%
Defining
%
\[
F_{\sigma}(s)=-\frac{1}{e}+\int_{\frac{1}{e}}^{s}F_{\sigma}'(\xi)d\xi,
\]
%
we have
$F_{\sigma}\in C^{2}({\bf R})$ and $F_{\sigma}''\geq 0$.

Clearly, $\mathcal{F}_{\sigma\varepsilon M}$ has a lower
bound which is independent of $\sigma$ and $\varepsilon$.
We claim that $\mathcal{F}_{\sigma\varepsilon M}$ can also be
bounded from below independently of $M$. To prove this fact, we just 
have to
estimate $g_{M}(\theta) +h_{M}(c,\theta)$. We have
%
\begin{align*}
 g_{M}(\theta) +h_{M}(c,\theta)&=
\frac{\delta}{4}h_{2}^{2}(M;\theta)
+h_{1}(c)h_{2}(M;\theta)\\
 &=
\frac{\delta}{4}h_{2}(M;\theta)\big[h_{2}(M;\theta)
+\frac{4}{\delta}h_{1}(c)\big]\geq
-\frac{h_{1}^{2}(c)}{\delta}\geq
-\frac{W_{0}^{2}}{\delta }
\end{align*}
%
Therefore,
%
\begin{equation} \label{tpge3.1}
\begin{split}
&-U_{0}-\frac{W_{0}^{2}}{\delta }-\frac{2}{e}\leq
 \mathcal{F}_{\sigma \varepsilon  M}(c,\theta)\quad \mbox{in } {\bf 
R}^{2}, \\
&\mathcal{F}_{\sigma \varepsilon M}(c,\theta)<
 U_{0}+g_{M}(\theta)-h_{M}(c,\theta)
\quad \mbox{in } \mathop{\rm cl}\,{I}.
\end{split}
\end{equation}
%
Then, the perturbed systems we will consider are
%
\begin{equation} \label{tpgepds}
\begin{gathered}
\partial_{t} c =
D(\partial_{c}
\mathcal{F}_{\sigma \varepsilon M}(c,\theta)-\kappa_{c} c_{xx}
)_{xx}, \quad (x,t)\in \Omega_{T}\\
\partial_{t} \theta =-L[\partial_{\theta}
\mathcal{F}_{\sigma \varepsilon M}(c,\theta)-\kappa\theta_{xx} ],
\quad (x,t)\in \Omega_{T}\\
\partial_{{\bf n}}c= \partial_{{\bf n}}
(\partial_{c}\mathcal{F}_{\sigma \varepsilon M}(c,\theta)
-\kappa_{c}c_{xx})=\partial_{{\bf n}}\theta=0
\quad (x,t)\in S_{T}\\
 c(x,0)= c_{0}(x),\quad \theta(x,0)=\theta_{0}(x),
\quad x\in \Omega
\end{gathered}
\end{equation}
%
To solve the above problem, we shall use the next proposition which is 
an
existence result stated by Passo et
al.~\cite{passo1299} for the system
%
\begin{equation} \label{tpgepl}
\begin{gathered}
\partial_{t} u = [q_{1}(u,v)(
f_{1}(u,v)-\kappa_{1} u_{xx}
)_{x}]_{x},\quad (x,t)\in \Omega_{T}\\
\partial_{t} v =-q_{2}(u,v)[
f_{2}(u,v)-\kappa_{2}v_{xx} ], \quad  (x,t)\in \Omega_{T}\\
\partial_{{\bf n}}u=\partial_{{\bf n}}u_{xx}= \partial_{{\bf n}}v=0 
\quad
 (x,t)\in S_{T}\\
u(x,0)= u_{0}(x),\quad v(x,0)=v_{0}(x), \quad  x\in \Omega
\end{gathered}
\end{equation}
%
where $q_{i}$ and $f_{i}$ satisfy the following hypotheses:
\begin{itemize}

\item[(H1)] $q_{i}\in C({\bf R}^{2},{\bf R}^{+})$, with
$q_{{\rm min}}\leq q_{i}\leq q_{{\rm max}}$ for some
$0<q_{{\rm min}}\leq q_{{\rm max}}$;

\item[(H2)] $f_{1}\in C^{1}({\bf R}^{2},{\bf R})$ and
$f_{2}\in C({\bf R}^{2},{\bf R})$, with
$\|f_{1}\|_{C^{1}}+\|f_{2}\|_{C^{0}}\leq F_{0}$ for some $F_0>0$.
\end{itemize}

\begin{proposition}\label{tpgt2.1}
Assuming (H1), (H2) and that $u_{0}, \; v_{0}\in H^{1}(\Omega)$, there
exists a pair of functions $(u,v)$ such that
%
\begin{enumerate}
\item $u\in L^{\infty}(0,T,H^{1}(\Omega))\cap L^{2}(0,T,H^{3}(\Omega))
\cap C([0,T];H^{\lambda}(\Omega))$, $\lambda<1$

\item $v\in L^{\infty}(0,T,H^{1}(\Omega))\cap L^{2}(0,T,H^{2}(\Omega))
\cap C([0,T];H^{\lambda}(\Omega))$, $\lambda<1$

\item $\partial_{t} u \in L^{2}(0,T,[H^{1}(\Omega)]'),\quad
\partial_{t} v \in L^{2}(\Omega_{T})$

\item $u(0)=u_{0}$ and $v(0)=v_{0}$ in $L^{2}(\Omega)$

\item ${\partial_{{\bf n}}u}_{|_{S_{T}}}=
{\partial_{{\bf n}}v}_{|_{S_{T}}}=0$ in $L^{2}(S_{T})$

\item $(u,v)$ solves (\ref{tpgepl}) in the following sense
\begin{gather*}
\int_{0}^{t}\langle \partial_{t} u ,\phi\rangle =
-\iint_{\Omega_{t}}q_{1}(u,v)(f_{1}(u,v)-\kappa_{1}u_{xx})_{x}\phi_{x},\quad
\forall \phi\in L^{2}(0,T,H^{1}(\Omega))\\
\iint_{\Omega_{t}}\partial_{t} v\psi =
-\iint_{\Omega_{t}}q_{2}(u,v)(f_{2}(u,v)-\kappa_{2}v_{xx})\psi,\quad
\forall \phi\in L^{2}(\Omega_{T}).
\end{gather*}
where $\Omega_t=\Omega\times(0,t)$ and $\iint_{\Omega_{t}}$ is the
integral over $\Omega_t$.
\end{enumerate}
\end{proposition}

\begin{remark} {\rm The regularity of the test functions with respect 
to $t$
allow us to obtain the integrals over $(0,t)$, instead of $(0,T)$ as
originally presented by Passo et al.~\cite{passo1299}.}
\end{remark}

Applying the above proposition, for each $\varepsilon,\sigma ,
M>0$ there exists a solution $(c_{\sigma\varepsilon
M},\theta_{\sigma\varepsilon M})$ of Problem~(\ref{tpgepds}) in the
following sense
%
\begin{align}
\label{tpgepdsa}
&\int_{0}^{t}\langle \partial_{t} c_{\sigma\varepsilon M},\phi\rangle =
-\iint_{\Omega_{t}}D(\partial_{c}
\mathcal{F}_{\sigma \varepsilon M}(c_{\sigma\varepsilon 
M},\theta_{\sigma\varepsilon M})
-\kappa_{c}[c_{\sigma\varepsilon M}]_{xx})_{x}\phi_{x},\\
\intertext{for $\phi\in L^{2}(0,T,H^{1}(\Omega))$ and}
\label{tpgepdsb}
&\iint_{\Omega_{t}}\partial_{t} \theta_{\sigma\varepsilon M} \psi =
-\iint_{\Omega_{t}}L(\partial_{\theta}
\mathcal{F}_{\sigma \varepsilon M}(c_{\sigma\varepsilon 
M},\theta_{\sigma\varepsilon M})
-\kappa [\theta_{\sigma\varepsilon M}]_{xx})\psi,
\end{align}
%
for $\psi\in L^{2}(\Omega_{T})$.
Let us observe that equation for $c$ in equation~(\ref{tpgepdsa})
implies that the mean value of $c_{\sigma\varepsilon M}$ in $\Omega$
is
%
\begin{equation}
\label{tpge3.2}
\overline{c_{\sigma\varepsilon M}(t)}=\overline{c_{0}}\in (0,1)
\end{equation}

\section{Limit as $M\rightarrow\infty$}\label{limm}

In this section we obtain some a priori estimates that will allow us to 
take
the limit in the parameter $M$. Actually, some of these estimates
are also independent of the parameters $\sigma$ and $\varepsilon$ and
will be useful in next sections.
%
\begin{lemma} \label{tpgl3.1M}
There exists a constant $C_{1}$
independent of $M$ (sufficiently large), $\sigma$ (sufficiently
small) and $\varepsilon$ such that
%
\begin{enumerate}
\item \label{tpgl3.1Mi1} $\|c_{\sigma \varepsilon M}
\|_{L^{\infty}(0,T,H^{1}(\Omega))} \leq C_{1}$

\item \label{tpgl3.1Mi2} $\|\theta_{\sigma \varepsilon M}
\|_{L^{\infty}(0,T,H^{1}(\Omega))}\leq C_{1}$

\item \label{tpgl3.1Mi3} $\|(\partial_{c}\mathcal{F}_{\sigma 
\varepsilon M} -\kappa_{c}
(c_{\sigma\varepsilon M})_{xx})_{x}\|_{L^{2}(\Omega_{T})}\leq C_{1}$

\item \label{tpgl3.1Mi4} $\|\partial_{\theta}\mathcal{F}_{\sigma 
\varepsilon M} -\kappa
(\theta_{\sigma\varepsilon M})_{xx}\|_{L^{2}(\Omega_{T})}\leq C_{1}$

\item \label{tpgl3.1Mi5} $\|\partial_{t}c_{\sigma \varepsilon M}
\|_{L^{\infty}(0,T,[H^{1}(\Omega)]')}\leq C_{1}$

\item \label{tpgl3.1Mi6} $\|\partial_{t}\theta_{\sigma \varepsilon M}
\|_{L^{2}(\Omega_{T})}\leq C_{1}$

\item \label{tpgl3.1Mi7} $\|\mathcal{F}_{\sigma \varepsilon 
M}(c_{\sigma \varepsilon M},
\theta_{\sigma \varepsilon M})
\|_{L^{\infty}(0,T,L^{1}(\Omega))}\leq C_{1}$
\end{enumerate}
\end{lemma}

\begin{proof} To obtain items~\ref{tpgl3.1Mi3},
\ref{tpgl3.1Mi4} and
\ref{tpgl3.1Mi7}, we argue as Passo et al.~\cite{passo1299} and Elliott 
and Garcke
\cite{elliott2796}. First, we observe that
by the regularity of $c_{\sigma \varepsilon M}$ and
$\theta_{\sigma\varepsilon M}$, we could take
%
\[
\partial_{c}\mathcal{F}_{\sigma \varepsilon M} -\kappa_{c}
(c_{\sigma\varepsilon M})_{xx}
\quad \mbox{and} \quad
\partial_{\theta}\mathcal{F}_{\sigma \varepsilon M} -\kappa
(\theta_{\sigma\varepsilon M})_{xx}
\]
%
as test
functions in the equations (\ref{tpgepdsa}) and (\ref{tpgepdsb}),
respectively. By adding the resulting identities, we obtain
%
\begin{equation} \label{aux}
\begin{split}
&\int_{0}^{t}\langle \partial_{t} c_{\sigma\varepsilon M},
\partial_{c}\mathcal{F}_{\sigma \varepsilon M} -\kappa_{c}
(c_{\sigma\varepsilon M})_{xx}\rangle
+\iint_{\Omega_{t}}\partial_{t} \theta_{\sigma\varepsilon M}
\partial_{\theta}\mathcal{F}_{\sigma \varepsilon M} -\kappa
(\theta_{\sigma\varepsilon M})_{xx}\\
&= - \iint_{\Omega_{t}}D[(\partial_{c}\mathcal{F}_{\sigma\varepsilon 
M}-
\kappa_{c} (c_{\sigma\varepsilon M})_{xx})_{x}]^{2}
-\iint_{\Omega_{t}}
L[\partial_{\theta}\mathcal{F}_{\sigma\varepsilon M}-
\kappa (\theta_{\sigma\varepsilon M})_{xx}]^{2}.
\end{split}
\end{equation}
%
Now, given a small $h>0$, we consider the functions
%
\[
c_{\sigma\varepsilon M h}(x,t)
=\frac{1}{h}\int_{t-h}^{t}c_{\sigma\varepsilon M}(\tau,x)d\tau
\]
%
where we set $c_{\sigma\varepsilon M}(x,t)=c_{0}(x)$
for $t\leq 0$. Since $\partial_{t}c_{\sigma\varepsilon M h}(x,t)\in 
L^{2}(\Omega_{T})$,
we have
%
\begin{align*}
&\int_{0}^{T}\langle (c_{\sigma\varepsilon M h})_{t},
[\partial_{c}\mathcal{F}_{\sigma \varepsilon M h} -\kappa_{c}
(c_{\sigma\varepsilon M h})_{xx}]\rangle dt\\
&+\iint_{\Omega_{T}}(\theta_{\sigma\varepsilon
M})_{t}[\partial_{\theta}\mathcal{F}_{\sigma \varepsilon Mh}
-\kappa (\theta_{\sigma \varepsilon M})_{xx}]\\
&= \int_{\Omega}
[
\frac{\kappa_{c}}{2}|[c_{\sigma\varepsilon M h}(t)]_{x}|^{2}
+\frac{\kappa}{2}|[\theta_{\sigma\varepsilon M}]_{x}(t)|^{2}
+\mathcal{F}_{\sigma \varepsilon Mh}(t)
]\\
&\quad-\int_{\Omega} [\frac{\kappa_{c}}{2}|[c_{0}]_{x}|^{2}
+\frac{\kappa}{2}|[\theta_{0}]_{x}|^{2}
+\mathcal{F}_{\sigma \varepsilon M}(c_{0},\theta_{0})].
\end{align*}
%
Taking the limit as $h$ tends to
zero in the above expression and using the result in (\ref{aux}), we 
obtain
%
\begin{align*}
 \iint_{\Omega_{t}}
&D[(\partial_{c}\mathcal{F}_{\sigma\varepsilon M}-
\kappa_{c} (c_{\sigma\varepsilon M})_{xx})_{x}]^{2}
+\iint_{\Omega_{t}}
L[\partial_{\theta}\mathcal{F}_{\sigma\varepsilon M}-
\kappa (\theta_{\sigma\varepsilon M})_{xx}]^{2}\\
& +\frac{\kappa_{c}}{2}\|[c_{\sigma\varepsilon M 
}]_{x}(t)\|_{L^{2}(\Omega)}^{2}
+\frac{\kappa}{2}\|[\theta_{\sigma\varepsilon 
M}]_{x}(t)\|_{L^{2}(\Omega)}^{2}
+\int_{\Omega}\mathcal{F}_{\sigma \varepsilon M}(t)\\
& =\frac{\kappa_{c}}{2}\|[c_{0}]_{x}\|_{L^{2}(\Omega)}^{2}
+\frac{\kappa}{2}\|[\theta_{0}]_{x}\|_{L^{2}(\Omega)}^{2}
+\int_{\Omega}\mathcal{F}_{\sigma \varepsilon M}(c_{0},\theta_{0})
\end{align*}
%
for almost every $t\in (0,T]$. Using (\ref{tpge1.4}) and
(\ref{tpge3.1}), we could choose $M_{0}$ and $\sigma_{0}$, depending 
only on the initial
conditions, to obtain for all $M>M_{0}$ and all $\sigma <\sigma_{0}$
%
\begin{equation}
\begin{split}
 \iint_{\Omega_{T}}
&D[(\partial_{c}\mathcal{F}_{\sigma\varepsilon M}-
\kappa_{c} (c_{\sigma\varepsilon M})_{xx})_{x}]^{2}
+\iint_{\Omega_{T}}
L[\partial_{\theta}\mathcal{F}_{\sigma\varepsilon M}-
\kappa (\theta_{\sigma\varepsilon M})_{xx}]^{2}
\\
&+
\frac{\kappa_{c}}{2}\|[c_{\sigma\varepsilon M
}]_{x}(t)\|_{L^{2}(\Omega)}^{2}
+\frac{\kappa}{2}\|[\theta_{\sigma\varepsilon
    M}]_{x}(t)\|_{L^{2}(\Omega)}^{2}
+\int_{\Omega}\mathcal{F}_{\sigma \varepsilon M}(t)
\leq C_{1}
\end{split}
\label{tpge3.3M}
\end{equation}
%
which implies items~\ref{tpgl3.1Mi3}, \ref{tpgl3.1Mi4}
and \ref{tpgl3.1Mi7}
since we have (\ref{tpge3.1}). Using the Poincar\'e's
inequality, (\ref{tpge3.1}) and
(\ref{tpge3.2}), Item~\ref{tpgl3.1Mi1} is also verified.

To prove
Item~\ref{tpgl3.1Mi6}, we choose
$\psi=\partial_{t}\theta_{\sigma \varepsilon
M}$ as a test function in (\ref{tpgepdsb}),
which yields
%
\begin{align*}
\iint_{\Omega_{T}}[\partial_{t}\theta_{\sigma \varepsilon M}]^{2}
&=-\iint_{\Omega_{T}}(\partial_{\theta}\mathcal{F}_{\sigma \varepsilon 
M} -\kappa
(\theta_{\sigma\varepsilon M})_{xx})\partial_{t}\theta_{\sigma 
\varepsilon M}\\
&\leq
\Big(\iint_{\Omega_{T}}(\partial_{\theta}\mathcal{F}_{\sigma 
\varepsilon M} -\kappa
(\theta_{\sigma\varepsilon M})_{xx})^{2}\Big)^{1/2}
\Big(\iint_{\Omega_{T}}[\partial_{t}\theta_{\sigma \varepsilon 
M}]^{2}\Big)^{1/2}.
\end{align*}
%
Since
%
\[
 \int_{\Omega}\theta_{\sigma \varepsilon M}^{2}\leq
2\int_{\Omega}|\theta_{0}|^{2}
+2t\iint_{\Omega_{T}}(\partial_{t}\theta_{\sigma \varepsilon
M})^{2}d\tau\leq C_{2},
\]
%
Item \ref{tpgl3.1Mi6} and (\ref{tpge3.3M}), it follows that
 Item \ref{tpgl3.1Mi2} is verified.
Finally, Item \ref{tpgl3.1Mi5} follows since
%
\[
\big|\int_{0}^{T}\langle \partial_{t}
c_{\sigma \varepsilon M},\phi\rangle\big|\leq
\Big(\iint_{\Omega_{T}}[(\partial_{c}\mathcal{F}_{\sigma \varepsilon M} 
-\kappa_{c}
(c_{\sigma\varepsilon M })_{xx})_{x}]^{2}\Big)^{1/2}
\Big(\iint_{\Omega_{T}}(\phi_{x})^{2}\Big)^{1/2}
\]
%
for all $\phi \in L^{2}(0,T,H^{1}(\Omega))$.
\end{proof}

\begin{remark} \rm
From (\ref{tpge3.3M}), using (\ref{tpge3.1}), we obtain
%
\begin{equation}
\begin{split}
 &\iint_{\Omega_{T}}
D[(\partial_{c}\mathcal{F}_{\sigma\varepsilon M}-
\kappa_{c} (c_{\sigma\varepsilon M})_{xx})_{x}]^{2}
+\iint_{\Omega_{T}}
L[\partial_{\theta}\mathcal{F}_{\sigma\varepsilon M}-
\kappa (\theta_{\sigma\varepsilon M})_{xx}]^{2}\\
&+ \frac{\kappa_{c}}{2}\|[c_{\sigma\varepsilon M 
}(t)]_{x}\|_{L^{2}(\Omega)}^{2}
+\frac{\kappa}{2}\|[\theta_{\sigma\varepsilon 
M}]_{x}(t)\|_{L^{2}(\Omega)}^{2}
\leq C_{1} .
\end{split}
\label{tpge3.3Mres}
\end{equation}
\end{remark}

%
\begin{lemma} \label{tpgl3.4M}
For $M$ sufficiently large and $\sigma$ sufficiently small,
there exist a constant $C_{3}$ independent of $\sigma, M$ and 
$\varepsilon$
and a constant $C_{3}'(\sigma)$ independent of $M$ and $\varepsilon$ 
such that
\begin{enumerate}
\item \label{tpgl3.4Mi1}$\|\partial_{c}\mathcal{F}_{\sigma\varepsilon 
M}
\|_{L^{2}(0,T,H^{1}(\Omega))}\leq C_{3}'$,

\item
\label{tpgl3.4Mi3}$\|\partial_{\theta}\mathcal{F}_{\sigma\varepsilon M}
\|_{L^{2}(\Omega_{T})}\leq C_{3}$,

\item \label{tpgl3.4Mi2}$\|[c_{\sigma\varepsilon 
M}]_{xx}\|_{L^{2}(\Omega_{T})}\leq
C_{3}$,

\item \label{tpgl3.4Mi4}
$\|[\theta_{\sigma\varepsilon M}]_{xx}\|_{L^{2}(\Omega_{T})}\leq 
C_{3}$,
\end{enumerate}
\end{lemma}

\begin{proof} First, we prove items~\ref{tpgl3.4Mi3} and 
\ref{tpgl3.4Mi4}.
From Item~\ref{tpgl3.1Mi4} of Lemma~\ref{tpgl3.1M}, we have
%
\begin{equation} \label{tpge3.9M}
\iint_{\Omega_{T}}(\partial_{\theta}\mathcal{F}_{\sigma\varepsilon 
M})^{2}
-2\kappa
\iint_{\Omega_{T}}\partial_{\theta}\mathcal{F}_{\sigma\varepsilon M}
[\theta_{\sigma\varepsilon M}]_{xx}+
\kappa^{2}
\iint_{\Omega_{T}}[\theta_{\sigma\varepsilon M}]_{xx}^{2}\leq C_{3}.
\end{equation}
%
Since
%
\begin{align*}
\partial_{\theta}\mathcal{F}_{\sigma\varepsilon M}
[\theta_{\sigma\varepsilon M}]_{xx} &=
[g_{M}'(\theta_{\sigma\varepsilon M})+\partial_{\theta}h_{M}
(c_{\sigma\varepsilon M},\theta_{\sigma\varepsilon M})]
[\theta_{\sigma\varepsilon M}]_{xx}\\
&= [g_{M}'(\theta_{\sigma\varepsilon M})+h_{1}(c_{\sigma\varepsilon M})
h_{2}'(M;\theta_{\sigma\varepsilon M})]
[\theta_{\sigma\varepsilon M}]_{xx},
\end{align*}
%
using (\ref{limitacoes2}) and (\ref{limitacoes3}), we obtain
%
\[
2\kappa \partial_{\theta}\mathcal{F}_{\sigma\varepsilon M}
[\theta_{\sigma\varepsilon M}]_{xx} \leq
\frac{\kappa^{2}}{2}[\theta_{\sigma\varepsilon M}]_{xx}^{2}
+C_{3}[\theta_{\sigma\varepsilon 
M}^{6}+W_{0}^{2}\theta_{\sigma\varepsilon M}^{2}].
\]
%
Thus, from
Item~\ref{tpgl3.1Mi2} of Lemma~\ref{tpgl3.1M},
it follows from~(\ref{tpge3.9M}) that
%
\begin{equation}
\label{tpge3.9Ma}
\iint_{\Omega_{T}}(\partial_{\theta}\mathcal{F}_{\sigma\varepsilon 
M})^{2}
+
\frac{\kappa^{2} }{2}
\iint_{\Omega_{T}}[\theta_{\sigma\varepsilon M}]_{xx}^{2}\leq C_{3}.
\end{equation}
%
Now, we prove Item~\ref{tpgl3.4Mi2}. Defining, $H_{\sigma\varepsilon M}
=\partial_{c}\mathcal{F}_{\sigma\varepsilon M}
-\kappa_{c}[c_{\sigma\varepsilon M}]_{xx}$, since
${[c_{\sigma\varepsilon M}]_{x}}_{|S_{T}}=0$, we have
%
\[
\iint_{\Omega_{T}}H_{\sigma\varepsilon M}=
\iint_{\Omega_{T}}\partial_{c}\mathcal{F}_{\sigma\varepsilon M},
\]
%
and from Item~\ref{tpgl3.1Mi3} of Lemma~\ref{tpgl3.1M},
%
\[
\iint_{\Omega_{T}}[H_{\sigma\varepsilon M}]_{x}^{2}\leq C_{1}.
\]
%
Using the definition of $\mathcal{F}_{\sigma\varepsilon M}$, given in
(\ref{energia2}), and an integration by
parts, we obtain
%
\begin{align*}
&\iint_{\Omega_{T}}H_{\sigma\varepsilon M}^{2}\\
&=\iint_{\Omega_{T}}(\partial_{c}\mathcal{F}_{\sigma\varepsilon M})^{2}
 +2\kappa_{c}\varepsilon
\iint_{\Omega_{T}}
[F_{\sigma}''(c_{\sigma\varepsilon
    M})+F_{\sigma}''(1-c_{\sigma\varepsilon M})]
[c_{\sigma\varepsilon M}]_{x}^{2}\\
&\quad -2\kappa_{c} L
\iint_{\Omega_{T}}(f'(c_{\sigma\varepsilon M})+
h_{1}'(c_{\sigma\varepsilon M})h_{2}(M;\theta_{\sigma\varepsilon M}))
[c_{\sigma\varepsilon M}]_{xx}
+ \kappa_{c}^{2}
\iint_{\Omega_{T}}[c_{\sigma\varepsilon M}]_{xx}^{2}.
\end{align*}
%
On the other hand, we can write
%
\begin{align*}
\iint_{\Omega_{T}}H_{\sigma\varepsilon M}^{2}
&=\iint_{\Omega_{T}}[H_{\sigma\varepsilon
    M}-\overline{H_{\sigma\varepsilon M}}]^{2}
+\iint_{\Omega_{T}}\overline{H_{\sigma\varepsilon M}}^{2}\\
&
\leq C_{P}\iint_{\Omega_{T}}[H_{\sigma\varepsilon M}]_{x}^{2}+
\iint_{\Omega_{T}}(\partial_{c}\mathcal{F}_{\sigma\varepsilon M})^{2}
\end{align*}
%
where $C_{P}$ denotes the constant appearing in Poincar\'e's 
inequality. From
these two last results,
Item \ref{tpgl3.4Mi2} follows recalling that
$[F_{\sigma}''(c_{\sigma\varepsilon M})+
F_{\sigma}''(1-c_{\sigma\varepsilon M})]\geq 0$ and
using (\ref{limitacoes1}), (\ref{limitacoes2}), (\ref{limitacoes3}) and
Item~\ref{tpgl3.1Mi2} of Lemma~\ref{tpgl3.1M}.

Finally, recalling that for each $\sigma$, $F_{\sigma}'(s)$ is
bounded in ${\bf R}$, using again the definition of $f$ and $h_{M}$ and
Item~\ref{tpgl3.1Mi2} of Lemma~\ref{tpgl3.1M}, we obtain
%
\begin{align*}
\|\partial_{c}\mathcal{F}_{\sigma\varepsilon
M}\|_{L^{2}(\Omega_{T})}^{2}
&\leq C\iint_{\Omega_{T}}\{[f'(c_{\sigma\varepsilon M})]^{2}
+[h_{1}'(c_{\sigma\varepsilon
M})]^{2}[h_{2}(M;\theta_{\sigma\varepsilon M})]^{2}\\
&\quad +\varepsilon^{2}
[F_{\sigma}'(c_{\sigma\varepsilon
    M})-F_{\sigma}'(1-c_{\sigma\varepsilon M})]\}\\
&\leq C\{[U_{0}^{2}|\Omega_{T}|+W_0^2\|\theta_{\sigma\varepsilon
M}\|_{L^{4}}^{4}]+C(\sigma)\}\leq C_{3}'(\sigma).
\end{align*}
%
A similar  argument shows that
$\|[\partial_{c}\mathcal{F}_{\sigma\varepsilon 
M}]_{x}\|_{L^{2}(\Omega_{T})}^{2}$
is also bounded by a constant which depends only on
$\sigma$. Thus, we have proved the Item~\ref{tpgl3.4Mi1}.
\end{proof}

We can now state the following result.
%
\begin{proposition} \label{tpgp3.5M}
For $\sigma$ (sufficiently small), there exists a pair
$(c_{\sigma\varepsilon},\theta_{\sigma\varepsilon})$ such that:
%
\begin{enumerate}
\item \label{tpgp3.5Mi1}$c_{\sigma\varepsilon}
\in L^{\infty}(0,T,H^{1}(\Omega))\cap L^{2}(0,T,H^{3}(\Omega))$

\item \label{tpgp3.5Mi1a}$\theta_{\sigma\varepsilon}
\in L^{\infty}(0,T,H^{1}(\Omega))\cap L^{2}(0,T,H^{2}(\Omega))$

\item \label{tpgp3.5Mi3}$\partial_{t} c_{\sigma\varepsilon}
\in L^{2}(0,T,[H^{1}(\Omega)]'),
\quad \partial_{t} \theta_{\sigma\varepsilon} \in L^{2}(\Omega_{T})$

\item \label{tpgp3.5Mi6}$
\partial_{c}\mathcal{F}_{\sigma\varepsilon}
(c_{\sigma\varepsilon},\theta_{\sigma\varepsilon}),\;
\partial_{\theta}\mathcal{F}_{\sigma\varepsilon}
(c_{\sigma\varepsilon},\theta_{\sigma\varepsilon})
\in L^{2}(\Omega_{T})$

\item \label{tpgp3.5Mi8}$c_{\sigma\varepsilon}(0)=c_{0}$ and
$\theta_{\sigma\varepsilon}(0)=\theta_{0}$ in $L^{2}(\Omega)$

\item \label{tpgp3.5Mi9}${[c_{\sigma\varepsilon}]_{x}}_{|_{S_{T}}}
={[\theta_{\sigma\varepsilon}]_{x}}_{|_{S_{T}}}=0$ in $L^{2}(S_{T})$

\item
\label{tpgp3.5Mi10}$(c_{\sigma\varepsilon},\theta_{\sigma\varepsilon})$
solves the perturbed system (\ref{tpgepds}) in the following sense:
%
\begin{equation}
\label{tpge3.11Mp}
\int_{0}^{T}\langle \partial_{t}c_{\sigma\varepsilon} ,\phi\rangle =
-\iint_{\Omega_{T}}D[
\partial_{c}\mathcal{F}_{\sigma\varepsilon}
(c_{\sigma \varepsilon},\theta_{\sigma\varepsilon})
-\kappa_{c} (c_{\sigma\varepsilon})_{xx}]_{x}\phi_{x}
\end{equation}
%
for all $\phi\in L^{2}(0,T,H^{1}(\Omega))$, and
%
\begin{equation}
\label{tpge3.12Mp}
\iint_{\Omega_{T}}\partial_{t}\theta_{\sigma\varepsilon}\psi =
-\iint_{\Omega_{T}}L(\partial_{\theta}\mathcal{F}_{\sigma\varepsilon}
(c_{\sigma\varepsilon},\theta_{\sigma\varepsilon })-
\kappa (\theta_{\sigma\varepsilon })_{xx})\psi
\end{equation}
%
for all $\psi\in L^{2}(\Omega_{T})$, and
$\mathcal{F}_{\sigma\varepsilon}$ is given by
%
\[
\mathcal{F}_{\sigma \varepsilon}(c,\theta)=
f(c)+\frac{\delta}{4}\theta^{4}+h_{1}(c)
\theta^{2}
+\varepsilon
[F_{\sigma}(c)+ F_{\sigma}(1-c)].
\]
\end{enumerate}
\end{proposition}

\begin{proof} First, let us observe that from
Item~\ref{tpgl3.1Mi3} of Lemma~\ref{tpgl3.1M} and
Item~\ref{tpgl3.4Mi1} of Lemma~\ref{tpgl3.4M}, the norm of
$[c_{\sigma\varepsilon M}]_{xxx}$ in $L^{2}(\Omega_{T})$
is bounded by a constant which does not depend
on $M$. This fact, the estimates of Lemmas~\ref{tpgl3.1M}
and~\ref{tpgl3.4M} together with a compactness argument imply that
there exists a subsequence (still denoted
by $\{(c_{\sigma\varepsilon M},\theta_{\sigma\varepsilon M})\}$)
that satisfies (as $M$ goes to infinity)
%
\begin{gather*}
c_{\sigma\varepsilon M},\;\theta_{\sigma\varepsilon M}
\quad \mbox{converge weakly-* to} \quad
c_{\sigma\varepsilon},\theta_{\sigma\varepsilon}
\quad \mbox{in} \quad L^{\infty}(0,T,H^{1}(\Omega)),\\
c_{\sigma\varepsilon M},
\quad \mbox{converges weakly to} \quad
c_{\sigma\varepsilon}
\quad \mbox{in} \quad L^{2}(0,T,H^{3}(\Omega)),\\
\theta_{\sigma\varepsilon M},
\quad \mbox{converges weakly to} \quad
\theta_{\sigma\varepsilon}
\quad \mbox{in} \quad L^{2}(0,T,H^{2}(\Omega)),\\
\partial_{t}c_{\sigma\varepsilon M},
\quad \mbox{converges weakly to} \quad
\partial_{t} c_{\sigma\varepsilon}
\quad \mbox{in} \quad L^{2}(0,T,[H^{1}(\Omega)]'),\\
\partial_{t} \theta_{\sigma\varepsilon M},
\quad \mbox{converges weakly to} \quad
\partial_{t} \theta_{\sigma\varepsilon}
\quad \mbox{in} \quad L^{2}(\Omega_{T})\\
c_{\sigma\varepsilon M},\;\theta_{\sigma\varepsilon M}
\quad \mbox{converge to} \quad
c_{\sigma\varepsilon},\theta_{\sigma\varepsilon}
\quad \mbox{in} \quad L^{2}(\Omega_{T}).
\end{gather*}
%
By recalling Lemmas~\ref{tpgl3.1M} and~\ref{tpgl3.4M},
items~\ref{tpgp3.5Mi1}--\ref{tpgp3.5Mi3} now
follow. Now, items \ref{tpgl3.4Mi1} and \ref{tpgl3.4Mi3} of 
Lemma~\ref{tpgl3.4M}
imply that
%
\begin{gather*}
\partial_{c}\mathcal{F}_{\sigma \varepsilon M}
(c_{\sigma\varepsilon M},\theta_{\sigma\varepsilon M})
\quad \mbox{converges weakly to} \quad \mathcal{G}
\quad \mbox{in} \quad L^{2}(\Omega_{T}),\\
\partial_{\theta}\mathcal{F}_{\sigma \varepsilon M}
(c_{\sigma\varepsilon M},\theta_{\sigma\varepsilon M})
\quad \mbox{converges weakly to} \quad \mathcal{H}
\quad \mbox{in} \quad L^{2}(\Omega_{T}).
\end{gather*}
%
Since the strong convergence of the sequence $(c_{\sigma\varepsilon
M})$ implies that (at least for a subsequence)
$\partial_{c}\mathcal{F}_{\sigma \varepsilon M}(c_{\sigma\varepsilon
M},\theta_{\sigma\varepsilon M})$ converges pointwise in $\Omega_{T}$,
it follows from Lions \cite[Lemma~1.3]{lions69}, that
$\mathcal{G}=
\partial_{c}\mathcal{F}_{\sigma
\varepsilon}(c_{\sigma\varepsilon},\theta_{\sigma\varepsilon})$.
Similarly, we have
$\mathcal{H}=
\partial_{\theta}\mathcal{F}_{\sigma
\varepsilon}(c_{\sigma\varepsilon},\theta_{\sigma\varepsilon})$. Thus
Item~\ref{tpgp3.5Mi6} is proved.

Item~\ref{tpgp3.5Mi8} is straightforward. Now, by compactness we have
that
%
\begin{align*}
&c_{\sigma\varepsilon M}
\quad \mbox{converges to} \quad
c_{\sigma\varepsilon}
\quad \mbox{in} \quad L^{2}(0,T,H^{2-\lambda}(\Omega)),\quad 
\lambda>0,\\
&\theta_{\sigma\varepsilon M}
\quad \mbox{converges to} \quad
\theta_{\sigma\varepsilon}
\quad \mbox{in} \quad L^{2}(0,T,H^{2-\lambda}(\Omega)),\quad \lambda>0,
\end{align*}
%
which imply Item~\ref{tpgp3.5Mi9}.
To prove Item~\ref{tpgp3.5Mi10}, by using the previous convergences,
we pass to the limit as $M$ goes to
infinity in the equations~(\ref{tpgepdsa}) and (\ref{tpgepdsb}).
\end{proof}

\section{Limit as $\sigma\rightarrow 0^{+}$}\label{lims}

In this section we obtain some a priori estimates that allow taking
the limit in the parameter $\sigma$.

First, let us note that (\ref{tpge3.11Mp}) implies that the mean value
of $c_{\sigma \varepsilon}$ in $\Omega$ is given by
%
\begin{equation}
\label{tpge3.2s}
\overline{c_{\sigma \varepsilon}(t)}=\overline{c_{0}}\in (0,1),
\end{equation}
%
We start with the following Lemma.

\begin{lemma} \label{tpgl3.1s}
There exists a constant $C_{1}$ independent of $\varepsilon$ and
$\sigma$ (sufficiently small) such that
\begin{enumerate}
\item \label{tpgl3.1si1}$\|c_{\sigma \varepsilon }
\|_{L^{\infty}(0,T,H^{1}(\Omega))}\leq C_{1}$

\item \label{tpgl3.1si2}$\|\theta_{\sigma \varepsilon }
\|_{L^{\infty}(0,T,H^{1}(\Omega))}\leq C_{1}$

\item \label{tpgl3.1si3}$\|[\partial_{c}\mathcal{F}_{\sigma 
\varepsilon} -\kappa_{c}
(c_{\sigma\varepsilon})_{xx}]_{x}\|_{L^{2}(\Omega_{T})}\leq C_{1}$

\item \label{tpgl3.1si4}$\|\partial_{\theta}\mathcal{F}_{\sigma 
\varepsilon } -\kappa
(\theta_{\sigma\varepsilon})_{xx}\|_{L^{2}(\Omega_{T})}\leq C_{1}$

\item \label{tpgl3.1si5}$\|\partial_{t}c_{\sigma \varepsilon }
\|_{L^{\infty}(0,T,[H^{1}(\Omega)]')}\leq C_{1}$

\item \label{tpgl3.1si6}$\|\partial_{t}\theta_{\sigma \varepsilon }
\|_{L^{2}(\Omega_{T})}\leq C_{1}$

\item \label{tpgl3.1si7}$\|
\mathcal{F}_{\sigma \varepsilon}(c_{\sigma \varepsilon},\theta_{\sigma 
\varepsilon})
\|_{L^{\infty}(0,T,L^{1}(\Omega))}\leq C_{1}$
\end{enumerate}
\end{lemma}

\begin{proof}
Let us observe that in the proof of Proposition~\ref{tpgp3.5M}, we
have identified the weak limits, when $M$ goes to infinity, of the 
sequences
$\partial_{c}\mathcal{F}_{\sigma \varepsilon M}$
and $\partial_{\theta}\mathcal{F}_{\sigma \varepsilon M}$ as
$\partial_{c}\mathcal{F}_{\sigma \varepsilon}$
and $\partial_{\theta}\mathcal{F}_{\sigma \varepsilon}$,
respectively. Thus, by taking the inferior limit as $M$ goes to
infinity, of estimate (\ref{tpge3.3Mres}), we obtain
%
\begin{equation}
\label{tpge3.3s}
\begin{split}
&\frac{\kappa_{c}}{2}\|(c_{\sigma \varepsilon})_{x}
\|_{L^{\infty}(0,T,L^{2}(\Omega))}^{2}
+\frac{\kappa}{2}\|(\theta_{\sigma\varepsilon})_{x}
\|_{L^{\infty}(0,T,L^{2}(\Omega))}^{2}\\
&+
D\|[\partial_{c}\mathcal{F}_{\sigma \varepsilon} -\kappa_{c}
(c_{\sigma\varepsilon})_{xx}]_{x}\|_{L^{2}(\Omega_{T})}^{2}
+L\|\partial_{\theta}\mathcal{F}_{\sigma\varepsilon }-
\kappa (\theta_{\sigma\varepsilon})_{xx}\|_{L^{2}(\Omega_{T})}^{2}
\leq C_{1}.
\end{split}
\end{equation}
%
The items~\ref{tpgl3.1si3}~and~\ref{tpgl3.1si4} follow from 
(\ref{tpge3.3s}).
Using (\ref{tpge3.3s}), Poincar\'e's inequality and
(\ref{tpge3.2s}), we obtain Item~\ref{tpgl3.1si1}.
To prove
items~\ref{tpgl3.1si2}, \ref{tpgl3.1si5} and~\ref{tpgl3.1si6},
we just take the inferior
limit of items~\ref{tpgl3.1Mi2}, \ref{tpgl3.1Mi5}
and~\ref{tpgl3.1Mi6} of Lemma~\ref{tpgl3.1M}. Finally, using 
(\ref{limitacoes1}),
(\ref{limitacoes2}), (\ref{limitacoes3}) and the estimates of 
Lemma~\ref{tpgl3.1M}, we can estimate
$\|f(c)+\frac{\delta}{4}\theta^{4}+h_{1}(c)
\theta^{2}\|_{L^{\infty}(0,T,L^{1}(\Omega))}$. Also with the estimates 
of 
Lemma~\ref{tpgl3.1M}, we get
the strong convergence of a subsequence of $(c_{\sigma\varepsilon})$. 
Using this convergence and the 
Fatou's Lemma, we
get a bound for \linebreak $\|\varepsilon
[F_{\sigma}(c)+ F_{\sigma}(1-c)]\|_{L^{\infty}(0,T,L^{1}(\Omega))}$
which together with the previous estimate yield
Item~\ref{tpgl3.1si7}.
\end{proof}

As Passo et al.~\cite{passo1299}, by arguing in a standard way
(see Bernis and Friedman~\cite{bernis8390} for a proof, p. 183), we 
obtain

\begin{corollary} \label{tpgc3.2s}
There exists a constant $C_{2}$ independent of $\varepsilon$ and
$\sigma$ (sufficiently small)
such that
%
\begin{equation}
\label{tpge3.4s}
\|c_{\sigma \varepsilon }
\|_{C^{0,\frac{1}{2},\frac{1}{8}}(\mathrm{cl}\,\Omega_{T})}\leq C_{2}
\quad \mbox{and} \quad
\|\theta_{\sigma \varepsilon }
\|_{C^{0,\frac{1}{2},\frac{1}{8}}(\mathrm{cl}\,\Omega_{T})}\leq C_{2}
\end{equation}
\end{corollary}
%
By Corollary~\ref{tpgc3.2s}, we can extract a subsequence (still
denoted by $(c_{\sigma \varepsilon },\theta_{\sigma \varepsilon})$) 
such that
%
\begin{equation}
\begin{split}
&\mbox{$(c_{\sigma \varepsilon},\theta_{\sigma \varepsilon})$
converges uniformly to
$(c_{\varepsilon},\theta_{\varepsilon})$
in $\mathrm{cl}\,\Omega_{T}$
as $\sigma$ approaches zero},\\
&c_{\varepsilon }\in 
C^{0,\frac{1}{2},\frac{1}{8}}(\mathrm{cl}\,\Omega_{T})
\quad \mbox{and} \quad
\theta_{\varepsilon }\in 
C^{0,\frac{1}{2},\frac{1}{8}}(\mathrm{cl}\,\Omega_{T}).
\end{split}
\label{tpge3.5s}
\end{equation}
%
We now demonstrate that the limit $c_{\varepsilon}$ lies within the
interval
\[
I=\{c\in \mathbb{R}, 0<c<1\}.
\]

\begin{lemma} \label{tpgl3.3s}
$|\Omega_{T}\setminus \mathcal{B}(c_{\varepsilon})|=0$
with $\mathcal{B}(c)=\{(x,t)\in \mathrm{cl}\,\Omega_{T},\; c(x,t)\in 
I\}$.
\end{lemma}

\begin{proof} Arguing as Passo et al.~\cite{passo1299}, let $N$ denote 
the
operator defined as minus the inverse of
the Laplacian with zero Neumann boundary
conditions. That is, given $f\in [H^{1}(\Omega)]_{null}'=\{f\in
[H^{1}(\Omega)]',\;\langle f,1\rangle =0\}$, we define $Nf \in
H^{1}(\Omega)$ as the unique solution of
%
\begin{equation} \label{opgreen}
\int_{\Omega}(Nf)'\psi'=\langle f,\psi\rangle,\quad \forall \psi \in 
H^{1}(\Omega)
\quad \mbox{and} \quad
\int_{\Omega}Nf =0.
\end{equation}
%
By (\ref{tpge3.2s}) and Item~\ref{tpgl3.1si1} of Lemma~\ref{tpgl3.1s},
$N(c_{\sigma \varepsilon }- \overline{c_{\sigma \varepsilon }})$
is well defined. Choosing\linebreak
$\phi = N(c_{\sigma \varepsilon }-
\overline{c_{\sigma \varepsilon }})$ as a test function in the
equation (\ref{tpge3.11Mp}), we have
%
\begin{align*}
&\int_{0}^{T}\langle \partial_{t}c_{\sigma\varepsilon},
N(c_{\sigma \varepsilon }- \overline{c_{\sigma \varepsilon}})\rangle dt 
\\
&=-\iint_{\Omega_{T}} D[\partial_{c}\mathcal{F}_{\sigma \varepsilon} 
-\kappa_{c}
(c_{\sigma\varepsilon})_{xx}]_{x}
[N(c_{\sigma \varepsilon }- \overline{c_{\sigma \varepsilon }})]_{x}\\
&= -\iint_{\Omega_{T}}
D(c_{\sigma \varepsilon }- \overline{c_{\sigma \varepsilon }})
\partial_{c}\mathcal{F}_{\sigma\varepsilon}(c_{\sigma\varepsilon},\theta_{\sigma\varepsilon})
-D\kappa_{c}\iint_{\Omega_{T}} [(c_{\sigma\varepsilon})_{x}]^{2}
\end{align*}
%
Now, estimates in Lemma~\ref{tpgl3.1s} and the definition of $N$
imply
%
\begin{equation} \label{tpge3.6s}
\iint_{\Omega_{T}}(c_{\sigma \varepsilon }- \overline{c_{\sigma 
\varepsilon }})
\partial_{c}\mathcal{F}_{\sigma\varepsilon}
(c_{\sigma\varepsilon},\theta_{\sigma\varepsilon})\leq C_{4}.
\end{equation}
%
We observe that the following identity holds for any $m\in {\bf R}$,
%
\begin{equation}
\begin{split}
&(c-m)\partial_{c}\mathcal{F}_{\sigma\varepsilon}(c,\theta)\\
&= (c-m)[f'(c)+h_{1}'(c)\theta^{2}
+\varepsilon (F_{\sigma}'(c)-F_{\sigma}'(1-c))]\\
&=\varepsilon\{c[F_{\sigma}'(c)-1]
+(1-c)[F_{\sigma}'(1-c)-1] + 2\}
\\
& \quad +(c-m)[f'(c)+h_{1}'(c)\theta^{2}]
-\varepsilon -\varepsilon m F_{\sigma}'(c)
-\varepsilon (1-m)F_{\sigma}'(1-c)
\end{split}
\label{tpge3.7s}
\end{equation}
%
We observe that the terms inside the braces are bounded from below
since for any $\sigma\in (0,1/e)$, we have
%
\begin{gather*}
-1/e\leq \sigma \ln \sigma
\leq s[F_{\sigma}'(s)-1]\leq 0,
\quad s\leq \sigma,\\
-1/e\leq s\ln s = s[F_{\sigma}'(s)-1]
\leq 0,\quad \sigma\leq s \leq 1-\sigma,\\
-2\leq s[F_{\sigma}'(s)-1]
\leq 0, \quad 1-\sigma \leq s\leq 2,\\
0 = s[F_{\sigma}'(s)-1],
\quad s\geq 2.
\end{gather*}
%
We now recall that the mean value of
$c_{\sigma\varepsilon}$ in $\Omega$ is conserved and is equal to
$\overline{c_{0}}$ which belongs to the interval $(0,1)$. Thus, since
$f'$, $h_{1}'$ are uniformly bounded, using the estimates in 
Lemma~\ref{tpgl3.1s},
by setting $m=\overline{c_{\sigma\varepsilon}}=\overline{c_{0}}$
in (\ref{tpge3.7s}) and noting $F_{\sigma}'\leq 1$, it follows from
(\ref{tpge3.6s}) that
%
\begin{equation} \label{tpge3.8s}
-\varepsilon\iint_{\Omega_{T}}
[F_{\sigma}'(c_{\sigma\varepsilon})+
F_{\sigma}'(1-c_{\sigma\varepsilon})]\leq C_{4}
\end{equation}
%
To complete the proof, suppose by contradiction that the set
$\Omega_{T}\setminus \mathcal{B}(c_{\varepsilon})$ has a positive
measure. We start  supposing that
%
\[
A=\{(x,t)\in \Omega_{T},\; c_{\varepsilon}\leq 0\}
\]
%
has positive measure. Since $F_{\sigma}'\leq 1$, the estimate
(\ref{tpge3.8s}) gives
%
\[
-\varepsilon\iint_{A}F_{\sigma}'(c_{\sigma\varepsilon})
\leq C_{4}.
\]
%
Note, however, that the uniform convergence of
$c_{\sigma\varepsilon}$ implies that
%
\[
\forall \lambda >0, \; \exists \sigma_{\lambda},\quad
c_{\sigma\varepsilon}\leq \lambda,\quad \forall
(x,t)\in A,\;\;\sigma <\sigma_{\lambda}
\]
%
therefore, due to the convexity of $F_{\sigma}$, we have
$F_{\sigma}'(c_{\sigma\varepsilon})\leq F_{\sigma}'(\lambda)$. Hence
%
\[
-\varepsilon|A|(\ln \lambda + 1)=
-\varepsilon\lim_{\sigma\rightarrow 0^+}\iint_{A}F_{\sigma}'(\lambda)
\leq C_{4}
\]
%
which leads to a contradiction for $\lambda\in (0,1)$ sufficiently
small. The same argument shows that $B=\{(x,t)\in \Omega_{T},\;
c_{\varepsilon}\geq 1\}$ has zero measure.
\end{proof}

In the next lemma we derive additional estimates which allow us to
pass to the limit as $\sigma$ tends to zero. Its proof
follows directly from the estimates of Lemma~\ref{tpgl3.1s}.
%
\begin{lemma} \label{tpgl3.4s}
There exists a constant $C_{3}$ which is independent of $\varepsilon$
and $\sigma$ (sufficiently small) such that
\begin{enumerate}
\item
$\|\partial_{\theta}\mathcal{F}_{\sigma\varepsilon}\|_{L^{2}(\Omega_{T})}
\leq C_{3}$,

\item
  \label{tpgl3.4si2}
$\|[c_{\sigma\varepsilon}]_{xx}\|_{L^{2}(\Omega_{T})}\leq
C_{3}$,

\item
$\|[\theta_{\sigma\varepsilon}]_{xx}\|_{L^{2}(\Omega_{T})}\leq C_{3}$,
\end{enumerate}
\end{lemma}

To pass to the limit as $\sigma$ goes to zero, we need an
estimate of $\partial_{c}\mathcal{F}_{\sigma\varepsilon}$ that
is independent of $\sigma$. We cannot repeat the argument that we used
in Lemma~\ref{tpgl3.4M} because there we obtained with a
constant that depends on
$\sigma$. The desired estimate will be obtained by using the next 
lemma,
presented by Copetti and Elliott~\cite[p.~48]{copetti6392},
and by Elliott and
Luckhaus~\cite[p.~23]{elliott88791}.

\begin{lemma} \label{tmcl2.4}
Let $v\in L^{1}(\Omega)$ such that there exist positive constants
$\delta_{1}$ and $\delta_{1}'$ satisfying
%
\begin{gather}
8\delta_{1} <\frac{1}{|\Omega|}\int_{\Omega}vdx <1-8\delta_{1}, \\
\label{tmcl2.4e2}
\frac{1}{|\Omega|}\int_{\Omega}([v-1]_{+}+[-v]_{+})dx<\delta_{1}'.
\end{gather}
%
If $ 16\delta_{1}'<\delta_{1}^{2}$ then
%
\[
|\Omega_{\delta_{1}}^{+}|=
\left|\left\{x\in \Omega,\quad v(x)>1-2\delta_{1}\right\}\right|
<(1-\delta_{1})|\Omega|
\]
%
and
%
\[
|\Omega_{\delta_{1}}^{-}|=
\left|\left\{x\in \Omega,\quad
v(x)<2\delta_{1}\right\}\right|<(1-\delta_{1})|\Omega|.
\]
\end{lemma}


Our task is now to verify the hypothesis of this lemma for the
functions $c_{\sigma\varepsilon}$. To
obtain (\ref{tmcl2.4e2}), we note that items \ref{tpgl3.1si2}
and~\ref{tpgl3.1si7} of Lemma~\ref{tpgl3.1s}, (\ref{limitacoes1}) and
(\ref{limitacoes2}) imply that, for almost every
$t\in [0,T]$,
%
\begin{align*}
&\varepsilon\int_{\Omega}
[F_{\sigma}(c_{\sigma\varepsilon})+F_{\sigma}(1-c_{\sigma\varepsilon})]dx\\
&\leq C_{1} + \left\|f(c_{\sigma\varepsilon})
+\delta\theta_{\sigma\varepsilon}^{4}/4
+h_{1}(c_{\sigma\varepsilon})\theta_{\sigma\varepsilon}^{2}
\right\|_{L^{\infty}(0,T,L^{1}(\Omega))}\leq C.
\end{align*}
%
 From the definition of $F_{\sigma}(s)$ for $s<\sigma$
(see page~\pageref{fsigma}),
we obtain
%
\begin{align*}
&\int_{\{c_{\sigma\varepsilon}< 0\}}
F_{\sigma}(c_{\sigma\varepsilon})dx\\
&\geq |\ln \sigma|\int_{\Omega}[-c_{\sigma\varepsilon}(\cdot,t)]_{+}dx
-\sigma[|\ln \sigma| +2\sigma]|\Omega|
-\sigma \|c_{\sigma\varepsilon}(t)\|_{L^{2}(\Omega)}|\Omega|^{1/2}.
\end{align*}
%
Hence, since $F_{\sigma}(s)\geq -1/e$, we have
%
\begin{align*}
&\int_{\Omega}F_{\sigma}(c_{\sigma\varepsilon})dx\\
&\geq|\ln \sigma|\int_{\Omega}[-c_{\sigma\varepsilon}(\cdot,t)]_{+}dx
-\sigma[|\ln \sigma| +2\sigma +e^{-1}]|\Omega|
-\sigma\|c_{\sigma\varepsilon}(t)\|_{L^{2}(\Omega)}|\Omega|^{1/2}.
\end{align*}
%
In the same way, we have
%
\begin{align*}
\int_{\Omega}F_{\sigma}(1-c_{\sigma\varepsilon})dx
&\geq |\ln \sigma|
\int_{\Omega}[c_{\sigma\varepsilon}(\cdot,t)-1]_{+}dx
-\sigma[|\ln \sigma| +2\sigma -1\\
&\quad +e^{-1}]|\Omega|-\sigma\|c_{\sigma\varepsilon}(t)
\|_{L^{2}(\Omega)}|\Omega|^{1/2}.
\end{align*}
%
Thus, using the above estimates and
Item~\ref{tpgl3.1si1} of Lemma~\ref{tpgl3.1s}, we obtain
%
\[
\int_{\Omega}[c_{\sigma\varepsilon}(\cdot,t)-1]_{+}dx
+\int_{\Omega}[-c_{\sigma\varepsilon}(\cdot,t)]_{+}dx
\leq \frac{C}{\varepsilon|\ln \sigma|}
\]
%
The equation (\ref{tpge3.2s}) says that the mean value of
$c_{\sigma\varepsilon}$ is equal to $\overline{c_{0}}$ which
belongs to $(0,1)$. Thus there exists $\delta_{1}>0$, such that
$8\delta_{1} < \overline{c_{0}}<1-8\delta_{1}$. Using
Lemma~\ref{tmcl2.4}, for
$\sigma$ sufficiently small, we have for
almost every $t\in [0,T]$
%
\begin{equation} \label{tmcemed}
\begin{gathered}
 |\Omega_{\sigma \delta_{1}}^{+}|=\left\{
x\in \Omega,\quad
c_{\sigma\varepsilon}(x,t)>1-2\delta_{1}
\right\}<(1-\delta_{1})|\Omega|,\\
|\Omega_{\sigma \delta_{1}}^{-}|=\left\{
x\in \Omega,\quad
c_{\sigma\varepsilon}(x,t)<2\delta_{1}
\right\}<(1-\delta_{1})|\Omega|.
\end{gathered}
\end{equation}

We are now in position to estimate 
$\partial_{c}\mathcal{F}_{\sigma\varepsilon}
(c_{\sigma\varepsilon},\theta_{\sigma\varepsilon})$.

\begin{lemma} \label{tmcl2.5s}
There exists a constant $C_{4}$ which is independent of $\varepsilon$
and $\sigma$ (sufficiently small) such that
\[
\|\partial_{c}\mathcal{F}_{\sigma\varepsilon}
(c_{\sigma\varepsilon},\theta_{\sigma\varepsilon})\|_{L^{2}(\Omega_{T})}\leq 
C_{4}.
\]
\end{lemma}
%
\begin{proof}
First, let us recall that
%
\[
\partial_{c}\mathcal{F}_{\sigma\varepsilon}=
f'(c_{\sigma\varepsilon})
+h_{1}'(c_{\sigma\varepsilon})\theta_{\sigma\varepsilon}^{2}
+\varepsilon[F_{\sigma}'(c_{\sigma\varepsilon})-F_{\sigma}'(1-c_{\sigma\varepsilon})].
\]
%
In view of Item~\ref{tpgl3.1si2} of
Lemma~\ref{tpgl3.1s}, (\ref{limitacoes1})
and (\ref{limitacoes2}), to
obtain the desired estimate it is enough to obtain a  bound for
the norm of
$\varepsilon[F_{\sigma}'(c_{\sigma\varepsilon})
-F_{\sigma}'(1-c_{\sigma\varepsilon})]$
in $L^{2}(\Omega_{T})$. Arguing as Copetti and
Elliott~\cite{copetti6392}, we obtain this
bound by using the next equality
%
\begin{equation} \label{tmcl2.5sep1}
\begin{split}
&\left\|\varepsilon[F_{\sigma}'(c_{\sigma\varepsilon})
-F_{\sigma}'(1-c_{\sigma\varepsilon})]-
\overline{
\varepsilon[F_{\sigma}'(c_{\sigma\varepsilon})- 
F_{\sigma}'(1-c_{\sigma\varepsilon})]}
\right\|_{L^{2}(\Omega_{T})}^{2}\\
&=
\left\|\varepsilon[F_{\sigma}'(c_{\sigma\varepsilon})
-F_{\sigma}'(1-c_{\sigma\varepsilon})]\right\|_{L^{2}(\Omega_{T})}^{2}-
\iint_{\Omega_{T}}
\Big(\overline{\varepsilon[F_{\sigma}'(c_{\sigma\varepsilon})-
  F_{\sigma}'(1-c_{\sigma\varepsilon})]} \Big)^{2}
\end{split}
\end{equation}
%
and estimating the term at the left hand side and the last
term at the right hand side of the above equation.

Let us note that using Poincar\'e's inequality and
Item~\ref{tpgl3.1si3} of Lemma~\ref{tpgl3.1s}, we obtain
\[
 \|\partial_{c}\mathcal{F}_{\sigma\varepsilon}
(c_{\sigma\varepsilon},\theta_{\sigma\varepsilon})-
\kappa_{c}(c_{\sigma\varepsilon})_{xx}
-\overline{
\partial_{c}\mathcal{F}_{\sigma\varepsilon}
(c_{\sigma\varepsilon},\theta_{\sigma\varepsilon})
-\kappa_{c}(c_{\sigma\varepsilon})_{xx}}\|_{L^{2}(\Omega_{T})}\leq 
C_{1}.
\]
%
Recalling that ${c_{x}}_{|S_{T}}=0$, we have
%
\[
\overline{\partial_{c}\mathcal{F}_{\sigma\varepsilon}
(c_{\sigma\varepsilon},\theta_{\sigma\varepsilon})
-\kappa_{c}(c_{\sigma\varepsilon})_{xx}}=
\overline{f'(c_{\sigma\varepsilon})
+h_{1}'(c_{\sigma\varepsilon})\theta_{\sigma\varepsilon}^{2}
+\varepsilon[F_{\sigma}'(c_{\sigma\varepsilon})
-F_{\sigma}'(1-c_{\sigma\varepsilon})]}.
\]
%
Thus, using the estimates for $(c_{\sigma\varepsilon})_{xx}$ in
Item~\ref{tpgl3.4si2} of Lemma~\ref{tpgl3.4s} and for
$\theta_{\sigma\varepsilon}$ in
Item~\ref{tpgl3.1si2} of Lemma~\ref{tpgl3.1s} together 
(\ref{limitacoes1})
and (\ref{limitacoes2}),
we obtain
%
\begin{equation}\label{tmcl2.5sep2}
\|\varepsilon [F_{\sigma}'(c_{\sigma\varepsilon})-
  F_{\sigma}'(1-c_{\sigma\varepsilon})]-
\overline{\varepsilon
[F_{\sigma}'(c_{\sigma\varepsilon})-
F_{\sigma}'(1-c_{\sigma\varepsilon})]}\|_{L^{2}(\Omega_{T})}
\leq \widetilde{C}_{1}
\end{equation}
%
We now use the monotonicity of
$F_{\sigma}'(s)- F_{\sigma}'(1-s)$ and (\ref{tmcemed}) to obtain for
almost every $t\in [0,T]$:
%
\begin{align*}
& \overline{\varepsilon[F_{\sigma}'(c_{\sigma\varepsilon})-
  F_{\sigma}'(1-c_{\sigma\varepsilon})]}\\
&=\varepsilon|\Omega|^{-1}\int_{\Omega_{\sigma \delta_{1}}^{+}}
[F_{\sigma}'(c_{\sigma\varepsilon})-
  F_{\sigma}'(1-c_{\sigma\varepsilon})]+
\varepsilon|\Omega|^{-1}\int_{[\Omega_{\sigma \delta_{1}}^{+}]^{c}}
[F_{\sigma}'(c_{\sigma\varepsilon})-
F_{\sigma}'(1-c_{\sigma\varepsilon})]\\
&\leq (1-\delta_{1})^{1/2}|\Omega|^{-1/2}
\|\varepsilon[F_{\sigma}'(c_{\sigma\varepsilon})
-F_{\sigma}'(1-c_{\sigma\varepsilon})]\|_{L^{2}(\Omega)}\\
&\quad +\varepsilon[ F_{\sigma}'(1-2\delta_{1})
-F_{\sigma}'(2\delta_{1})].
\end{align*}
%
In the same way, observing that $F_{\sigma}'(2\delta_{1})
-F_{\sigma}'(1-2\delta_{1})<0$, we have
%
\begin{align*}
\overline{\varepsilon[F_{\sigma}'(c_{\sigma\varepsilon})
- F_{\sigma}'(1-c_{\sigma\varepsilon})]}
&\geq -(1-\delta_{1})^{1/2}|\Omega|^{-1/2}
\|\varepsilon[F_{\sigma}'(c_{\sigma\varepsilon})\\
&\quad - F_{\sigma}'(1-c_{\sigma\varepsilon})]\|_{L^{2}(\Omega)}
+\varepsilon[ F_{\sigma}'(2\delta_{1})
-F_{\sigma}'(1-2\delta_{1})].
\end{align*}
%
Therefore, by using that $(a+b)^{2}\leq
a^{2}(1+\frac{1}{\delta_{1}})+
b^{2}(1+\delta_{1})$, we have
%
\begin{equation}
\begin{split}
&\big(\overline{\varepsilon[F_{\sigma}'(c_{\sigma\varepsilon})-
  F_{\sigma}'(1-c_{\sigma\varepsilon})]}\big)^{2}\\
&\leq \varepsilon\big(1+\frac{1}{\delta_{1}}\big)
[F_{\sigma}'(1-2\delta_{1})
- F_{\sigma}'(2\delta_{1})
]^{2}
(1-\delta_{1}^{2})\frac{1}{|\Omega|}
\|\varepsilon[F_{\sigma}'(c_{\sigma\varepsilon})
- F_{\sigma}'(1-c_{\sigma\varepsilon})]\|_{L^{2}(\Omega)}^{2}.
\end{split}
\label{tmce2.10}
\end{equation}
%
Multiplying the above estimate by $|\Omega|$, integrating it in $t$
and using (\ref{tmcl2.5sep1}) and (\ref{tmcl2.5sep2}),
it results that for $\sigma$ sufficiently small,
we have
%
\begin{align*}
&\delta_{1}^{2}
\|\varepsilon[F_{\sigma}'(c_{\sigma\varepsilon})-
F_{\sigma}'(1-c_{\sigma\varepsilon})]\|_{L^{2}(\Omega_{T})}^{2}\\
& \leq
\varepsilon|\Omega_{T}|(1+\frac{1}{\delta_{1}})
[ F_{\sigma}'(2\delta_{1})
- F_{\sigma}'(1-2\delta_{1})]^{2}\\
&\quad
+\left\|\varepsilon[F_{\sigma}'(c_{\sigma\varepsilon})
-F_{\sigma}'(1-c_{\sigma\varepsilon})] -
\overline{\varepsilon[F_{\sigma}'(c_{\sigma\varepsilon})
- F_{\sigma}'(1-c_{\sigma\varepsilon})]}
\right\|_{L^{2}(\Omega_{T})}^{2}\\
& \leq
\varepsilon|\Omega_{T}|(1+\frac{1}{\delta_{1}})
[ F'(2\delta_{1})
- F'(1-2\delta_{1})]^{2}
+\widetilde{C}_{1}\leq \widetilde{C}_{2}.
\end{align*}
%
\end{proof}

We define
\[
w_{\sigma\varepsilon}=D(\partial_{c}\mathcal{F}_{\sigma\varepsilon}
(c_{\sigma\varepsilon},\theta_{\sigma\varepsilon})
-\kappa_{c}[c_{\sigma\varepsilon}]_{xx}).
\]
%
Then the estimates in Lemmas~\ref{tpgl3.1s},
\ref{tpgl3.4s} and \ref{tmcl2.5s} and Lemma~\ref{tpgl3.3s}
imply that
$w_{\sigma\varepsilon}$ converge weakly to
$w_{\varepsilon}$ in $L^{2}(0,T,H^{1}(\Omega))$,
where
\[
w_{\varepsilon}=D(\partial_{c}\mathcal{F}_{\varepsilon}
(c_{\varepsilon},\theta_{\varepsilon})-\kappa_{c}[c_{\varepsilon}]_{xx}),
\]
and where $\mathcal{F}_{\varepsilon}$ is defined as in the next
Proposition. To identify the limit of the nonlinear term,  
we use Lemmas~\ref{tpgl3.1s} and ~\ref{tpgl3.3s} to see that
$\partial_{c}\mathcal{F}_{\varepsilon}(c_{\varepsilon},\theta_{\varepsilon})$ 
is the pointwise limit of a subsequence of
$\partial_{c}\mathcal{F}_{\sigma\varepsilon}(c_{\sigma\varepsilon},\theta_{\sigma\varepsilon})$. 
This fact with Lemma~ \ref{tmcl2.5s} and Fatou's Lemma imply
$\partial_{c}\mathcal{F}_{\varepsilon}
(c_{\varepsilon},\theta_{\varepsilon})\in L^2(\Omega_T)$. Finally,
we use Lion's Lemma (\cite{lions69}, p.~12) to identify the weak limit 
of
$\partial_{c}\mathcal{F}_{\sigma\varepsilon}(c_{\sigma\varepsilon},\theta_{\sigma\varepsilon})$ 
as
$\partial_{c}\mathcal{F}_{\sigma\varepsilon}(c_{\sigma\varepsilon},\theta_{\sigma\varepsilon})$.
Therefore, arguing as in Proposition~\ref{tpgp3.5M},
we can pass to the limit as $\sigma$ goes to zero to
obtain the following result.
%
\begin{proposition} \label{tpgp3.5s}
There exists a triplet
$(c_{\varepsilon},w_{\varepsilon},\theta_{\varepsilon})$ such that:
%
\begin{enumerate}
\item $ c_{\varepsilon}, \theta_{\varepsilon}\in
L^{\infty}(0,T,H^{1}(\Omega))$,

\item $\partial_{t} c_{\varepsilon}\in L^{2}(0,T,[H^{1}(\Omega)]')
\quad \mbox{and} \quad \partial_{t}\theta_{\varepsilon}\in
L^{2}(\Omega_{T})$,

\item $[c_{\varepsilon}]_{xx}, [\theta_{\varepsilon}]_{xx}\in
L^{2}(\Omega_{T})$,

\item $|\Omega_{T}\setminus \mathcal{B}(c_{\varepsilon})|=0$,

\item
  $\partial_{c}\mathcal{F}_{\varepsilon}
(c_{\varepsilon},\theta_{\varepsilon}),
\partial_{\theta}\mathcal{F}_{\varepsilon}
(c_{\varepsilon},\theta_{\varepsilon})\in L^{2}(\Omega_{T})$,

\item $w_{\varepsilon}\in L^{2}(0,T,H^{1}(\Omega))$

\item $c_{\varepsilon}(0)=c_{0}(x),\quad \theta_{\varepsilon
M}(0)=\theta_{0}(x)$

\item ${[c_{\varepsilon}]_{x}}_{|S_{T}}={[\theta_{\varepsilon
M}]_{x}}_{|S_{T}}=0$ in $L^{2}(S_{T})$


\item $(c_{\varepsilon},w_{\varepsilon},\theta_{\varepsilon})$
satisfies
\begin{gather}
\int_{0}^{T}\langle \partial_{t}c_{\varepsilon},\phi\rangle dt=
-\iint_{\Omega_{T}}[w_{\varepsilon}]_{x}\phi_{x},\quad \forall \phi \in
L^{2}(0,T,H^{1}(\Omega))\label{tpge3.11p}
\\
w_{\varepsilon}=D[
\partial_{c}\mathcal{F}_{\varepsilon}
(c_{\varepsilon},\theta_{\varepsilon})
-\kappa_{c} (c_{\varepsilon})_{xx}]\label{tpge3.13p}
\\
\iint_{\Omega_{T}}\partial_{t}\theta_{\varepsilon}\psi=
-\iint_{\Omega_{T}}
L(\partial_{\theta}\mathcal{F}_{\varepsilon}
(c_{\varepsilon},\theta_{\varepsilon })-
\kappa (\theta_{\varepsilon })_{xx})\psi,\quad \forall \psi \in
L^{2}(\Omega_{T})\label{tpge3.12p}
\end{gather}
%
where, since $c_{\varepsilon}\in (0,1)$ a.e in $\Omega_{T}$,
\begin{align*}
\mathcal{F}_{\varepsilon}(c,\theta)
&=-\frac{A}{2}(c-c_{m})^{2}+\frac{B}{4}(c-c_{m})^{4}
+\frac{D_{\alpha}}{4}(c-c_{\alpha})^{4}\\
&\quad + \frac{D_{\beta}}{4}(c-c_{\beta})^{4}
+\frac{\delta}{4}\theta^{4}-\frac{\gamma}{2}c^{2}
\theta^{2}+\varepsilon[F(c)+ F(1-c)]
\end{align*}
%
with $F(s)=s\ln s$.
\end{enumerate}
\end{proposition}

\section{Limit as $\varepsilon\rightarrow 0^{+}$}\label{lime}

In this section we finally prove Theorem~\ref{tpt1} when the spatial 
dimension is one
and the number of crystallographic orientations is  $p=1$. We recall 
that we treat
this simpler case just for simplicity of notation and  exposition, 
since, as we will
show in  Section~\ref{dimmaior}, the necessary changes to extend the 
results of
Theorem~\ref{tpt1}  to higher  spatial dimensions and $p>1$ are simple 
ones.

We start by observing that, as before, we have the mean value of 
$c_{\varepsilon}$ in
$\Omega$ given by
%
\begin{equation}
\label{tpge3.2e}
\overline{c_{\varepsilon}}=\overline{c_{0}}\in (0,1),
\end{equation}
%
Since the estimates obtained for $\sigma$ in Lemmas~\ref{tpgl3.1s},
\ref{tpgl3.4s} and \ref{tmcl2.5s} do not depend on
$\varepsilon$, we have

\begin{lemma} \label{tpgl3.1e}
There exists a constant $C_{1}$ independent of $\varepsilon$ such that
\begin{enumerate}
\item \label{tpgl3.1ei1}$\|c_{\varepsilon
}\|_{L^{\infty}(0,T,H^{1}(\Omega))}\leq C_{1}$

\item \label{tpgl3.1ei2}$\|\theta_{\varepsilon
}\|_{L^{\infty}(0,T,H^{1}(\Omega))}
\leq C_{1}$

\item \label{tpgl3.1ei3}$\|(w_{\varepsilon
})_{x}\|_{L^{2}(\Omega_{T})}\leq C_{1}$

\item
  \label{tpgl3.1ei4}$\|\partial_{\theta}\mathcal{F}_{\varepsilon }
-\kappa (\theta_{\varepsilon})_{xx}\|_{L^{2}(\Omega_{T})}\leq C_{1}$

\item \label{tpgl3.1ei5}$\|\partial_{t}c_{\varepsilon }
\|_{L^{\infty}(0,T,[H^{1}(\Omega)]')}\leq C_{1}$

\item \label{tpgl3.1ei6}
$\|\partial_{t}\theta_{\varepsilon}\|_{L^{2}(\Omega_{T})}
\leq C_{1}$

\item $\|\partial_{c}\mathcal{F}_{\varepsilon}
(c_{\varepsilon},\theta_{\varepsilon})\|_{L^{2}(\Omega_{T})}\leq
C_{1}$,

\item $\|\partial_{\theta}\mathcal{F}_{\varepsilon}
(c_{\varepsilon},\theta_{\varepsilon})\|_{L^{2}(\Omega_{T})}\leq 
C_{1}$,

\item $\|[c_{\varepsilon}]_{xx}\|_{L^{2}(\Omega_{T})}\leq
C_{1}$,

\item $\|[\theta_{\varepsilon}]_{xx}\|_{L^{2}(\Omega_{T})}\leq C_{1}$
\end{enumerate}
\end{lemma}

Now, we complete the proof of Theorem~\ref{tpt1}.

\begin{proof}[Proof of the case $\mathbf{d=1}$ and $\mathbf{p=1}$]
We recall
\[
\begin{split}
\mathcal{F}(c,\theta)
&=-\frac{A}{2}(c-c_{m})^{2}+\frac{B}{4}(c-c_{m})^{4}
+\frac{D_{\alpha}}{4}(c-c_{\alpha})^{4} \\
&\quad + \frac{D_{\beta}}{4}(c-c_{\beta})^{4}
-\frac{\gamma}{2}(c-c_{\alpha})^{2}\theta^{2} 
+\frac{\delta}{4}\theta^{4}
\end{split}
\]
%
Then, we argue as Elliott and
Luckhaus~\cite{elliott88791}, p.~35. For this, let
%
\[
\phi^{\rho}\in K^{+}=\{\phi \in H^{1}(\Omega),\;\; 0<\phi<1\}
\quad \mbox{and} \quad
\rho <\phi^{\rho}<1-\rho
\]
%
for some small positive $\rho$. We
have $[F'(\phi^{\rho})-F'(1-\phi^{\rho})]\in
L^{2}(\Omega)$ because $\rho <\phi^{\rho}<1-\rho$. Hence it follows 
from
(\ref{tpge3.13p}) that
%
\begin{align*}
&\int_{0}^{T}\xi(t)(w_{\varepsilon},\phi^{\rho}-c_{\varepsilon})dt\\
&= D\int_{0}^{T}\xi(t)(\partial_{c}\mathcal{F}
(c_{\varepsilon},\theta_{\varepsilon}) +\varepsilon
[F'(c_{\varepsilon})- F'(1-c_{\varepsilon})]
-\kappa_{c} (c_{\varepsilon})_{xx},\phi^{\rho}-c_{\varepsilon})dt.
\end{align*}
%
Integrating by parts and rewriting, we obtain
%
\begin{align*}
&\int_{0}^{T} \xi(t)
\left\{\kappa_{c} D(\nabla c_{\varepsilon},\nabla \phi^{\rho})
-(w_{\varepsilon}-D\partial_{c}\mathcal{F}
(c_{\varepsilon},\theta_{\varepsilon}),
\phi^{\rho}-c_{\varepsilon}) \right\}dt\\
&=\int_{0}^{T}\xi(t)
\kappa_{c} D(\nabla c_{\varepsilon},\nabla c_{\varepsilon})dt\\
&\quad + \varepsilon D\int_{0}^{T}\xi(t)([F'(\phi^{\rho})- 
F'(1-\phi^{\rho})]-
[F'(c_{\varepsilon})- 
F'(1-c_{\varepsilon})],\phi^{\rho}-c_{\varepsilon})dt\\
&\quad - \varepsilon D\int_{0}^{T}\xi(t)([F'(\phi^{\rho})- 
F'(1-\phi^{\rho})]
,\phi^{\rho}-c_{\varepsilon})dt.
\end{align*}
%
By using the monotonicity of
$F'(\cdot)-F'(1-\cdot)$ and the convergence properties
of $(c_{\varepsilon},w_{\varepsilon},\theta_{\varepsilon})$ we may
pass to the limit and obtain for $\xi\in C[0,T],\, \xi \geq 0$, that
%
\begin{align*}
&\int_{0}^{T}\xi(t)\left\{\kappa_{c} D(\nabla c,\nabla \phi^{\rho})
-(w-D\partial_{c}\mathcal{F}(c,\theta),
\phi^{\rho}-c)\right\}dt\\
&\geq \limsup_{\varepsilon\rightarrow 0^+}\int_{0}^{T}\xi(t)
\left\{\kappa_{c}D (\nabla c_{\varepsilon},\nabla \phi^{\rho})
-(w-D\partial_{c}\mathcal{F}(c_{\varepsilon},\theta_{\varepsilon}),
\phi^{\rho}-c_{\varepsilon}) \right\}dt\\
&\geq \liminf_{\varepsilon\rightarrow 0^+}\int_{0}^{T}\xi(t)
\kappa_{c} D(\nabla c_{\varepsilon},\nabla c_{\varepsilon})dt\\
&\quad -\lim_{\varepsilon\rightarrow 0^+}
\varepsilon D\int_{0}^{T}\xi(t)([F'(\phi^{\rho})- F'(1-\phi^{\rho})]
,\phi^{\rho}-c_{\varepsilon})dt\\
&\geq \int_{0}^{T}\xi(t) \kappa_{c} D(\nabla c,\nabla c)dt.
\end{align*}
%
Furthermore, since any $\phi\in K=\{\phi \in H^{1}(\Omega),\;
0\leq\phi\leq 1,\,\overline{\phi}=\overline{c_{0}}\}$
can be approximated by
$\phi^{\rho}\in K^{+}$, for small $\rho$ with $\rho
<\phi^{\rho}<1-\rho$, we may pass to the limit as
$\rho$ goes to zero in the left hand side of the above
inequality and obtain
%
\begin{equation}
\int_{0}^{T}\xi(t)\left\{\kappa_{c} (\nabla c,\nabla \phi-\nabla c)
-(w-\partial_{c}\mathcal{F}(c,\theta),
\phi-c)\right\}dt\geq 0
\label{tpge3.12pe}
\end{equation}
%
for $\xi\in C[0,T),\, \xi \geq 0$, and $
\phi \in K $.
Arguing as in the previous sections we also obtain
%
\begin{gather}
\int_{0}^{T}\langle \partial_{t} c ,\phi\rangle dt=
-\iint_{\Omega_{T}}w_{x}\phi_{x},\quad \forall \phi \in
L^{2}(0,T,H^{1}(\Omega)), \label{tpge3.11pe}
\\
\iint_{\Omega_{T}}\partial_{t} \theta \psi=
-\iint_{\Omega_{T}}
L(\partial_{\theta}\mathcal{F}(c,\theta)-
\kappa \theta_{xx})\psi,\quad \forall \psi \in
L^{2}(\Omega_{T}). \label{tpge3.13pe}
\end{gather}
%
Thus, for spatial dimension one and $p=1$, Theorem~\ref{tpt1} is a
direct consequence of Lemma~\ref{tpgl3.1e}, (\ref{tpge3.12pe}),
(\ref{tpge3.11pe}) and (\ref{tpge3.13pe}).
\end{proof}

\section{Proof of the Case of Higher Spatial Dimensions and $p>1$}
\label{dimmaior}

In the case of higher spatial dimensions and  $p>1$, we have to
slightly change the previously presented arguments.
Firstly, we show the changes to be done when the spatial dimension 
satisfies
$2\leq d\leq 3$. We start by remarking that, as observed by Passo et 
al.~\cite{passo1299}, Proposition~\ref{tpgt2.1} is valid for any
dimension. Also, in higher dimensions,
we use an argument of elliptic regularity of the Laplacian to
obtain estimates in $L^{2}(0,T,H^{2}(\Omega))$ and in 
$L^{2}(0,T,H^{3}(\Omega))$.
Furthermore, all of our previous arguments hold for dimensions $2\leq 
d\leq 3$,
except the result of Corollary~\ref{tpgc3.2s}, where the fact that
dimension was one was essential. This result was only used in the proof
of Lemma~\ref{tpgl3.3s} to extract an uniformly convergent subsequence
and conclude that the measure of the set
$\Omega_{T}\setminus \mathcal{B}(c_{\varepsilon})$ is zero
(where $\mathcal{B}(c)=\{(x,t)\in \mathrm{cl}\,\Omega_{T},\; c(x,t)\in 
I\}$). 
Once  Lemma~\ref{tpgl3.3s} is
stated for $2\leq d\leq 3$, all results but Corollary~\ref{tpgc3.2s} 
are also valid for $2\leq d\leq 3$.
 On the next Lemma, we show how to state the same result of
Lemma~\ref{tpgl3.3s} when the dimension $d$ satisfies $2\leq d\leq 3$. 
As mentioned before, all the results stated before Corollary~5.1
does not depend
upon dimension one and can be extended for dimensions $2\leq d\leq 3$. 
We will use this fact on the proof of
the next Lemma.

\begin{lemma} \label{tpgl3.3sd}
$|\Omega_{T}\setminus \mathcal{B}(c_{\varepsilon})|=0$
with $\mathcal{B}(c)=\{(x,t)\in \mathrm{cl}\,\Omega_{T},\; c(x,t)\in 
I\}$.
\end{lemma}

\begin{proof}  Arguing exactly as in the proof of Lemma~\ref{tpgl3.3s}, 
we obtain
\begin{equation} \label{tpge3.8sd}
-\varepsilon\iint_{\Omega_{T}}
[F_{\sigma}'(c_{\sigma\varepsilon})+
F_{\sigma}'(1-c_{\sigma\varepsilon})]\leq C_{4}
\end{equation}
%
To complete the proof of the lemma, suppose by contradiction that the 
set
$\Omega_{T}\setminus \mathcal{B}(c_{\varepsilon})$ has a positive
measure. Suppose by instance that
%
\[
A=\{(x,t)\in \Omega_{T},\; c_{\varepsilon}\leq 0\}
\]
%
has positive measure.
Using Item~\ref{tpgl3.1si1} of Lemma~\ref{tpgl3.1s}, we can extract a 
subsequence of
$(c_{\sigma\varepsilon})$ which converges almost everywhere to 
$c_{\varepsilon}$ on $\Omega_T$.
Now, by using Egoroff's Theorem, we may conclude that  such subsequence 
also converges almost
uniformly on $\Omega_T$. Thus, there exists a set $B\subset \Omega_T$ 
such that  $|B|\leq \dfrac{1}{2}|A|$
and  $c_{\sigma\varepsilon}$ converges uniformly to $c_{\varepsilon}$ 
on $\Omega_T\setminus B$.
Let $C=A\cap(\Omega_T\setminus B)$. We can see that $|C|>0$.
Since $F_{\sigma}'\leq 1$, the estimate  (\ref{tpge3.8s}) gives
%
\[
-\varepsilon\iint_{C}F_{\sigma}'(c_{\sigma\varepsilon})
\leq C_{4}.
\]
%
Note, however, that the uniform convergence of
$c_{\sigma\varepsilon}$ implies that
%
\[
\forall \lambda >0, \; \exists \sigma_{\lambda},\quad
c_{\sigma\varepsilon}\leq \lambda,\quad \forall
(x,t)\in C,\;\;\sigma <\sigma_{\lambda}
\]
%
therefore, due to the convexity of $F_{\sigma}$, we have
$F_{\sigma}'(c_{\sigma\varepsilon})\leq F_{\sigma}'(\lambda)$. Hence
%
\[
-\varepsilon|C|(\ln \lambda + 1)=
-\varepsilon\lim_{\sigma\rightarrow 0^+}\iint_{C}F_{\sigma}'(\lambda)
\leq C_{4}
\]
%
which leads to a contradiction for $\lambda\in (0,1)$ sufficiently
small. The same argument shows that $B=\{(x,t)\in \Omega_{T},\;
c_{\varepsilon}\geq 1\}$ has zero measure.
\end{proof}

We remark that In this last proof we have just repeated the 
contradiction argument presented
in the proof of Lemma~\ref{tpgl3.3s} with the only difference that now 
we have
supposed by contradiction that there exists a subset of 
$\Omega_{T}\setminus
\mathcal{B}(c_{\varepsilon})$ that has positive measure and where the
convergence is uniform.

Now we explain the necessary modifications to be done  when the number 
of
crystallographic orientations is larger than one. In this case, the
local free energy density is
%
\begin{align*}
\mathcal{F}(c,\theta_{1},\dots,\theta_{p}) 
=&-\frac{A}{2}(c-c_{m})^{2}+\frac{B}{4}(c-c_{m})^{4}
+\frac{D_{\alpha}}{4}(c-c_{\alpha})^{4}+
\frac{D_{\beta}}{4}(c-c_{\beta})^{4}\\
&+\sum_{i=1}^{p}[-\frac{\gamma}{2}(c-c_{\alpha})^{2}\theta_{i}^{2}
+\frac{\delta}{4} \theta_{i}^{4}]+\sum_{i\not=j=1}^{p}
\frac{\varepsilon_{ij}}{2}\theta_{i}^{2}\theta_{j}^{2}.
\end{align*}
%
The introduction of the mixed terms depending only on
the $\theta_{i}$'s (the last terms) will
not change greatly the arguments presented in the case
when $p$ was equal to one.
In fact, in the following we will point out how our
previous estimates can be extended for this case.

The main feature of the perturbed systems in
Section~\ref{sistap} is that their corresponding local
free energy density have lower bounds that do not depend on the
truncation parameter $M$. Since the extended local free energy
just introduces nonnegative terms,
we can define a similar truncation that maintains the same
property, with such perturbed systems it is then possible to
similarly establish Lemma~\ref{tpgl3.1M}.

As for Lemma~\ref{tpgl3.4M}, we treat the new terms by using the
immersion of $H^{1}(\Omega)$ in $L^{4}(\Omega)$ and the estimates for
the orientation field variables given in Lemma~\ref{tpgl3.1M}.

After we have extended the results of Lemmas~\ref{tpgl3.1M}
and~\ref{tpgl3.4M}, all the other lemmas are their direct consequence
without any significant change due to the introduction of the new 
terms.


\section{Final Remarks}

\begin{remark} \rm
Observe that (\ref{tpge1.7}) implies that for almost
all $t \in (0,T]$ there holds
\begin{align*}
&\kappa_{c}D (\nabla c(\cdot, t) ,\nabla \phi-\nabla c(\cdot, t))\\
&-(w-D\partial_{c}\mathcal{F}(c (\cdot,t),\theta_{1}(\cdot, t),\dots,
\theta_{p}(\cdot,t)), \phi-c(\cdot, t))\geq 0,
\end{align*}
for all $\phi\in K=\{\eta \in H^{1}(\Omega),\; 0\leq \eta\leq 1,\,
\overline{\eta}=\overline{c_{0}}\}$.

Moreover, since $c \in L^2 (0,T, H^2 (\Omega))$ and $ 1 \leq d \leq 3$,
by standard Sobolev imbeddings, we can assume that
$c(x,t)$ is a continuous
function of $x$ for the same times $t$ as above.
Fix such a $t$ and assume that in a neighborhood $\mathcal V$ of a 
point
$\overline{x} \in \Omega$ we have that $ 0 < c(x,t) < 1$ for $x$ in the
closure of $\mathcal V$.
Thus, for a given $C^\infty$--function $\varphi$ of compact support in
$\mathcal V$ satisfying $\overline{\varphi} = 0$,
$c(\cdot, t) + \lambda \varphi (\cdot) \in K$ for small
enough real $\lambda$.
By taking $\phi = c + \lambda \varphi$ back in the last inequality, and
observing that $\lambda$ assumes positive and negative
values, we conclude that
for any  $C^\infty$--function $\varphi$ of compact support in $\mathcal 
V$
such that $\overline{\varphi} = 0$,
\[
\kappa_{c}D (\nabla c(\cdot, t) ,\nabla \varphi)
-(w-D\partial_{c}\mathcal{F}(c (\cdot,
t),\theta_{1}(\cdot, t),\dots,
\theta_{p}(\cdot,t)), \varphi) =  0.
\]
We conclude that in regions such $ 0 < c(x,t) < 1$, for almost all 
times
$t \in (0,T]$, $w=D \nabla ( \partial_{c}
  \mathcal{F}-\kappa_{c} \Delta c)$,
up to a function of time. Substituting back in the first equation of
(\ref{tpgep}), we obtain that
$\partial_{t} c=\nabla \cdot[
D\nabla (
\partial_{c} \mathcal{F}-\kappa_{c} \Delta c
)]$ in such regions.
In this sense, the obtained solution is a generalized solution of the
original problem.
\end{remark}

\begin{remark} \rm
When $\gamma=\delta=\varepsilon_{ij}=0$,
we obtain $c$ that solves a generalized formulation of
the Cahn--Hilliard equation, and for which the estimate
$0\leq c\leq 1$ is still valid. But, we are able to
guarantee
that the classical Cahn-Hilliard equation is satisfied
only in regions where
$ 0 < c < 1$. Thus, unfortunately we are not able to reach this kind of
estimate for the classical Cahn-Hilliard equation with polynomial
free-energy.

The physical model we are considering in this paper assumes that the
coefficients appearing in the free-energy functions are positive.
In mathematical  terms, we could have a more general situation.
For instance, $\varepsilon_{ij}\geq 0$ is enough to guarantee that,
in the case when $p>1$ and the cross terms
$\theta_i^2\theta_j^2$ are present, the free energy
functional $\mathcal{F}_{\sigma\varepsilon M}$ is still
bounded below by a constant that does not depend on
$M$, as indicated in formula (\ref{tpge3.1}),
and the results in the paper are true.
The coefficient $\gamma$ could have any sign. When $\gamma \leq 0$, the
problem in fact is simpler because in this case the corresponding term 
in
the free-energy functional has the ``right sign''
In the case that $\gamma>0$, to get the first
inequality in (\ref{tpge3.1}), the coefficient $\delta$
must be positive.
\end{remark}

\begin{remark} \rm
In our first attempts to study the problem with the
free-energy functional presented in
the paper, we tried to use Galerkin method. However, we
were not able to get the necessary
estimates (basically due to presence of the term
$c^2\theta_i^2$) except in the special
case in which the coefficients are such that the
original free-energy functional is bounded
from below. In this special case it is possible to
solve the problem using Galerkin method.
Even in such special case, however, we cannot identify
such solution with the generalized solution presented
in this paper. This is due to the following facts: our
uniqueness result
is based upon the fact that the $c$ presented in the
paper is an element of the set
$K=\{\eta \in H^{1}(\Omega),\; 0\leq \eta\leq 1,\,
\overline{\eta}=\overline{c_{0}}\}$;
however, we are not able to get a $L^\infty$--bound for
the solution obtained by the
Galerkin method in the special case, and thus we do not
know whether such solution belongs
to $K$. Therefore, even in the special case
we are not able to compare the
         generalized solution and the solution obtained
         by the Galerkin method.
\end{remark}

\begin{remark} \rm
The physical meaning of $c$ (concentration of one of
the two materials in the mixture)
requires that $0 \leq c \leq 1$. On the other hand,
from physical arguments, it is
expected that asymptotically in time the concentration
approach certain values of
the minimizers of the function $f$ appearing in
$\mathcal F$ (maybe different ones in
different regions of $\Omega$.) In terms of
mathematical possibilities, it may occur
that in certain situations two of these minimizer
(exactly two in the special case of
two-wells potentials) correspond to pure materials
($c=0$ or $c=1$.) However, in
physical terms, these minimizers may be associated in fact
to values of $c$ corresponding
to mixtures of materials (say, $c_1$ and $c_2$, $0
<c_1<c_2 < 1$,) although, maybe,
with clear predominance of one of them (even in cases
with two-wells potentials.)
In these cases, it is perfectly possible to have a
initial condition for the
concentration, $c_0$ such that the values of $c_0(x)$
are not confined to the interval
$[c_1, c_2]$. As before, one expects that
asymptotically in time the
concentration evolves in such way to approach either
$c_1$ or $c_2$,
depending on the region of $\Omega$. But, if it does
so, it cannot not satisfy
$c_1 \leq c(x,t) \leq c_2$ for all time $t$
and $x$, although on physical
grounds it should satisfy $0 \leq c(x,t) \leq 1$.
Thus, the location of the minimizers of the potential
is in fact independent of the
required physical range of $c$.
Here, we considered the problem at this level of
possibilities, and our generalized
solutions satisfy this physical requirement.

A more precise and difficult question related to this
situation is the following:
suppose $c_0$ satisfies $c_1 \leq c_0 (x) \leq c_2$ for all $x$,
does $c$ satisfy $c_1 \leq c(x,t) \leq c_2$ for all $x$ and $t$?
For certain parabolic (scalar) equations, one expect
this to be true. However,
for general systems, where interactions between the
unknown variables play a
significant role, this may be not so. Even in homogeneous situations, 
when
spatial variables play no significant role, the
trajectories could be spirals
approaching the equilibrium points of the corresponding
ordinary differential
system. We do not know whether this is the case of the
system considered in this
article. Certainly this is an interesting point to investigate.

Another interesting point to consider is the
following. We placed the singularities
of the logarithmic perturbation at $0$ and $1$ to comply with the
required physical restriction $0 \leq c \leq 1$ on the concentration.
However, suppose that we have an initial condition for the 
concentration
satisfying $ 1/3 \leq c_0 \leq 2/3$; if we repeat the
argument of the paper,
but with the logarithmic singularities placed at $1/3$
and $2/3$, we obtain
a solution satisfying $1/3 \leq c \leq 2/3$. The same sort of reasoning
would imply that it is always possible to construct a solution  
satisfying
$\min c_0 \leq c \leq \max c_0$. Moreover, if we consider that we have
uniqueness, this would be the solution, taking us to the conclusion
that every solutions satisfy $\min c_0 \leq c \leq \max c_0$. But this 
is
strange because, as we said above, on physical grounds one expects the
values of $c$ approach either $c_1$ or $c_2$. This raises the
question whether there is something wrong.

The key to ease this discomfort is the observation that, when we change
where we place the logarithmic singularities, in fact we are changing
the problem to be solved. For instance, in the previous example of
changing the placement of the singularities, the differential 
inequality
(5) would require that $c$ and the test functions $\phi$ belong to
modified $\widetilde {K}=\{\eta \in H^{1}(\Omega),\;
1/3 \leq \eta \leq 2/3,\, \overline{\eta}=\overline{c_{0}}\}$.
 Moreover, the solutions of these two
different problems are not comparable, since it is clear that the
uniqueness stated in Lemma \ref{unicidade} holds for exactly the same 
problem.
Thus, we cannot reason as in the previous paragraph and
its ``conclusions'' do not hold.

A final point must be considered. Since different placements of the
logarithmic singularities introduce different generalized problems, and
therefore different solutions, which is the ``right'' one to be picked?
We argue that the one in this paper is the ``right'' one based on
two reasons.
First, the only reasonable physical restriction to the concentration is
$0 \leq c \leq 1$, which requires the placement of the singularities
as we chose, and not in different places. 
Second, as we explained
in a previous remark, our generalized solutions
satisfy the classical Cahn-Hilliard equation in regions
described basically by $0 < c(x,t) < 1$, leaving out, maybe, only the
regions of pure materials, where the physical mechanisms for
the mixture no longer apply.
The same sort of reasoning, when applied for a problem obtained
with perturbation with singularities placed for instance at $1/3$ and
$2/3$ would guarantee the Cahn-Hilliard equation only in regions such
that $1/3 < c(x,t) < 2/3 $, leaving out, maybe, regions
where in fact the mechanisms for mixtures still apply, which is not
physically reasonable.
\end{remark}

\subsection*{Acknowledgment}
The authors would like to express their gratitude to the referee for 
his/her helpful
suggestions that surely improved this article.

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