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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 04, pp. 1--15.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/04\hfil Anisotropic Cahn-Hilliard and Allen-Cahn systems]
{Well-posedness for one-dimensional anisotropic Cahn-Hilliard and
 Allen-Cahn systems}

\author[A. Makki, A. Miranville \hfil EJDE-2015/04\hfilneg]
{Ahmad Makki, Alain Miranville}  % in alphabetical order

\address{Ahmad Makki \newline
Universit\'e de Poitiers,
Laboratoire de Math\'ematiques et Applications,
UMR CNRS 7348 - SP2MI, Boulevard Marie et Pierre Curie
- T\'el\'eport 2,  F-86962 Chasseneuil Futuroscope Cedex, France}
\email{ahmad.makki@math.univ-poitiers.fr}

\address{Alain Miranville \newline
 Universit\'e de Poitiers,
 Laboratoire de Math\'ematiques et Applications,
 UMR CNRS 7348 - SP2MI, Boulevard Marie et Pierre Curie
- T\'el\'eport 2, F-86962 Chasseneuil Futuroscope Cedex, France}
\email{alain.miranville@math.univ-poitiers.fr}

\thanks{Submitted December 20, 2014. Published January 5, 2015.}
\subjclass[2000]{35B45, 35K55}
\keywords{Cahn-Hilliard equation; Allen-Cahn equation; well-posedness;
\hfill\break\indent  Willmore regularization}

\begin{abstract}
 Our aim  is to prove the existence and uniqueness of solutions for
 one-dimensional Cahn-Hilliard and Allen-Cahn type equations based on
 a modification of the Ginzburg-Landau free energy proposed in \cite{k1}.
 In particular, the free energy contains an additional term called
 Willmore regularization and takes into account strong anisotropy effects.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}

The original Ginzburg-Landau free energy
\begin{equation}\label{e1}
\Psi_{GL}=\int_{\Omega} \big(\frac12 |\nabla u|^2+F(u)\big)\,dx
\end{equation}
plays a fundamental role in phase separation and transition, 
see,  \cite{c2,g1}. Here, $u$ is the order parameter, $\Omega$ 
is the domain occupied by the material (we assume that it is a bounded 
and regular domain of $\mathbb{R}^N$),
\begin{gather}\label{e2}
F(s)=\frac14 (s^2-1)^2, \\
\label{e3}
f(s)=s^3-s.
\end{gather}

In \cite{c5} (also in \cite{t2}), the authors proposed the following modification
of the Ginzburg-Landau free energy which takes into account strong 
anisotropy effects arising during the growth and coarsening of thin films, namely,
\begin{equation}\label{e4}
\Psi_{MGL}=\int_{\Omega} \Big(\gamma(n)(\frac12 |\nabla u|^2+F(u))
+\frac{\beta}{2} \omega^2\Big) \,dx,
\end{equation}
where
\begin{equation}\label{e5}
n=\frac{\nabla u}{|\nabla u|},\quad \omega=f(u)-\Delta u,\quad F'=f.
\end{equation}
Here, $\gamma(n)$ accounts for anisotropy effects (we also refer the reader to, 
e.g., \cite{c4} for a different approach to account for anisotropy effects 
in phase-field models) and $G(u)=\omega^2$ is called nonlinear Willmore 
regularization. Such a regularization is relevant, e.g., 
in determining the equilibrium shape of a crystal in its own liquid matrix, 
when anisotropy effects are strong. Indeed, in that case, the equilibrium 
interface may not be a smooth curve, but may present facets and 
corners with slopes of discontinuities (see, e.g., \cite{t1}). 
In particular, the corresponding Cahn-Hilliard equation
$$
\frac{\partial u}{\partial t}=\Delta \frac{D \Psi_{MGL}}{D u}
$$
(where $\frac{D}{D u}$ denotes a variational derivative) is an ill-posed 
problem and requires regularization. The author in \cite{m1} proved 
the well-posedness for a one-dimensional Allen-Cahn system based on \eqref{e4}.

In \cite{k1}, the author introduced another modification of the Ginzburg-Landau 
free energy, namely,
 \begin{equation}\label{e6}
\Psi_{AMGL}=\int_{\Omega}\big[\frac{1}{2}|\gamma(n)\nabla u|^2+F(u)
+\frac{1}{2}\omega^2\big]\,dx.
\end{equation}
This model describes dendritic pattern formations and plays an important 
role in crystal growth.

To the best of our knowledge, there is no mathematical result concerning 
the Cahn-Hilliard (resp. Allen-Cahn) model associated with the free 
energy \eqref{e6}.

In this article, we consider the one dimensional case, i.e., 
taking $\Omega=(-L,L)$, \eqref{e6} reads
\begin{equation}\label{a1}
\Psi=\int_{\Omega}\big[\frac{1}{2}|\gamma(n)u_x|^2+F(u)+\frac{1}{2}\omega^2\big]\,dx,
\end{equation}
where
\begin{equation}\label{a2}
n=\frac{u_x}{|u_x|},\quad \omega=f(u)-u_{xx}, \quad  F'=f.
\end{equation}

In \cite{c5,w1}, the authors proposed efficient energy stable schemes 
for the Cahn-Hilliard equation based on \eqref{e4} and \eqref{e6}; 
actually, in \cite{c5}, the authors considered a slightly different problem 
and also considered a second regularization, based on the bi-Laplacian, and, 
in that case, studied the isotropic case $\gamma(n)=1$ as well. 
We also mention that, in \cite{m2} (resp. \cite{m3}), the Cahn-Hilliard 
(resp. Allen-Cahn) equation based on the Willmore regularization is 
studied in the isotropic case. There, well-posedness results are obtained.

Our aim in this article is to prove the existence and uniqueness of 
solutions for the Cahn-Hilliard and Allen-Cahn systems associated 
with the Ginzburg-Landau free energy \eqref{a1}.

\subsection*{Assumptions and notation}
As far as the nonlinear term $f$ is concerned, we assume more generally
 that $f$ is of class $C^4$ and that
\begin{gather}\label{a3}
f(0)=0,\quad f'(s)\geq -c_0,\quad c_0\geq 0,\quad s\in \mathbb{R}, \\
\label{a4}
f(s)s\geq c_1 F(s)-c_2\geq -c'_2,\quad c_1>0,\quad c_2, c'_2\geq 0,\quad 
s\in \mathbb{R}, 
\end{gather}
where $ F(s)=\int_0^s f(\tau)\,d\tau$,
\begin{gather}\label{a5}
sf(s)f'(s)-f(s)^2\geq c_3 f(s)^2-c_4,\quad c_3>0,\quad c_4\geq 0,\quad
 s\in \mathbb{R}, \\
\label{a6}
|f'(s)|\leq \epsilon|f(s)|+c_5,\quad \forall \epsilon >0,\; c_5\geq 0,\;
 s \in \mathbb{R}, \\
\label{a7}
sf''(s)\geq 0,\quad s \in \mathbb{R}.
\end{gather}
Note that these assumptions are satisfied by the cubic nonlinear term \eqref{e3}.

As far as the bounded function $\gamma$ is concerned, we introduce 
the following functions:
\begin{equation}\label{a8}
g(s)=\begin{cases}
\gamma^2(-1)s^2 & s<0,\\
0 &s=0,\\
\gamma^2(1)s^2 &s>0,
\end{cases}
\end{equation}
$g$ being a $C^1$-function, with $g'(0)=0$,
and
\begin{equation}\label{a9}
h(s)=\begin{cases}
\gamma^2(-1)s &s<0,\\
0 &s=0,\\
\gamma^2(1)s &s>0.
\end{cases}
\end{equation}
Thus, $h$ is a $C^0$-function, with $h'\in L^{\infty}(\mathbb{R})$.

\begin{lemma} \label{lem1} 
The function $h$ is Lipschitz continuous on $(-L,L)$.
\end{lemma}

\begin{proof}
 Let $s_1$ and $s_2$ belong to $\mathbb{R}$.
We have two cases, depending on the sign of $s_1$ and $s_2$:
\begin{itemize}
\item  If $s_1$ and $s_2$ have the same sign (or vanish), then it is clear that
$$
|h(s_1)-h(s_2)|\leq \max \{\gamma^2(1),\gamma^2(-1)\}|s_1-s_2|.
$$

\item If $s_1$ and $s_2$ have opposite signs, then, assuming that $s_1>0$ and 
$s_2<0$ (the case $s_1<0$ and $s_2>0$ is similar),
\begin{align*}
|h(s_1)-h(s_2)|
&=\gamma^2(1)s_1-\gamma^2(-1)s_2\\
&\leq \max \{\gamma^2(1),\gamma^2(-1)\}(s_1-s_2)\\
&=\max \{\gamma^2(1),\gamma^2(-1)\}|s_1-s_2|.
\end{align*}
\end{itemize}
The result follows.
\end{proof}


We denote by $((\cdot,\cdot))$ the usual $L^2$-scalar product, with 
associated norm $\|\cdot\|$, and we set 
$\| \cdot \|_{-1}=\|(-\Delta)^{-1/2} \cdot\|$, where $(-\Delta)^{-1}$ 
is the inverse minus Laplace operator associated with Neumann boundary 
conditions and acting on functions with null average.

We set, whenever it makes sense,
 $\langle \cdot \rangle=\frac{1}{\operatorname{Vol}(\Omega)}\int_{\Omega} \cdot \,dx$,
 being understood that, for 
$\varphi \in H^{-1}(\Omega), \langle\varphi\rangle
=\frac{1}{\operatorname{Vol}(\Omega)}\langle\varphi,1\rangle_{H^{-1}(\Omega),
H^1(\Omega)}$, and we note that
$$
\varphi \mapsto \left(\|\varphi-\langle\varphi\rangle^2\|^2_{-1}
+\langle\varphi\rangle^2\right)^{1/2}
$$
is a norm on $H^{-1}(\Omega)$ which is equivalent to the usual one.

Throughout this article, the same letter $c$ (and sometimes $c'$) denotes 
constants which may vary from line to line. Similarly, 
the same letter $Q$ denotes monotone increasing (with respect to each argument) 
functions which may vary from line to line.

\begin{remark} \label{rmk1} 
We can write, formally, for a small variation,
\begin{align*}
D \Psi
&= \int_{-L}^L \big[ \left(\gamma(n) u_{x}\right)D(\gamma(n) u_x)
+F'(u)D u +\omega D \omega \big]\,dx\\
&= \int_{-L}^L \big[ \gamma(n) u_x D(\gamma(n) u_x)+f(u) D u+\omega f'(u)D u
-\omega_{xx} D u \big]\,dx.
\end{align*}
We then note that
$$
\Big(\gamma\big(\frac{s}{|s|}\big)s\Big)'
=\gamma\big(\frac{s}{|s|}\big)\quad \text{in } \mathcal{D}'.
$$
Indeed, we have
$$
\Big(\gamma\big(\frac{s}{|s|}\big)s\Big)'
=s \gamma'\big(\frac{s}{|s|}\big) \big(\frac{s}{|s|}\big)'
+\gamma\big(\frac{s}{|s|}\big) \quad
\text{in } \mathcal{D}'.
$$
Now, it is sufficient to prove that
$$
s \gamma'\big(\frac{s}{|s|}\big) \big(\frac{s}{|s|}\big)'=0 \quad\text{in } \mathcal{D}'.
$$
To do so, we let $\varphi \in \mathcal{D}(-L,L)$ and have
\begin{align*}
\langle \big(\frac{s}{|s|}\big)',\varphi\rangle_{\mathcal{D}',\mathcal{D}}
&=-\langle \frac{s}{|s|} , \varphi'\rangle_{\mathcal{D}',\mathcal{D}}
 =-\int_{-L}^L \frac{s}{|s|} \varphi'(s) \,ds\\
&=-\int_0^L \varphi'(s) \,ds+\int_{-L}^0 \varphi'(s) \,ds\\
&=[\varphi(s)]_{-L}^0+[-\varphi(s)]_0^L\\
&=2\varphi(0)=2\langle \delta_0,\varphi\rangle_{\mathcal{D}',\mathcal{D}},
\end{align*}
so that
$$
s \gamma'\big(\frac{s}{|s|}\big) \big(\frac{s}{|s|}\big)'
=2s \delta_0 \gamma'\big(\frac{s}{|s|}\big) \text{\  in } \mathcal{D}'.
$$
Since
$s \delta_0=0$  in $\mathcal{D}'$, we obtain
\begin{equation}\label{a10}
\Big(\gamma\big(\frac{s}{|s|}\big)s\Big)'=\gamma\big(\frac{s}{|s|}\big)\quad\text{in }
\mathcal{D}'.
\end{equation}
Thus, owing to \eqref{a10}, we obtain, formally,
\begin{align*}
D \Psi&=\int_{-L}^L \big[\gamma^2(n) u_x D (u_x)+f(u) D u
+\omega f'(u)D u-\omega_{xx} D u \big]\,dx\\
&=\int_{-L}^L \big[-(\gamma^2(n) u_x)_x+f(u)+\omega f'(u)-\omega_{xx} \big]D u \,dx
\end{align*}
and the variational derivative of $\Psi$ with respect to $u$ reads
$$
\frac{D \Psi}{D u}=-\left(h(u_x)\right)_x+f(u)+\omega f'(u)-\omega_{xx}.
$$
\end{remark}

\section{Cahn-Hilliard system}

 The Cahn-Hilliard equation is an equation of mathematical physics which 
describes the evolution of different material phases via an order parameter 
(or multiple order parameters).
The equation was initially derived as a model for spinodal decomposition 
in solid materials \cite{c1,c3} and has since been extended to many other
 physical systems.

\subsection*{Setting of the problem} 
Writing mass conservation, i.e., $\frac{\partial u}{\partial t}=-h_x$,
where $h$ is the mass flux which is related to the chemical potential $\mu$ 
by the constitutive relation $h=-\mu_x$, and that the chemical potential 
is the variational derivative of $\Psi$ with respect to $u$, we end up 
with the following sixth-order Cahn-Hilliard system
\begin{gather}\label{b1}
\frac{\partial u}{\partial t}=\mu_{xx}, \\
\label{b2}
\mu=-(h(u_x))_x+f(u)+\omega f'(u)-\omega_{xx},\\
\label{b3}
\omega=f(u)-u_{xx},
\end{gather}
together with the Neumann boundary conditions
\begin{equation}\label{b4}
u_x\big|_{\pm L}=\mu_x\big|_{\pm L}=\omega_x\big|_{\pm L}=0
\end{equation}
and the initial condition
\begin{equation}\label{b5}
u\big|_{t=0}=u_0.
\end{equation}

\subsection{A priori estimates}
We first note that, integrating (formally) \eqref{b1} over $\Omega$, 
we obtain the conservation of mass, namely,
\begin{equation}\label{b6}
\langle u(t)\rangle=\langle u_0\rangle,\quad t\geq 0.
\end{equation}
Multiplying \eqref{b1} by $(-\Delta)^{-1}\frac{\partial u}{\partial t}$,
 we have, integrating over $\Omega$ and by parts,
\begin{equation}\label{b7}
\|\frac{\partial u}{\partial t}\|_{-1}^2
=-((\mu, \frac{\partial u}{\partial t})).
\end{equation}
We then multiply \eqref{b2} by $\frac{\partial u}{\partial t}$ and 
integrate over $\Omega$ to obtain
\begin{equation}\label{b8}
\begin{aligned}
&((\mu,\frac{\partial u}{\partial t}))\\
&=\int_{\Omega} h(u_x) \frac{\partial u_x}{\partial t}\,dx
 +\frac{d}{dt}\int_{\Omega} F(u)\,dx 
 +((\omega f'(u),\frac{\partial u}{\partial t}))
-((\omega_{xx},\frac{\partial u}{\partial t} )).
\end{aligned}
\end{equation}
Noting that  from \eqref{b3} it follows that
\begin{equation}\label{b9}
((\omega f'(u),\frac{\partial u}{\partial t}))-((\omega_{xx}, 
\frac{\partial u}{\partial t}))=\frac{1}{2}\frac{d}{dt}\|\omega\|^2,
\end{equation}
we have, owing to \eqref{a8},
\begin{equation}\label{b10}
\int_{\Omega} h(u_x) \frac{\partial u_x}{\partial t} \,dx
=\frac12 \frac{d}{dt} \int_{\Omega} g(u_x) \,dx.
\end{equation}
We finally deduce from \eqref{b7}-\eqref{b10} that
\begin{equation}\label{b11}
\frac{d}{dt}\Big[\int_{\Omega} g(u_x) \,dx
+2\int_{\Omega} F(u) \,dx+\|\omega\|^2\Big]
+2\|\frac{\partial u}{\partial t}\|_{-1}^2=0.
\end{equation}
In particular, \eqref{b11} yields that the free energy decreases
 along the trajectories, as expected.

We now multiply \eqref{b1} by $(-\Delta)^{-1}\bar{u}$, where 
$\bar{u}=u-\langle u\rangle$, and integrate over $\Omega$. We obtain, owing to 
\eqref{b6},
\begin{equation}\label{b12}
\frac{1}{2}\frac{d}{dt}\|\bar{u}\|_{-1}^2
=-((\mu,u))+\operatorname{Vol}(\Omega)\langle\mu\rangle\langle u_0\rangle,
\end{equation}
where, owing to \eqref{b2},
\begin{equation}\label{b13}
\langle\mu\rangle = \langle f(u)\rangle+\langle f'(u)\rangle.
\end{equation}
Multiplying then $\eqref{b2}$ by $u$ and integrating over $\Omega$, 
we have, owing to \eqref{b3},
\begin{equation}\label{b14}
\begin{aligned}
((\mu,u))&=\int_{\Omega} g(u_x) \,dx +((f(u), u))+((f(u) f'(u), u))\\
&\quad -((f'(u) u_{xx}, u))-((f(u)_{xx}, u))+\|u_{xx}\|^2.
\end{aligned}
\end{equation}
Noting that
\begin{gather*}
((f'(u)u_{xx}, u))=-((f'(u) u_x, u_x))-((uf''(u)u_x, u_x)),\\
((f(u)_{xx}, u))=-((f'(u)u_x, u_x)),
\end{gather*}
we obtain
\begin{align*}
((\mu, u))&=\int_{\Omega} g(u_x) \,dx+((f(u), u))+\|\omega\|^2+((uf''(u)u_x, u_x))\\
&\quad +\int_{\Omega}\left(f(u)f'(u)u-f^2(u)\right)\,dx
\end{align*}
and finally, owing to \eqref{a4}, \eqref{a5}, \eqref{a7} and \eqref{b12}, we obtain
\begin{equation}\label{b15}
\begin{aligned}
&\frac{d}{dt}\|\bar{u}\|_{-1}^2+c\Big[\int_{\Omega} g(u_x)\,dx
+2\int_{\Omega} F(u) \,dx+\|\omega\|^2+\|f(u)\|^2\Big]\\
&\leq 2 \operatorname{Vol}(\Omega) \langle\mu\rangle\langle u_0\rangle+c',\quad c>0.
\end{aligned}
\end{equation}
We now assume that
\begin{equation}\label{b16}
|\langle u_0\rangle|\leq M\quad \text{(hence, $|\langle u(t)\rangle|\leq M$,
$t\geq 0$)},\quad M\geq 0.
\end{equation}
Therefore, owing to $\eqref{a6}$ and $\eqref{b13}$,
\begin{equation}\label{b17}
\begin{aligned}
|2 \operatorname{Vol}(\Omega)\langle u_0\rangle\langle\mu\rangle|
&\leq c_M\left(|\langle f(u)\rangle |+|\langle\omega f'(u)\rangle|\right)\\ 
&\leq \frac{c}{2}\Big(\int_{\Omega}f(u)^2\,dx+\int_{\Omega} \omega^2 \,dx \Big)
+c_M',
\end{aligned}
\end{equation}
where $c$ is the constant appearing in \eqref{b15}, and we deduce 
from \eqref{b15} and \eqref{b17} that
\begin{equation}\label{b18}
\frac{d}{dt}\|\bar{u}\|_{-1}^2
+c\Big[\int_{\Omega} g(u_x) \,dx+2\int_{\Omega} F(u)\,dx +\|\omega\|^2\Big]\leq c_M'.
\end{equation}
Combining \eqref{b11} and \eqref{b18}, we have an inequality of the form
\begin{equation}\label{b19}
\frac{dE}{dt}+c(E+\|\frac{\partial u}{\partial t}\|_{-1}^2)\leq c'_M,
\end{equation}
where
\begin{equation}\label{b20}
E=\| \bar{u} \|_{-1}^2+ \langle u\rangle ^2+\int_{\Omega}g(u_x) \,dx
+2\int_{\Omega} F(u) \,dx+\|\omega\|^2.
\end{equation}
In particular, we deduce from \eqref{b19} and Gronwall's Lemma that
\begin{equation}\label{b21}
E(t)\leq E(0) e^{-ct}+c_M',\quad c>0,\; t\geq 0.
\end{equation}
Noting that, owing to $\eqref{a3}$,
\begin{equation}\label{b22}
\|\omega\|^2\geq \|f(u)\|^2+\|u_{xx}\|^2-2c_0\|u_x\|^2,
\end{equation}
we finally deduce from \eqref{b20}-\eqref{b22} and the boundedness of $\gamma(n)$ that
\begin{equation}\label{b23}
\|u\|_{H^2(\Omega)}^2+\|f(u)\|^2 \leq Q(\|u_0\|_{H^2(\Omega)})e^{-ct}+c_{M}'.
\end{equation}
Rewriting \eqref{b1} in the equivalent form
\begin{equation}\label{b24}
\mu=\langle\mu\rangle-\left(-\Delta\right)^{-1}\frac{\partial u}{\partial t},
\end{equation}
we obtain
\begin{equation}\label{b25}
\|\mu_x\|\leq c\|\frac{\partial u}{\partial t}\|_{-1}.
\end{equation}
Noting that, proceeding as in \eqref{b17},
\[
|\langle\mu\rangle|\leq c\left(\|u\|_{H^2(\Omega)}^2+\|f(u)\|^2+1\right),
\]
we finally find
\begin{equation}\label{b26}
\|\mu\|_{H^1(\Omega)}\leq c\big(\|\frac{\partial u}{\partial t}\|_{-1}
+\|u\|_{H^2(\Omega)}^2+\|f(u)\|^2+1\big).
\end{equation}
Now, owing to $\eqref{b2}$, we have
$$
\omega_{xx}=-(h(u_x))_x-\mu+f(u)+\omega f'(u)
$$
and, owing to $\eqref{a6}$, there holds
\begin{equation}\label{b27}
\begin{aligned}
\|\omega_{xx}\|
&\leq c\left(\|(h(u_x))_x\|+\|f(u)\|^2+\|\omega\|^2+\|\mu\|\right) \\
&\leq c\left(\|h(u_x)\|_{H^1(\Omega)}+\|f(u)\|^2+\|\omega\|^2+\|\mu\|\right),
\end{aligned}
\end{equation}
where we have used the fact that
$$
\left\{ \begin{array}{c}
h(u_x)=\gamma^2(n)u_x\in L^2(\Omega)\\
(h(u_x))'=h'(u_x) u_{xx} \in L^2(\Omega)
\end{array}
\right\}\Rightarrow h(u_x) \in H^1(\Omega).
$$
Recall that $h$ is Lipschitz continuous, with $h(0)=0$, and note that
$$
\|h(u_x)\|_{H^1(\Omega)}\leq c\|u\|_{H^2(\Omega)}.
$$
We then have, owing to \eqref{a8} and \eqref{b26}-\eqref{b27},
\begin{equation}\label{b28}
\|\omega\|_{H^2(\Omega)}
\leq c\Big(\|\frac{\partial u}{\partial t}\|_{-1}+\|u\|_{H^2(\Omega)}^2
+\|f(u)\|^2+1\Big).
\end{equation}
We now multiply \eqref{b1} by $u$ and integrate over $\Omega$ to get
\begin{equation}\label{b29}
\frac{1}{2} \frac{d}{dt} \|u\|^2 = -((\mu_{x},u_x)).
\end{equation}
Multiplying then \eqref{b2} by $-u_{xx}$ and integrating over $\Omega$,
we obtain, in view of \eqref{b3},
\begin{equation}\label{b30}
\begin{aligned}
((\mu_x, u_x))
&= \int_{\Omega} h(u_x) u_{xxx}\,dx+((f'(u)u_x, u_x))-((\omega f'(u), u_{xx}))\\
&\quad -((f(u)_{xx}, u_{xx}))+\|u_{xxx}\|^2.
\end{aligned}
\end{equation}
We note that
\begin{equation} \label{b31}
\begin{aligned}
|((\omega f'(u),u_{xx}))|
&\leq \|f'(u)\|_{L^{\infty}(\Omega)}\|\omega\|\|u_{xx}\|\\
&\leq \frac{1}{2}\|u_{xx}\|^2 +Q\left(\|u\|_{H^2(\Omega)}\right)\|\omega\|^2,
\end{aligned}
\end{equation}
where $Q$ is continuous (here, we have used the fact that $H^2(\Omega)$
is continuously embedded into $C(\bar{\Omega})$), and, proceeding similarly,
\begin{equation} \label{b32}
\begin{aligned}
\big\vert((f(u)_{xx}, u_{xx}))\big\vert
&= \big\vert((f'(u)u_x,u_{xxx}))\big\vert \\
&\leq \frac{1}{2}\|u_{xxx}\|^2+Q\left(\|u\|_{H^2(\Omega)}\right) \|u_x\|^2.
\end{aligned}
\end{equation}
Finally,
\begin{equation}\label{b33}
\big\vert\int_{\Omega} h(u_x) u_{xxx}\,dx\big\vert
\leq c[\|u_x\|^2+\|u_{xxx}\|^2].
\end{equation}
It thus follows from \eqref{a3} and \eqref{b29}-\eqref{b33} that
\begin{equation}\label{b34}
\frac{d}{dt}\|u\|^2+\|u\|_{H^3(\Omega)}^2
\leq Q(\|u\|_{H^2(\Omega)})\big(\|u\|_{H^1(\Omega)}^2+\|\omega\|^2\big),
\end{equation}
where $Q$ is continuous.

\subsection{Existence and uniqueness of solutions}

\begin{theorem} \label{thm2.1}
Assume that \eqref{b16} holds and that $u_0\in H^2(\Omega)$, with 
$\frac{\partial u_0}{\partial x}\big|_{\pm L}=0$. 
Then \eqref{b1}-\eqref{b5} admits a unique (variational) solution such that
\begin{gather*}
u\in L^{\infty}(\mathbb{R}^{+};H^2(\Omega))\cap L^2(0,T;H^3(\Omega)), \quad
\frac{\partial u}{\partial t}\in L^2(0,T;H^{-1}(\Omega)), \\
\mu \in L^2(0,T;H^1(\Omega)), \quad  
\omega \in L^{\infty}(\mathbb{R}^+;L^2(\Omega))\cap L^2(0,T;H^2(\Omega))
\end{gather*} 
for all $T>0$.
\end{theorem}

\begin{proof} 
(a) Existence:
The proof of existence is based on a classical Galerkin scheme and on
 the \textit{a priori} estimates derived in the previous section.
We can note that a weak (variational) formulation of \eqref{b1}-\eqref{b5} reads
\begin{gather}\label{b35}
((\frac{\partial u}{\partial t}, v))=((\mu_{xx}, v)),\quad 
\forall v\in H^1(\Omega), \\
\begin{gathered}
((\mu, v))=((h(u_x), v_x))+((\omega f'(u), v))
+((f(u), v))-((\omega_{xx}, v)),  \\ \forall v\in H^1(\Omega),
\end{gathered} \label{b36} \\
\label{b37}
((\omega, v))=((f(u), v))-((u_{xx}, v)),\quad \forall v\in H^1(\Omega), \\
\label{b38}
u\big|_{t=0}=u_0.
\end{gather}

Let $v_0, v_1,\dots$ be an orthonormal (in $L^2(\Omega)$) and orthogonal 
(in $H^1(\Omega)$) family associated with the eigenvalues 
$0=\lambda_0< \lambda_1 \leq\cdot \cdot \cdot$ of the operator 
$-\Delta$ associated with Neumann boundary conditions (note that $v_0$ 
is a constant). We set
$$
V_m=\operatorname{Span} \{v_0,v_1,\dots,v_m\}
$$
and consider the approximate problem: 

Find $(u_m,\mu_m,\omega_m): [0,T]\to V_m \times V_m \times V_m$ such that
\begin{gather} \label{b39}
((\frac{\partial u_m}{\partial t}, v))=-(({\mu_m}_{x}, v)),\quad \forall v\in V_m,\\
\begin{aligned}
((\mu_m, v))&=((h({u_m}_x), v_x))+((\omega f'(u_m), v))\\
&\quad +((f(u_m), v))-(({\omega_m}_{xx}, v)), \quad \forall v\in V_m,
\end{aligned}\label{b40} \\
\label{b41}
((\omega_m, v))=((f(u_m), v))-(({u_m}_{xx}, v)),\quad \forall v \in V_m, \\
\label{b42}
u_m\big|_{t=0}=u_{0,m},
\end{gather}
where $u_{0,m}=P_m u_0$, $P_m$ being the orthogonal projector from 
$L^2(\Omega)$ onto $V_m$.

The existence of a local (in time) solution to \eqref{b39}-\eqref{b42} 
is standard. Indeed, we have to solve a Lipschitz continuous finite-dimensional 
system of ODE's to find $u_m$, which yields $\omega_m$ and then $\mu_m$.

The \textit{a priori} estimates derived in the previous section, which are 
now justified within the Galerkin approximation, yield that the solution 
is global and that, up to a subsequence which we do not relabel and owing 
to classical Aubin-Lions compacteness results,
\begin{gather*}
u_m \to u \quad\text{weak star in $L^{\infty}(0,T;H^2(\Omega))$,
strongly in  $ C([0,T];H^{2-\varepsilon}(\Omega))$, and a.e.},
\\
\frac{\partial u_m}{\partial t}\to \frac{\partial u}{\partial t} \quad
\text{weakly in } L^2(0,T;H^{-1}(\Omega)), \\
\mu_m \to \mu \quad \text{weakly in } L^2(0,T;H^{1}(\Omega)), \\
\omega_m \to \omega \quad \text{weak star in } L^{\infty}(0,T; L^2(\Omega)) 
\text{ and weakly in } L^2(0,T;H^2(\Omega)),
\end{gather*}
as $m \to + \infty, \forall T>0$.

Note that, owing to \eqref{b19}, \eqref{b21} and \eqref{b23}, we have 
$u\in L^{\infty}(\mathbb{R}^+;H^2(\Omega))$ and, consequently, 
$\omega \in L^{\infty}(\mathbb{R}^+;L^2(\Omega))$.

As far as the passage to the limit is concerned, the most delicate part 
is to prove that
\begin{gather*}
\int_0^T \int_{\Omega} (\omega_m f'(u_m)-\omega f'(u)) \varphi \,dx \,dt\to 0\quad
\text{as } m \to +\infty, \\
\int_0^T \int_{\Omega} (h({u_m}_x)-h({u}_x)) \varphi_x \,dx \,dt \to 0\quad
\text{as } m \to +\infty,
\end{gather*}
for $\varphi$ regular enough.

We have, say, for $\varphi \in C^2([0,T] \times \bar{\Omega})$ such that 
$\varphi(T)=\varphi(0)=0$,
\begin{equation}\label{b43}
\begin{aligned}
&\int_0^T \int_{\Omega} \left(\omega_m f'(u_m)-\omega f'(u)\right)\varphi \,dx\,dt\\
&=\int_0^T \int_{\Omega}(\omega_m-\omega)f'(u)\varphi \,dx \,dt
+\int_0^T \int_{\Omega}\omega_m \left(f'(u_m)-f'(u)\right)\varphi \,dx \,dt.
\end{aligned}
\end{equation}
The passage to the limit in the first integral in the right-hand side 
of \eqref{b43} is straightforward, while the passage to the limit in the 
second one follows from the above convergences which yield, in particular, 
the inequality
\[
\big\vert \int_0^T \int_{\Omega} \omega_m \left(f'(u_m)-f'(u)\right)\varphi 
\,dx \,dt\big\vert \leq c\|u_m-u\|_{L^2((0,T)\times \Omega)}.
\]
Finally, recalling that $h$ is Lipschitz continuous, we have
\[
\big\vert \int_0^T \int_{\Omega}\big(h({u_m}_{x})-h(u_x)\big)\varphi_x \,dx \,dt 
\big\vert
\leq c\|{u_m}_x-u_x\|_{L^2((0,T)\times \Omega)}.
\]
\smallskip

\noindent(b) Uniqueness:
Let $(u_1,\mu_1,\omega_1)$ and $(u_2,\mu_2,\omega_2)$ be two solutions 
to \eqref{b1}-\eqref{b4} with initial data $u_{1,0}$ and $u_{2,0}$,
 respectively, such that
\begin{equation}\label{b44}
|\langle u_{i,0} \rangle|\leq M,\quad i=1,2.
\end{equation}
We set $(u,\mu,\omega)=(u_1,\mu_1,\omega_1)-(u_2,\mu_2,\omega_2)$ and 
$u_0=u_{1,0}-u_{2,0}$ and have
\begin{gather}\label{b45}
\frac{\partial u}{\partial t}=\mu_{xx} , \\
\begin{aligned}
\mu &=-\big(h({u_1}_x)\big)_x+\big(h( {u_2}_x)\big)_x+f(u_1)-f(u_2)\\
&\quad  +\omega_1 f'(u_1)-\omega_2 f'(u_2)-\omega_{xx},
\end{aligned} \label{b46}\\
\label{b47}
\omega= f(u_1)-f(u_2)-u_{xx}, \\
\label{b48}
 u_x\big|_{\pm L}= \mu_x \big|_{\pm L}= \omega_x \big|_{\pm L}=0, \\
\label{b49}
 u\big|_{t=0}=u_0.
\end{gather}
We multiply \eqref{b45} by $(-\Delta)^{-1}\bar{u}$ and obtain, integrating 
over $\Omega$ and by parts,
\begin{equation}\label{b50}
\frac12 \frac{d}{dt}\|\bar{u}\|_{-1}^2=-((\mu, u))
+\operatorname{Vol}(\Omega) \langle\mu\rangle \langle u\rangle ,
\end{equation}
where, owing to \eqref{b46},
\begin{equation}\label{b51}
\langle\mu\rangle=\langle f(u_1)-f(u_2)\rangle
+\langle\omega_1 f'(u_1)-\omega_2 f'(u_2)\rangle.
\end{equation}
We then multiply \eqref{b46} by $u$ and find, in view of \eqref{b47},
\begin{equation} \label{b52}
\begin{aligned}
((\mu, u))
&=\int_{\Omega} h({u_1}_x )u_x \,dx-\int_{\Omega} h({u_2}_x) u_x \,dx\\
&\quad +((f(u_1)-f(u_2), u))+((\omega_1 f'(u_1)-\omega_2 f'(u_2), u)) \\
&\quad -((f(u_1)-f(u_2), u_{xx}))+\|u_{xx}\|^2.
\end{aligned}
\end{equation}
We have, owing to \eqref{a3},
\begin{equation}\label{b53}
((f(u_1)-f(u_2), u))=((f'(u)u, u))\geq -c_0 \|u\|^2.
\end{equation}
Furthermore,
\begin{equation}\label{b54}
|((f(u_1)-f(u_2),u_{xx}))|
\leq \frac{1}{8}\|u_{xx}\|^2+Q(\|u_{1,0}\|_{H^2(\Omega)},
 \|u_{2,0}\|_{H^2(\Omega)})\|u\|^2
\end{equation}
and
\begin{equation}\label{b55}
\begin{aligned}
&\big\vert((\omega_1 f'(u_1)-\omega_2 f'(u_2), u))\big\vert\\
&\leq |((\omega_1(f'(u_1)-f'(u_2)), u))|+|((\omega f'(u_2), u))|\\
&\leq Q(\|u_{1,0}\|_{H^2(\Omega)},\|u_{2,0}\|_{H^2(\Omega)})
 \|\omega_1\|_{H^2(\Omega)} \|u\|^2 \\
&\quad +|((f'(u_2) u_{xx}, u))| +|((f'(u_2)(f(u_1)-f(u_2)), u))| \\
&\leq \frac{1}{8}\|u_{xx}\|^2+Q(\|u_{1,0}\|_{H^2(\Omega)},
 \|u_{2,0}\|_{H^2(\Omega)})(\|\omega_1\|_{H^2(\Omega)}+1)\|u\|^2.
\end{aligned}
\end{equation}
Similarly,
\begin{equation}\label{b56}
\begin{aligned}
&|\operatorname{Vol}(\Omega)\langle u\rangle  \langle\mu\rangle | \\
& \leq c (\int_{\Omega}|f(u_1)-f(u_2)|\,dx
 +\int_{\Omega}|\omega_1 f'(u_1)  -\omega_2 f'(u_2)|\,dx)|\langle u\rangle | \\
&\leq \Big(\int_{\Omega}|f(u_1)-f(u_2)||f'(u_2)|\,dx\Big)|\langle u\rangle | \\
&\quad +(\int_{\Omega} |\omega_1||f'(u_1)-f'(u_2)|\,dx
 +\int_{\Omega} |u_{xx}| |f'(u_2)|\,dx)|\langle u\rangle | \\
&\quad +Q(\|u_{1,0}\|_{H^2(\Omega)},\|u_{2,0}\|_{H^2(\Omega)})\|u\|
 |\langle u\rangle | \\
&\leq \frac{1}{8}\|u_{xx}\|^2+Q(\|u_{1,0}\|_{H^2(\Omega)},
 \|u_{2,0}\|_{H^2(\Omega)})(\|\omega_1\|+1)(\|u\|^2+|\langle u\rangle |^2).
\end{aligned}
\end{equation}
Recalling that $h$ is Lipschitz continuous, we have
\begin{equation} \label{b57}
|((h({u_1}_x)-h({u_2}_x), u_x))|
\leq \int_{\Omega} |h({u_1}_x)-h({u_2}_x)||u_x|\,dx
\leq c \|u_x\|^2.
\end{equation}
We finally deduce from \eqref{b50}, \eqref{b52}-\eqref{b57} and
the interpolation inequality
\begin{equation}\label{b58}
\| \bar{u}\|\leq c \|\bar{u}\|^{1/2}_{-1}\| \nabla \bar{u}\|^{1/2}
\leq c' \|\bar{u}\|^{1/2}_{-1}\|\Delta \bar{u}\|^{1/2}
\end{equation}
that
\begin{equation}\label{b59}
\begin{aligned}
&\frac{d}{dt}(\|\bar{u}\|_{-1}^2+\langle u\rangle ^2)+\|u_{xx}\|^2\\
&\leq Q(\|u_{1,0}\|_{H^2(\Omega)},\|u_{2,0}\|_{H^2(\Omega)})
(1+\|\omega_1\|+\|\omega_1\|_{H^2(\Omega)})
(\|\bar{u}\|^2_{-1}+|\langle u\rangle |^2).
\end{aligned}
\end{equation}
Gronwall's Lemma then yields, owing to \eqref{b19}, \eqref{b23}
and \eqref{b28} (written for $(u_1,\mu_1,\omega_1)$),
\begin{equation}\label{b60}
\|u(t)\|_{H^{-1}(\Omega)}\leq c e^{Q(\|u_{1,0}\|_{H^2(\Omega)},
\|u_{2,0}\|_{H^2(\Omega)})t}\|u_0\|_{H^{-1}(\Omega)},
\end{equation}
hence the uniqueness, as well as the continuous dependence with respect
 to the initial data in the $H^{-1}$-norm.

It follows from Theorem \ref{thm2.1} that we can define the continuous
 (for the $H^{-1}$-norm) semigroup
$$
S(t): \Phi_M \to \Phi_M,\quad u_0 \to u(t),\quad t\geq 0
$$
(i.e., $S(0)=Id$ and $S(t+s)=S(t) \circ S(s),\ t,s\geq 0$), where
$$
\Phi_M=\big\{v\in H^2(\Omega), \frac{\partial v}{\partial x}\big|_{\pm L}=0, 
|\langle v\rangle |\leq M\big\},\quad M\geq 0.
$$
We then deduce from \eqref{b23} that $S(t)$ is dissipative, i.e., 
it possesses a bounded absorbing set $\mathcal{B}_0\subset \Phi_M$ 
(in the sense that, for all $B \subset \Phi_{M}$ bounded, there exists 
$t_0=t_0(B)$ such that $t\geq t_0\Rightarrow S(t)B \subset \mathcal{B}_0$).
\end{proof}

\section{Allen-Cahn system}

The Allen-Cahn equation describes important processes related with 
phase separation in binary alloys, namely, the ordering of atoms in a
lattice (see \cite{a1}).

Assuming the relaxation dynamics 
$\frac{\partial u}{\partial t}=-\frac{D \psi}{D u}$, we obtain the 
 Allen-Cahn system
\begin{gather}\label{c1}
\frac{\partial u}{\partial t}-(h(u_x))_x+f(u)+\omega f'(u)-\omega_{xx}=0,\\
\label{c2}
\omega=f(u)-u_{xx},
\end{gather}
together with the Neumann boundary conditions
\begin{equation}\label{c3}
u_x\big|_{\pm L}=\omega_x\big|_{\pm L}=0
\end{equation}
and the initial condition
\begin{equation}\label{c4}
u\big|_{t=0}=u_0.
\end{equation}

\subsection{A priori estimates}
We Multiply $\eqref{c1}$ by $\frac{\partial u}{\partial t}$ and have, 
integrating over $\Omega$ and by parts,
\[
\|\frac{\partial u}{\partial t}\|^2+\int_{\Omega} h(u_x) 
\frac{\partial u_x}{\partial t} \,dx+\frac{d}{dt}
\int_{\Omega} F(u) \,dx+((\omega f'(u)-\omega_{xx}, 
\frac{\partial u}{\partial t}))=0,
\]
which yields, noting that it follows from \eqref{c2} that
\[
((\omega f'(u), \frac{\partial u}{\partial t}))
-((\omega_{xx}, \frac{\partial u}{\partial t}))
=\frac{1}{2}\frac{d}{dt}\|\omega\|^2
\]
and from \eqref{a8} that
\[
\int_{\Omega} h(u_x) \frac{\partial u_x}{\partial t}\,dx
=\frac12 \frac{d}{dt}\int_{\Omega} g(u_x)\,dx,
\]
the differential equality
\begin{equation}\label{c5}
\frac{d}{dt}\Big[\int_{\Omega} g(u_x)\,dx+2\int_{\Omega} F(u) \,dx
+\|\omega\|^2\Big]+2\|\frac{\partial u}{\partial t}\|^2=0.
\end{equation}
In particular, it follows from \eqref{c5} that the energy decreases 
along the trajectories, as expected.

We then multiply \eqref{c1} by $u$ and obtain, owing to \eqref{c2},
\begin{align*}
&\frac{1}{2}\frac{d}{dt}\|u\|^2+\int_{\Omega} g(u_x)\,dx+((f(u), u))
 +\int_{\Omega} uf(u) f'(u) \,dx \\
&\quad +2((f'(u) u_x, u_x))+((uf''(u) u_x, u_x))+\|u_{xx}\|^2=0,
\end{align*}
which yields, owing to \eqref{c2},
\begin{align*}
&\frac{1}{2}\frac{d}{dt}\|u\|^2+\int_{\Omega} g(u_x)\,dx+((f(u), u))+\|w\|^2\\
&+\int_{\Omega} (u f(u) f'(u)-f^2(u))\,dx+((u f''(u) u_x, u_x))=0,
\end{align*}
hence, in view of \eqref{a4}, \eqref{a5} and \eqref{a7},
\begin{equation}\label{c6}
\frac{d}{dt} \|u\|^2+c\Big[\int_{\Omega} g(u_x)\,dx
+2\int_{\Omega}F(u)\,dx+\|\omega\|^2\Big]\leq c',\quad c>0.
\end{equation}
Summing \eqref{c5} and \eqref{c6}, we find an inequality of the form
\begin{equation}\label{c7}
\frac{d E_1}{dt}+c\Big(E_1+\|\frac{\partial u}{\partial t}\|^2\Big)
\leq c',\quad c>0,
\end{equation}
where
\begin{equation}\label{c8}
E_1=\|u\|^2+\int_{\Omega} g(u_x)\,dx+2\int_{\Omega}F(u)\,dx+\|\omega\|^2.
\end{equation}
In particular, it follows from \eqref{c7} and Gronwall's Lemma that
\begin{equation}\label{c9}
E_1(t)\leq E_1(0)e^{-ct}+c',\quad c>0,
\end{equation}
hence, in view of \eqref{a3} (which yields that 
$\|\omega\|^2 \geq \|u_{xx}\|^2+\|f(u)\|^2-2c_0 \|u_x\|^2$), 
\eqref{c8} and classical elliptic regularity results,
\begin{equation}\label{c10}
\|u(t)\|_{H^2(\Omega)}\leq Q(\|u_0\|_{H^2(\Omega)})e^{-ct}+c',\quad c>0,\; t\geq 0.
\end{equation}

Next, we multiply \eqref{c1} by $-u_{xx}$ to have
\begin{equation}\label{c11}
\begin{aligned}
&-\int_{\Omega} \frac{\partial u}{\partial t} u_{xx} \,dx
-\int_{\Omega} h(u_x) u_{xxx}\,dx-\int_{\Omega} f(u) u_{xx}\,dx \\
&-\int_{\Omega} \omega f'(u) u_{xx}\,dx+\int_{\Omega}\omega_{xx} u_{xx}\,dx=0.
\end{aligned}
\end{equation}
It follows from \eqref{c2} that
\begin{equation}\label{c12}
\begin{aligned}
&\frac{1}{2}\frac{d}{dt}\|u_x\|^2-\int_{\Omega} h(u_x) u_{xxx}\,dx
+((f'(u) u_x, u_x)) \\
&-((\omega f'(u), u_{xx}))+(((f(u))_{xx}, u_{xx}))+\|u_{xxx}\|^2=0.
\end{aligned}
\end{equation}
Now, owing to the continuous embedding
 $H^2(\Omega) \subset \mathcal{C}(\bar{\Omega})$ and $\eqref{c2}$, 
there holds
\[
\big\vert((f'(u) u_x, u_x))\big\vert +\big\vert((\omega f'(u), u_{xx}))\big\vert 
+\big\vert(((f(u))_{xx}, u_{xx}))\big\vert\leq Q(\|u\|_{H^2(\Omega)})
\]
(indeed, it follows from $\eqref{c2}$ that $\|\omega\|\leq Q(\|u\|_{H^2(\Omega)})$)
 and
$$
\big|\int_{\Omega} h(u_x) u_{xxx}\,dx\big|
\leq c[\|u_x\|^2+\|u_{xxx}\|^2],
$$
hence
\begin{equation}\label{c13}
\frac{d}{dt}\|u_x\|^2+\|u\|^2_{H^3(\Omega)}\leq Q(\|u\|_{H^2(\Omega)}).
\end{equation}

\subsection{Existence and uniqueness of solutions}

\begin{theorem} \label{thm3.1}
Let $u_0 \in H^2(\Omega)\cap H_0^1(\Omega)$. Then, \eqref{c1}-\eqref{c4} 
admits a unique (variational) solution such that 
$u \in L^{\infty}(\mathbb{R}^+;H^2(\Omega)\cap H^1_0(\Omega))$ and 
$\frac{\partial u}{\partial t} \in L^2(0,T;L^2(\Omega))$. Furthermore,
 $\omega \in L^{\infty}(\mathbb{R}^+;L^2(\Omega))
\cap L^2(0,T; H^2(\Omega)\cap H_0^1(\Omega))$ for all $T>0$. 
Finally, the associated semigroup is dissipative in 
$H^2(\Omega)\cap H^1_0(\Omega)$.
\end{theorem}

\begin{proof} (a) Uniqueness:
Let $u_1$ and $u_2$ be two solutions to \eqref{c1}-\eqref{c3} with 
initial data $u_{1,0}$ and $u_{2,0}$ respectively, where $\omega_1$ and 
$\omega_2$ are defined from \eqref{c2}. We set $u=u_1-u_2$, 
$\omega=\omega_1-\omega_2$, $u_0=u_{1,0}-u_{2,0}$ and have
\begin{gather} \label{c14}
\begin{aligned}
&\frac{\partial u}{\partial t}-(h({u_1}_x))_x+(h({u_2}_x))_x+f(u_1)-f(u_2)\\
&+\omega_1f'(u_1)-\omega_2f'(u_2)-\omega_{xx}=0,
\end{aligned} \\
\label{c15}
\omega=f(u_1)-f(u_2)-u_{xx}, \\
\label{c16}
u_x\big|_{\pm L}=\omega_x\big|_{\pm L}=0, \\
\label{c17}
u\big|_{t=0}=u_0.
\end{gather}
We multiply \eqref{c14} by $u$ and  integrating over $\Omega$, we obtain
\begin{equation}\label{c18}
\begin{aligned}
&\frac12 \frac{d}{dt}\|u\|^2+((h({u_1}_x)-h({u_2}_x),u_x))+((f(u_1)-f(u_2),u))\\
&+((\omega_1f'(u_1)-\omega_2 f'(u_2), u))-((f(u_1)-f(u_2),u_{xx}))+\|u_{xx}\|^2=0.
\end{aligned}
\end{equation}
We note that, by \eqref{a3},
\[
((f(u_1)-f(u_2),u))\geq c_0 \|u\|^2
\]
and that, owing to \eqref{c15},
\begin{equation}\label{c19}
\begin{aligned}
& |((\omega_1f'(u_1)-\omega_2 f'(u_2), u))|\\
&\leq |((\omega f'(u_1),u))|+|((\omega_2 (f'(u_1)-f'(u_2)),u))|\\
&\leq Q(\|u_{1,0}\|_{H^2(\Omega)},\|u_{2,0}\|_{H^2(\Omega)})
 (\|\omega\|\|u\|+\|\omega_2\|\|u\|^2_{L^4(\Omega)}) \\
&\leq Q(\|u_{1,0}\|_{H^2(\Omega)},\|u_{2,0}\|_{H^2(\Omega)})
 (\|u_{xx}\|^2\|u\|+\|u_x\|^2) \\
&\leq \frac14 \|u_{xx}\|^2+Q(\|u_{1,0}\|_{H^2(\Omega)},
 \|u_{2,0}\|_{H^2(\Omega)})\|u_x\|^2
\end{aligned}
\end{equation}
and
\begin{equation}\label{c20}
|((f(u_1)-f(u_2),u_{xx}))|\leq \frac18 \|u_{xx}\|^2
+Q(\|u_{1,0}\|_{H^2(\Omega)},\|u_{2,0}\|_{H^2(\Omega)})\|u\|^2.
\end{equation}
Recalling that $h$ is Lipschitz continuous, we have
\begin{equation} \label{c21}
|((h({u_1}_x)-h({u_2}_x), u_x))|
\leq \int_{\Omega} |h({u_1}_x)-h({u_2}_x)||u_x|\,dx \\
\leq c \|u_x\|^2.
\end{equation}
We finally deduce from \eqref{c18}-\eqref{c21} and the interpolation inequality
$$
\|u_x\|\leq c\|u\|^{1/2}\|u_{xx}\|^{1/2}
$$
that
\begin{equation}\label{c22}
\frac{d}{dt}\|u\|^2+\|u_{xx}\|^2
\leq Q(\|u_{1,0}\|_{H^2(\Omega)},\|u_{2,0}\|_{H^2(\Omega)})\|u\|^2.
\end{equation}
Then Gronwall's Lemma  yields
\begin{equation}\label{c23}
\|u_1(t)-u_2(t)\|\leq c e^{Q(\|u_{1,0}\|_{H^2(\Omega)},
\|u_{2,0}\|_{H^2(\Omega)})t}\|u_0\|,
\end{equation}
hence the uniqueness, as well as the continuous dependence with respect
to the initial data in the $L^2$-norm.
\smallskip

\noindent (b) Existence:
The proof of existence of solutions is based on the \textit{a priori} 
estimates derived in the previous section and, e.g., a standard Galerkin scheme.

In particular, it follows from \eqref{c7}-\eqref{c8} and \eqref{c10} that 
we can construct a sequence of solutions $u_m$ to a proper approximated 
problem such that
\begin{gather*}
u_m \to u \quad \text{weak star in $L^{\infty}(0,T;H^2(\Omega))$, \
strongly in $C([0,T];H^{2-\varepsilon}(\Omega))$ and a.e.,}\\
\frac{\partial u_m}{\partial t}\to \frac{\partial u}{\partial t} \quad
\text{weakly in } L^2(0,T;L^2(\Omega)),\\
\omega_m \to \omega \quad \text{weak star in $L^{\infty}(0,T; L^2(\Omega))$
 and weakly in $L^2(0,T; H^2(\Omega))$},
\end{gather*}
as $m \to +\infty$ for all $T>0$.

The passage to the limit is then standard and can be done as in the previous section.
Furthermore, it follows from \eqref{c7}-\eqref{c8} and \eqref{c10} that
$$
u \in L^{\infty}(\mathbb{R}^{+}; H^2(\Omega)),\quad
\frac{\partial u}{\partial t}\in L^2(0,T;L^2(\Omega)),\ \forall T>0,
$$
and, consequently,
$\omega \in L^{\infty}(\mathbb{R}^+;L^2(\Omega))$.

It follows from Theorem \ref{thm3.1} that we can define the continuous 
(for the $L^2$-norm) semigroup
$$
S(t): \Phi \to \Phi,\quad u_0 \to u(t)$$
where
$\Phi=H^2(\Omega)\cap H^1_0(\Omega)$.
Finally, the dissipativity of $S(t)$ follows from \eqref{c10}.
\end{proof}


\begin{thebibliography}{00}

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\end{document}