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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 60, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2009/60\hfil A combustion model]
{A combustion model with unbounded thermal conductivity and reactant
 diffusivity in non-smooth domains}

\author[S. A. Sanni\hfil EJDE-2009/60\hfilneg]
{Sikiru Adigun Sanni}

\address{Sikiru Adigun Sanni \newline
  Department of Mathematics, University of Uyo,
  Uyo, Akwa Ibom State, Nigeria}
\email{sikirusanni@yahoo.com}

\thanks{Submitted June 25, 2008. Published May 4, 2009.}
\subjclass[2000]{35B40, 35K57, 80A25}
\keywords{Combustion model; reactant diffusivity; thermal conductivity;
\hfill\break\indent Banach fixed point theorem}

\begin{abstract}
 In this article, we present a strongly coupled quasilinear
 parabolic combustion model with unbounded thermal conductivity
 and reactant diffusivity in arbitrary non-smooth domains.
 A priori estimates are obtained, and the existence of a unique global
 strong solution is proved using a Banach fixed point theorem.
\end{abstract}

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

\section{Introduction}

We consider the  combustion model
\begin{gather}
\frac{\partial u}{\partial t}-\mathop{\rm div}(\phi\nabla u)
= Qwf(u),\quad \text{in } \Omega \times [0,\infty),\label{eqn:temp}\\
\frac{\partial w}{\partial t}-\mathop{\rm div}(\psi\nabla w)
= -wf(u),\quad \text{in }\Omega \times [0,\infty),\label{eqn:conc} \\
\frac{\partial u}{\partial n}
= \frac{\partial w}{\partial n}=0\quad \text{on } \partial \Omega\times
[0,\infty),\label{eqn:neum}\\
u(x, 0) = u_0(x),\quad w(x, 0) = w_0(x),\label{eqn:init}
\end{gather}
where
\begin{equation}
\phi=h^1(x,t,u,w), \quad \psi=h^2(x,t,u,w). \label{eqn:diff}
\end{equation}
Here $Qwf(u)$ and $-wf(u)$ are the reaction kinetics determined by a positive,
uniformly bounded, differentiable and Lipschitz continuous function
$f(u)$. Further, $f'(u)$ is also assumed to be uniformly bounded
and Lipschitz continuous. For our analysis, we shall take
\begin{gather}
0\leq f(u) \leq B,\quad |f(u)-f(\tilde{u})|\leq
L|u-\tilde{u}|,\label{eqn:func}\\
|f'(u)| \leq  B',\quad |f'(u)-f'(\tilde{u})|\leq
L'|u-\tilde{u}|\label{eqn:prif}.
\end{gather}

The system \eqref{eqn:temp}---\eqref{eqn:init} represents a one
step irreversible reaction, reactant $\to$ product. $w(x,t)$ is
assumed to be the mass fraction of the reactant, $1-w(x,t)$, the
mass fraction of the product, $u(x,t)$, the temperature in the
reaction vessel, $\phi(x,t,u,w)$, the thermal conductivity, and
$\psi(x,t,u,w)$, the reactant diffusivity. We assume that $\Omega$
is an open and bounded  arbitrary non-smooth domain in $\mathbb{R}^3$. In
theory, the reactant decomposes at a rate which is proportional to
$f(u)w(x,t)$, where $f(u)$ is the approximate number of molecules
that have sufficient energy for the reaction to begin.

We remark that  \eqref{eqn:temp} and \eqref{eqn:conc} represent an
approximate set of conservation equations for a single-step
irreversible reaction. For the complete set of conservation
equations, we refer to Fitzgibbon and Martin \cite{FM}. We further
refer the reader to Frank-Kamenetskii for more information on
chemical kinetics and combustion \cite{Kamen96}.

The literature on combustion models is quite rich. We mention here
the work of a few authors. Marion \cite{Marion85}, Terman
\cite{Terman90} and Wagner \cite{Wagner86} considered the
existence of travelling wave solutions, taking $\Omega=R$; while
their stability versus instability were analysed by Clavin
\cite{Clavin85} and Sivashinky \cite{Sivashinsky88}. Avrin
\cite{Avrin90,Avrin91} investigated nontravelling wave solutions
to the semilinear, constant coefficient case, obtaining existence
results on both $\Omega=R$ and $\Omega=\mathbb{R}^n$. With $\Omega$ a
bounded $C^\infty$ domain in $\mathbb{R}^n$, Fitzgibbon and Martin
\cite{FM} proved global existence to the system
\eqref{eqn:temp}-\eqref{eqn:init}, with bounded thermal
conductivity $\phi(x,t,u)$ and bounded reactant diffusivity
$\psi(x,t,w)$; both assumed to be
$C^2(\bar\Omega\times[0,\infty)\times R;R)$ functions. Further,
the  reaction kinetics  $Qwf(u)$ and $-wf(u)$ were determined by a
modified   Arrhenius rate function $f(u)$.

In this paper, we shall prove the existence of a unique, global
strong solution to the strongly coupled system
\eqref{eqn:temp}-\eqref{eqn:init}, with strictly positive,
unbounded Lipschitz continuous (with respect to $u$ and $w$)
$\phi$ and $\psi$. We remark that a similar situation arises in
\cite{LL}, where in analysing Reynolds Averaged Navier-Stokes
Problem model, the eddy viscosities in the fluid equation are
assumed to be nondecreasing, smooth and unbounded functions of the
turbulent kinetic energy.

For our analysis, we assume that, for all
$(x,t,u.w),(x,t,\tilde{u},\tilde{w})\in \bar{\Omega}\times
[0,T)\times R\times  R$,
\begin{gather}
\phi=h^1(x,t,u,w) \geq \sigma_1>0,\label{eqn:phi}\\
\psi=h^2(x,t,u,w) \geq \sigma_2>0,\label{eqn:psi}\\
|\phi(x,t,u,w)-\phi(x,t,\tilde{u},\tilde{w})|
\leq A_1(|u-\tilde{u}| + w-\tilde{w|}),\label{eqn:phil} \\
 |\psi(x,t,u,w)-\psi(x,t,\tilde{u},\tilde{w})|
\leq A_2(|u-\tilde{u}| +    |w-\tilde{w|}),\label{eqn:psil}\\
 \|\frac{\partial \phi}{\partial{t}}\|_{L^2[0,T; L^\infty(\Omega)]}
\leq  M,\quad
\|\frac{\partial \psi}{\partial{t}}\|_{L^2[0,T; L^\infty(\Omega)]}\leq M,
\label{eqn:phitltli}\\
\|\phi\|_{L^\infty[0,T; H^2(\Omega)]} \leq  N,\quad \|\psi\|_{L^\infty[0,T; H^2(\Omega)]}\leq
N,\label{eqn:philiht}\\
\|D^3\phi\|_{L^2[0,T; L^2(\Omega)]} \leq  N_1,\quad \|D^3\psi\|_{L^2[0,T; L^2(\Omega)]}\leq
N_1,\label{eqn:philtlt}\\
\|\frac{\partial \phi}{\partial{t}}\|_{L^\infty[0,T; H^1(\Omega)]} \leq  M_1,\quad
\|\frac{\partial \psi}{\partial{t}}\|_{L^\infty[0,T; H^1(\Omega)]}\leq
M_1,\label{eqn:phtliho}\\
\|D^2\partial_t\phi\|_{L^2[0,T; L^2(\Omega)]} \leq  M_2,\quad
\|D^2\partial_t\psi\|_{L^2[0,T; L^2(\Omega)]} \leq M_2.\label{eqn:phltlt}
\end{gather}
All first partial derivatives of $h^l(x,t,u,w)$ ($l=1,2$) with respect to its
arguments are assumed bounded by $\lambda_1$, and Lipschitz continuous (with
respect to $u$ and $w$) with Lipschitz constant $L_1$; while all second
partial derivatives, except $\frac{\partial^2 h^l}{\partial t^2}$, are assumed
to be bounded by $\lambda_2$, Lipschitz continuous (with respect to $u$ and
$w$) with Lipschitz constant $L_2$. For example,
\begin{gather}
|\frac{\partial h^l}{\partial t}| \leq \lambda_1, \quad
|\frac{\partial^2 h^l}{\partial t\partial u}|\leq
\lambda_2,\label{eqn:partial} \\
|\frac{\partial h^l}{\partial t}-\frac{\partial
  \tilde{h}^l}{\partial t}| \leq  L_1(|u-\tilde{u}|+|w-\tilde{w}|),
  \label{eqn:hflipschitz}\\
|\frac{\partial^2 h^l}{\partial u^2}-\frac{\partial^2
  \tilde{h}^l}{\partial u^2}| \leq
L_2(|u-\tilde{u}|+|w-\tilde{w}|),\label{eqn:hslipschitz}
\end{gather}
where
\begin{equation}
\quad\tilde{h}=h^l(x,t,\tilde{u},\tilde{w}).\label{eqn:htilde}
\end{equation}

The letter $C$ written, with or without one or more arguments,
shall represent constants which might differ from one step to the other.
We write the system \eqref{eqn:temp}-\eqref{eqn:init} in the
equivalent form
\begin{gather}
\frac{\partial u}{\partial t}-\mathop{\rm div}(\phi_0\nabla u)
= \mathop{\rm div}(f_u)+g_u,\quad \text{in }\Omega \times [0,\infty),\label{eqn:etemp}\\
\frac{\partial w}{\partial t}-\mathop{\rm div}(\psi_0\nabla w)
= \mathop{\rm div}(f_w)+g_w,\quad \text{in }
\Omega \times [0,\infty),\label{eqn:econc}\\
\frac{\partial u}{\partial n} = \frac{\partial w}{\partial n}=0,\quad
 \text{on }\partial \Omega\times [0,\infty),\label{eqn:eneum}\\
u(x, 0) = u_0(x),\quad w(x, 0) = w_0(x),\label{eqn:einit}
\end{gather}
where
\begin{gather}
\phi = h^1(x,t,u,w),\quad \psi=h^2(x,t,u,w), \label{eqn:ediff}\\
\phi_0 = h^1(x,0,u_0,w_0),\quad\psi_0=h^2(x,0,u_0,w_0), \label{eqn:oediff}\\
f_u = \int_0^t\partial_s\phi ds\nabla u,\quad g_u= Qwf(u),\label{eqn:fu}\\
f_w = \int_0^t\partial_s\psi ds\nabla w,\quad g_w= -wf(u).\label{eqn:fw}
\end{gather}


\section{A priori estimates}

First, we state and prove three useful Lemmas.

\begin{lemma}\label{eqn:first}
Let the conditions \eqref{eqn:phi}-\eqref{eqn:htilde} hold.
Then for $\theta=\phi,\psi$,
\begin{gather}
|\nabla(\theta-\tilde{\theta})|^2
\leq C(\lambda_1,L_1)\bigl(|u-\tilde{u}|^2 + |w-\tilde{w}|^2
 + |\nabla(u-\tilde{u})|^2
 + |\nabla(w-\tilde{w})|^2\bigr),\label{eqn:gradphi}
\\
|\partial_t(\theta-\tilde{\theta})|^2
\leq  C(\lambda_1,L_1)\bigl(|u-\tilde{u}|^2 +
 |w-\tilde{w}|^2 + |\partial_t(u-\tilde{u})|^2
 + |\partial_t(w-\tilde{w})|^2\bigr),\label{eqn:phit}
\\
\begin{aligned}
 |D^2(\theta-\tilde{\theta})|^2
&\leq C(\lambda_1,\lambda_2,L_1,L_2)\big[\big(1+|\nabla\tilde{u}|^2
 +|\nabla\tilde{u}|^4+|\nabla \tilde{u}|^2|\nabla \tilde{w}|^2  \\
&\quad +|D^2\tilde{u}|^2+|\nabla\tilde{w}|^2+|\nabla\tilde{w}|^4
 +|D^2\tilde{w}|^2\big)\big(|u-\tilde{u}|^2+|w-\tilde{w}|^2\big)  \\
&\quad+ \big(1+|\nabla u|^2+|\nabla\tilde{u}|^2+|\nabla
  w|^2+|\nabla\tilde{w}|^2\big)\big(|\nabla(u-\tilde{u})|^2 \\
&\quad+ \nabla(w-\tilde{w})|^2\big)+|D^2(u-\tilde{u})|^2
 +|D^2(w-\tilde{w})|^2\big],\label{eqn:d2phi}
\end{aligned}
\\
\begin{aligned}
|\partial_t\nabla(\theta-\tilde{\theta})|^2
&\leq C\left(\lambda_1,\lambda_2,L_1,L_2\right)
 \big[\big(1+|\partial_t\tilde{u}|^2+|\partial_t\tilde{w}|^2
 +|\nabla\tilde{u}|^2+|\nabla\tilde{w}|^2 \\
&\quad + |\partial_t\tilde{u}|^2|\nabla\tilde{u}|^2+
  |\partial_t\tilde{w}|^2|\nabla\tilde{u}|^2
 +|\partial_t\tilde{u}|^2|\nabla\tilde{w}|^2
 +|\partial_t\tilde{w}|^2|\nabla\tilde{w}|^2\\
&\quad + |\partial_t\nabla\tilde{u}|^2+|\partial_t\nabla\tilde{w}|^2
  \big)\big(|u-\tilde{u}|^2+|w-\tilde{w}|^2\big)+ \big(1+|\partial_t u|^2\\
&\quad  +|\partial_t   w|^2\big)\big(|\nabla(u-\tilde{u})|^2
 +|\nabla(w-\tilde{w})|^2\big)+\big(1+|\nabla\tilde{u}|^2 \\
&\quad +|\nabla\tilde{w}|^2\big)\big(|\partial_t(u-\tilde{u})|^2
 +|\partial_t(w-\tilde{w})|^2+|\partial_t\nabla(u-\tilde{u})|^2 \\
&\quad + |\partial_t\nabla(w-\tilde{w})|^2\big)\big].
\end{aligned}\label{eqn:dtphi}
\end{gather}
\end{lemma}

\begin{proof} The estimates are proved using the chain rule to obtain
  appropriate differential coefficients of $\theta-\tilde{\theta}$,
  rearranging and using the conditions \eqref{eqn:phi}-\eqref{eqn:htilde}.
For brevity, we illustrate with the proof of (\ref{eqn:gradphi}).
\begin{equation}
\begin{aligned}
|\nabla(\theta-\tilde{\theta})|^2
&=  \Big|\frac{\partial h^l}{\partial
 x_i}-\frac{\partial \tilde{h}^l}{\partial
 x_i} +\frac{\partial h^l}{\partial u}\frac{\partial u}{\partial
 x_i}-\frac{\partial \tilde{h}^l}{\partial \tilde{u}}\frac{\partial
 \tilde{u}}{\partial x_i}+\frac{\partial h^l}{\partial w}\frac{\partial
 w}{\partial x_i}-\frac{\partial \tilde{h}^l}{\partial
 \tilde{w}}\frac{\partial \tilde{w}}{\partial x_i}\Big|^2 \\
&= \Big|\frac{\partial h^l}{\partial
 x_i}-\frac{\partial \tilde{h}^l}{\partial
 x_i} +\frac{\partial h^l}{\partial u}\big(\frac{\partial u}{\partial
 x_i}-\frac{\partial
 \tilde{u}}{\partial x_i}\big)+\frac{\partial\tilde{u}}{\partial
 x_i}\big(\frac{\partial h^l}{\partial u}-\frac{\partial
 \tilde{h}^l}{\partial \tilde{u}}\big) \\
&\quad + \frac{\partial h^l}{\partial w}\big(\frac{\partial w}{\partial
 x_i}-\frac{\partial \tilde{w}}{\partial x_i}\big)
 +\frac{\partial\tilde{w}}{\partial
 x_i}\big(\frac{\partial h^l}{\partial w}-\frac{\partial
 \tilde{h}^l}{\partial \tilde{w}}\big)\Big|^2 \\
&\leq C(\lambda_1,L_1)\bigl(|u-\tilde{u}|^2 +
 |w-\tilde{w}|^2 +|\nabla(u-\tilde{u})|^2
 +|\nabla(w-\tilde{w})|^2\bigr).
\end{aligned} \label{eqn:pgradphi}
\end{equation}
\end{proof}

\begin{lemma}\label{eqn:lemma}
Let $\Omega\in \mathbb{R}^3$, be open and bounded set with a $C^1$ boundary;
and $u,\tilde{u},v,\tilde{v},w\in H^1(\Omega)$. Then there exists $\epsilon>0$,
$C=C(\Omega)$ such that
\begin{gather}
\int_\Omega |u|^2|v|^2dx \leq C\|u\|_{H^1(\Omega)}^2
\Big(\frac{1}{\epsilon}\|v\|_{L^2(\Omega)}^2 +
\epsilon\|v\|_{H^1(\Omega)}^2\Big),\label{eqn:athesis}
\\
\begin{aligned}
\int_\Omega |uv-\tilde{u}\tilde{v}|^2dx
&\leq C\|v\|_{H^1(\Omega)}^2\Big(\frac{1}{\epsilon}
\|u-\tilde{u}\|_{L^2(\Omega)}^2
+ \epsilon\|u-\tilde{u}\|_{H^1(\Omega)}^2\Big)  \\
&\quad + C\|\tilde{u}\|_{H^1(\Omega)}^2\Big(\frac{1}{\epsilon}
\|v-\tilde{v}\|_{L^2(\Omega)}^2 +
\epsilon\|v-\tilde{v}\|_{H^1(\Omega)}^2\Big),
\end{aligned}\label{eqn:bthesis}
\\
\int_\Omega |u|^2|v|^2|w|^2dx \leq C\|u\|_{H^1(\Omega)}^2
\|v\|_{H^1(\Omega)}^2\|w\|_{H^1(\Omega)}^2.
\label{eqn:cthesis}
\end{gather}
\end{lemma}

\begin{proof}
Estimates (\ref{eqn:athesis}) and (\ref{eqn:bthesis}) are proved in
  \cite{Sanni04}. We therefore render only the proof of
  (\ref{eqn:cthesis}). By H\"older and Sobolev's inequalities, we calculate
\begin{equation}
\begin{aligned}
\int_\Omega |u|^2|v|^2|w|^2dx
&\leq \Big(\int_\Omega|u|^6dx\Big)^{1/3}\Big(\int_\Omega|v|^6dx\Big)^{1/3}
\Big(\int_\Omega|w|^6dx\Big)^{1/3} \\
&\leq C\|u\|_{H^1(\Omega)}^2\|v\|_{H^1(\Omega)}^2\|w\|_{H^1(\Omega)}^2.
\end{aligned}
\end{equation}
\end{proof}

\begin{lemma}\label{eqn:lemma3}
Let $u_0\in H^3(\Omega)$; and \eqref{eqn:temp}, \eqref{eqn:conc},
\eqref{eqn:init} and \eqref{eqn:diff} be satisfied. Then we have
the estimate
\begin{equation}
\begin{aligned}
&\|\partial_t u(x,0)\|_{H^1(\Omega)}^2+\|\partial_t w(x,0)\|_{H^1(\Omega)}^2  \\
&\leq C\Big(\|u_0\|_{ H^3(\Omega)}^2+\|w_0\|_{
H^3(\Omega)}^2+\|u_0\|_{
H^2(\Omega)}^2\|w_0\|_{H^1(\Omega)}^2\Big)
\end{aligned}\label{eqn:iestimate}
\end{equation}
where $C=C(Q,\Omega,B,B',N,N_1)$.
\end{lemma}

\begin{proof}
1. Taking \eqref{eqn:temp}, \eqref{eqn:conc}, and \eqref{eqn:diff} at
 $t=0$ yield
\begin{gather}
\frac{\partial u_0}{\partial t}-\mathop{\rm div}(\phi_0\nabla u_0)
 = Qw_0f(u_0),\quad \text{in }\Omega,\label{eqn:itemp}\\
\frac{\partial w_0}{\partial t}-\mathop{\rm div}(\psi_0\nabla w_0)
 = -w_0f(u_0),\quad \text{in }\Omega ,\label{eqn:iconc} \\
u(x, 0) = u_0(x),\quad w(x, 0) = w_0(x),\label{eqn:iinit}
\end{gather}
where $\phi_0=h^1(x,0,u_0,w_0)$,
$\psi_0=h^2(x,0,u_0,w_0)$. %\label{eqn:idiff}

 2. Using (\ref{eqn:itemp}), we estimate
\begin{equation}
\begin{aligned}
\int_\Omega |\partial_t u_0|^2dx
&= \int_\Omega |\mathop{\rm div}\left(\phi_0\nabla u_0\right)+Qw_0f(u_0)|^2dx
 \\
&= \int_\Omega |\nabla \phi_0.\nabla u_0+\phi_0\Delta u_0+Qw_0f(u_0)|^2dx\\
& \leq C(\Omega)\Big(\|\phi_0\|_{H^2(\Omega)}^2\|u_0\|_{H^2(\Omega)}^2
 + \|\phi_0\|_{C^{0,\frac{1}{2}}(\Omega)}^2\|u_0\|_{H^2(\Omega)}^2\\
&\quad +Q^2|f(u_0)|^2\|w_0\|_{L^2(\Omega)}^2\Big) \\
&\leq C(Q,\Omega,B,N)\Big(\|u_0\|_{H^2(\Omega)}^2
 +\|w_0\|_{L^2(\Omega)}^2\Big),
\end{aligned}\label{eqn:iestimate1}
\end{equation}
where we have used
\eqref{eqn:cthesis} of Lemma \ref{eqn:lemma} and
 the Sobolev's embedding
$\|\phi_0\|_{C^{0,\frac{1}{2}}(\Omega)}\leq
C(\Omega)\|\phi_0\|_{H^2(\Omega)}$.

 3. Taking the gradient of (\ref{eqn:itemp}), we obtain
the estimate
\begin{equation}
\begin{aligned}
\int_\Omega |\nabla(\partial_t u_0)|^2dx
&=\int_\Omega|\nabla\left(\mathop{\rm div}(\phi_0\nabla u_0)\right)
 +\nabla(Qw_0f(u_0))|^2dx \\
&=\int_\Omega \big|\nabla^2\phi_0.\nabla u_0+\nabla \phi_0.\nabla^2 u_0
 +\nabla\phi_0\Delta u_0+\phi_0\nabla\left(\Delta u_0\right) \\
&\quad + Q\nabla w_0f(u_0)+w_0\nabla u_0f'(u_0)\big|^2dx \\
&\leq \int_\Omega C(\Omega)\Big(\|\phi_0\|_{H^3(\Omega)}^2\|u_0\|_{H^2(\Omega)}^2
 +\|\phi_0\|_{H^2(\Omega)}^2\|u_0\|_{H^3(\Omega)}^2  \\
&\quad +\|\phi_0\|_{C^{0,\frac{1}{2}}(\Omega)}^2\|u_0\|_{H^3(\Omega)}^2
 +Q^2|f(u)|^2\|w_0\|_{H^2(\Omega)}^2 \\
&\quad +Q^2|f'(u_0)|^2\|u_0\|_{H^2(\Omega)}^2\|w_0\|_{H^1(\Omega)}^2\Big)
\quad \text{(using (\ref{eqn:cthesis}))}\\
& \leq C\Big(\|u_0\|_{ H^3(\Omega)}^2+\|w_0\|_{
H^2(\Omega)}^2+\|u_0\|_{
H^2(\Omega)}^2\|w_0\|_{H^1(\Omega)}^2\Big)
\end{aligned}\label{eqn:iestimate2}
\end{equation}
where $C=C(Q,\Omega,B,B',N,N_1)$, and we have used the Sobolev's
embedding $\|\phi_0\|_{C^{0,\frac{1}{2}}(\Omega)}\leq
C(\Omega)\|\phi_0\|_{H^2(\Omega)}$.
 Combining (\ref{eqn:iestimate1}) and (\ref{eqn:iestimate2}) yields
(\ref{eqn:iestimate}),
\begin{equation}
\|\partial_t u_0\|_{H^1(\Omega)}^2\leq C\Big(\|u_0\|_{
H^3(\Omega)}^2+\|w_0\|_{ H^2(\Omega)}^2+\|u_0\|_{
H^2(\Omega)}^2\|w_0\|_{H^1(\Omega)}^2\Big)\label{eqn:iestimate3}
\end{equation}
where $C=(Q,\Omega,B,B',N,N_1)$

 4.
Using (\ref{eqn:iconc}), we have an analogous estimate to
(\ref{eqn:iestimate3}), namely
\begin{equation}
\|\partial_t w_0\|_{H^1(\Omega)}^2\leq C\left(\|w_0\|_{
H^3(\Omega)}^2+\|w_0\|_{ H^2(\Omega)}^2+\|u_0\|_{
H^2(\Omega)}^2\|w_0\|_{H^1(\Omega)}^2\right)\label{eqn:iestimate4}
\end{equation}
where $C=(\Omega,B,B',N,N_1)$. Hence, a combination of
(\ref{eqn:iestimate3}) and  (\ref{eqn:iestimate4}) gives
(\ref{eqn:iestimate}).
\end{proof}

\begin{theorem}\label{thm:main}
Let $u_o,w_o\in H^3(\Omega)$. Suppose there exists $(u,w)$ which
satisfies the system \eqref{eqn:etemp}-\eqref{eqn:fw}. Then
$(u,w)\in X \times X$, where
\begin{equation}
X={L^\infty[0,T; H^2(\Omega)]} \cap {L^2[0,T; H^3(\Omega)]} \cap W^{1,\infty}[0,T;{H^1(\Omega)}] \cap {H^1[0,T; H^2(\Omega)]} \label{eqn:space}
\end{equation}
and we have the estimate
\begin{equation}
\begin{aligned}
&\sup_{[0,T]}\left(\|u\|^2_{H^2(\Omega)}+\|w\|^2_{H^2(\Omega)}
+\|\partial_t u\|^2_{H^1(\Omega)}+\|\partial_t
  w\|^2_{H^1(\Omega)}\right) \\
&\quad +\int_0^T \int_\Omega|D^3u|^2\,dx\,dt +\int_0^T \int_\Omega|D^3w|^2\,dx\,dt
+ \int_0^T \int_\Omega|D^2(\partial_t u)|^2\,dx\,dt \\
&\quad + \int_0^T \int_\Omega|D^2(\partial_t w)|^2\,dx\,dt  \\
& \leq
  C(Q,\Omega,B,B',N)\Big(\|u_0\|^2_{H^2(\Omega)}\|w_0\|^2_{H^1(\Omega)}
  +\|u_0\|^2_{H^3(\Omega)} \\
&\quad +\|w_0\|^2_{H^3(\Omega)}\Big)e^{CT(C_1 + 1)} =: \Lambda.
\end{aligned} \label{eqn:estimate}
\end{equation}
where $C=C(\Omega,Q,L,L',L_1,L_2,\lambda_1,\lambda_2,M,N,N_1,M_1,M_2)$ and
$C_1$ is the bound in \eqref{eqn:ceterms}.
\end{theorem}

\begin{proof} 1. We multiply \eqref{eqn:etemp} by $u$ and integrate by parts
  over $\Omega$; and use (\ref{eqn:phi}) and (\ref{eqn:eneum}) to obtain
\begin{align*}
&\frac{1}{2}\frac{d}{dt}\Big(\int_\Omega u^2dx\Big)
 + \sigma_1\int_\Omega|\nabla u|^2dx \\
&\leq  -\int_\Omega\nabla uf_udx+\int_\Omega ug_udx  \\
&\leq \frac{\sigma_1}{2}\int_\Omega|\nabla u|^2dx+\frac{1}{2\sigma_1}
\int_\Omega f_u^2dx+  \frac{1}{2}\int_\Omega u^2dx+\frac{1}{2}\int_\Omega g_u^2dx
\end{align*}
or
\begin{equation}
\frac{d}{dt}\Big(\int_\Omega u^2dx\Big)+\int_\Omega|\nabla u|^2dx\leq C(\sigma_1)
\Big(\int_\Omega u^2dx+\int_\Omega f_u^2dx+\int_\Omega g_u^2dx\Big).\label{eqn:fephi}
\end{equation}
Similarly, we may use (\ref{eqn:psi}), (\ref{eqn:econc}), (\ref{eqn:eneum})
to obtain
\begin{equation}
\frac{d}{dt}\Big(\int_\Omega w^2dx\Big)+\int_\Omega|\nabla
w|^2dx\leq C(\sigma_2)\Big(\int_\Omega w^2dx+\int_\Omega f_w^2dx
+\int_\Omega g_w^2dx\Big).
\label{eqn:fepsi}
\end{equation}

2. We represent the ith differential quotient of size $h$ as
   $$
D_i^h\zeta=\frac{\zeta(x+he_i,t)-\zeta(x,t)}{h},\quad (i=1,\dots ,n)
$$
for
   $h\in R$, $0<|h|$; and define
   $D^h\zeta=(D_1^h\zeta,\dots ,D_n^h\zeta)$. Hence, we apply $D^h$ on
   \eqref{eqn:etemp} to get
\begin{equation}
\frac{\partial}{\partial
  t}(D^hu)-\mathop{\rm div}\left[\phi_0^h\nabla(D^hu)
\right]
=\mathop{\rm div}\left(D^hf_u+D^h\phi_0\nabla u\right)+D^hg_u\label{eqn:qphi}
 \end{equation}
Multiplying this equality  by   $D^hu$, integrating over $\Omega$ and
applying Young's inequality, we obtain
\begin{equation}
\begin{aligned}
&\frac{\partial}{dt}\Big(\int_\Omega|D^hu|^2dx\Big)+
\int_\Omega|\nabla(D^hu)|^2dx\\
&\leq C(\sigma_1)\Big(\epsilon\int_\Omega|D^{-h}(D^hu)|^2dx
+ \epsilon\int_\Omega|\nabla(D^hu)|^2dx\\
&\quad +\frac{1}{\epsilon}\int_\Omega|D^hf_u+D^h\phi_0\nabla
u|^2dx+\frac{1}{\epsilon}\int_\Omega g_u^2dx \\
&\quad +\int_{\partial \Omega}D^hu.\nabla(D^hu)\mathbf{n}dS
+\int_{\partial \Omega}D^hu.\mathbf{n}Qwf(u)dS\Big)
\end{aligned}\label{eqn:qephi}
\end{equation}
Similarly, we have
\begin{equation}
\begin{aligned}
&\frac{\partial}{dt}\Big(\int_\Omega|D^hw|^2dx\Big)
+ \int_\Omega|\nabla(D^hw)|^2dx\\
&\leq C(\sigma_2)\Big(\epsilon\int_\Omega|D^{-h}(D^hw)|^2dx
 + \epsilon\int_\Omega|\nabla(D^hw)|^2dx\\
&\quad +\frac{1}{\epsilon}\int_\Omega|D^hf_w+D^h\phi_0\nabla
w|^2dx+\frac{1}{\epsilon}\int_\Omega g_w^2dx \\
&\quad+\int_{\partial \Omega}D^hw.\nabla(D^hw)\mathbf{n}dS
-\int_{\partial \Omega}D^hw.\mathbf{n}wf(u)dS\Big)
\end{aligned}\label{eqn:qepsi}
\end{equation}

 3. Using \ref{eqn:fephi}, \ref{eqn:fepsi}, \ref{eqn:qephi} and
   \ref{eqn:qepsi}, we deduce, after taken the limit as $h\to 0$,
   chosen $\epsilon>0$ sufficiently small and simplifying, that
\begin{equation}
\begin{aligned}
&\frac{d}{dt}\Big(\int_\Omega u^2dx+\int_\Omega w^2dx+\int_\Omega|D_lu|^2dx
+ \int_\Omega|D_lw|^2dx\Big)\\
&\quad +\int_\Omega|\nabla(D_lu)|^2dx+
  \int_\Omega|\nabla(D_lw)|^2dx \\
&\leq C\Big(\int_\Omega u^2dx +\int_\Omega
w^2dx+\int_\Omega|f_u|^2dx+\int_\Omega|g_u|^2dx+\int_\Omega|f_w|^2dx \\
&\quad +\int_\Omega|g_w|^2dx+ \int_\Omega|D_lf_u + \nabla\phi_0\nabla u|^2dx
+ \int_\Omega|D_lf_w + \nabla\phi_0\nabla w|^2dx\Big),
\end{aligned}\label{eqn:cphipsi}
\end{equation}
where $D_lq=\lim_{h\to 0} D^hq$ and
$C=C(\sigma_1,\sigma_2)$.


4. Using (\ref{eqn:phitltli}), the definitions (\ref{eqn:fu}) and
   \eqref{eqn:fw}, we use H\"older inequality and (\ref{eqn:athesis}) of Lemma
   \ref{eqn:lemma} to evaluate some terms of (\ref{eqn:cphipsi}), namely
\begin{equation}
\begin{aligned}
&\int_\Omega|f_u|^2dx+\int_\Omega|g_u|^2dx+ \int_\Omega|D_lf_u +
 \nabla\phi_0\nabla u|^2dx +\int_\Omega|f_w|^2dx \\
& +\int_\Omega|g_w|^2dx+ \int_\Omega|D_lf_w + \nabla\phi_0\nabla w|^2dx \\
&\leq C(\Omega,M,N,B,Q)\Big[\int_\Omega w^2dx+
 \big(\frac{1}{T}+T\big)\Big(\int_\Omega|D_lu|^2dx \\
&\quad +\int_\Omega|D_lw|^2dx\Big)+ T\int_\Omega|\nabla (D_lu)|^2dx+T\int_\Omega|\nabla
(D_lw)|^2dx\Big].
\end{aligned}\label{eqn:aeterms}
\end{equation}
Using (\ref{eqn:aeterms}) in (\ref{eqn:cphipsi}), choosing $T$ sufficiently
small and simplifying we obtain
\begin{equation}
\begin{aligned}
&\frac{d}{dt}\Big(\int_\Omega u^2dx+\int_\Omega w^2dx+\int_\Omega|D_lu|^2dx +
  \int_\Omega|D_lw|^2dx\Big) \\
& +\int_\Omega|\nabla(D_lu)|^2dx+ \int_\Omega|\nabla(D_lw)|^2dx \\
&\leq C\Big(\int_\Omega u^2dx +\int_\Omega w^2dx+\int_\Omega|D_lu|^2dx +
  \int_\Omega|D_lw|^2dx\Big),
\end{aligned}\label{eqn:beterms}
\end{equation}
where  $C=C(\sigma_1,\sigma_2,\Omega,M,N,B,Q)$.
By applying the Gronwall's inequality and carrying out some simple
calculations, we deduce from (\ref{eqn:beterms}) the estimate
\begin{equation}
\begin{aligned}
&\sup_{[0,T]}\int_\Omega u^2dx+\sup_{[0,T]}\int_\Omega w^2dx
+\sup_{[0,T]}\int_\Omega|D_lu|^2dx +
 \sup_{[0,T]} \int_\Omega|D_lw|^2dx  \\
&+\int_0^T \int_\Omega|\nabla(D_lu)|^2\,dx\,dt+
  \int_0^T \int_\Omega|\nabla(D_lw)|^2\,dx\,dt \\
&\leq C\Big(\|u_0\|_{H^1(\Omega)}^2+\|w_0\|_{H^1(\Omega)}^2\Big)
 =: C_1 < \infty,
\end{aligned} \label{eqn:ceterms}
\end{equation}
where $C=C(\sigma_1,\sigma_2,\Omega,M,N,B,Q,T)$.

 5. Let $Y={L^\infty[0,T; L^2(\Omega)]}\cap{L^2[0,T; H^1_0(\Omega)]}$. For some $h>0$, we infer
from \eqref{eqn:ceterms} that
\begin{equation}
\begin{aligned}
\|D^h(u,w)\|_{Y\times Y}^2
&:= \sup_{[0,T]}\int_\Omega|D^hu|^2dx + \sup_{[0,T]}
  \int_\Omega|D^hw|^2dx  \\
&\quad +\int_0^T \int_\Omega|\nabla(D^hu)|^2\,dx\,dt
 +\int_0^T \int_\Omega|\nabla(D^hw)|^2\,dx\,dt \\
&\leq C_1
\end{aligned}\label{eqn:determs}
\end{equation}
Estimate (\ref{eqn:determs}) implies
\begin{equation}
\sup_{h}\|D^h(u,w)\|_{Y\times Y}<\infty. \label{eqn:eeterms}
\end{equation}
The space $Y\times Y$ is reflexive. Therefore, there exists by the weak
compactness theorem \cite{Evans98}, a subsequence $h_k\to 0$, and a
function $(U,W)\in Y\times Y$ such that
\begin{equation}
D^{h_k}(u,w)\rightharpoonup (U,W).\label{eqn:geterms}
\end{equation}
Thus, given a smooth function $\zeta \in {C_c^\infty(\Omega\times[0,T))}$,
we calculate
\begin{align*}
&\int_0^T \int_\Omega(u,w)\nabla \zeta \,dx\,dt\\
&=\lim_{h_k\to 0}\int_0^T \int_\Omega(u,w)D^{-h_k} \zeta\,dx\,dt
 - \lim_{h_k\to 0}\int_0^T \int_\Omega D^{h_k}(u,w)\zeta \,dx\,dt \\
&= -\int_0^T \int_\Omega(U,W)\zeta \,dx\,dt
\end{align*}
Thus $(U,W)=(\nabla u,\nabla w)$ in the weak sense, and so
$(\nabla u,\nabla w)\in Y\times Y$.
Hence, by the above deduction, \eqref{eqn:ceterms} implies
\begin{equation}
\begin{aligned}
&\sup_{[0,T]}\|u\|_{H^1(\Omega)}^2+\sup_{[0,T]}\|w\|_{H^1(\Omega)}^2
 +\int_0^T \int_\Omega|D^2u|^2\,dx\,dt+
  \int_0^T \int_\Omega|D^2w|^2\,dx\,dt. \\
&\leq C\left(\|u_0\|_{H^1(\Omega)}^2+\|w_0\|_{H^1(\Omega)}^2\right)
 =: C_1 < \infty
\end{aligned}\label{eqn:heterms}
\end{equation}
where $C=C(\sigma_1,\sigma_2,\Omega,M,N,B,Q,T)$

 6. We now set out to obtain higher a priori estimates needed to complete
part  of the proof of Theorem (\ref{thm:main}). In view of estimates
(\ref{eqn:heterms}), we may take the limit as $h\to 0$ in (\ref{eqn:qphi})
and apply   $D^h$ on the ensuing equation to obtain
\begin{equation}
\begin{aligned}
&\frac{\partial}{\partial t}\left(\nabla(D^hu)\right) -
\mathop{\rm div}\left(\phi_0^hD^2(D^hu)\right) \\
&= \mathop{\rm div}\big[\nabla\left(D^hf_u\right)
 +\nabla\left(D^h\phi_0\right)\nabla   u
 + \nabla \phi_0\nabla(D^hu)\big]+D^h(\nabla g)
\end{aligned} \label{eqn:hqphi}
\end{equation}
We take the dot product of (\ref{eqn:hqphi}) with $\nabla(D^hu)$ to estimate
\begin{equation}
\begin{aligned}
&\frac{d}{dt}\Big(\int_\Omega\big|\nabla(D^hu)\big|^2dx\Big)
+ \int_\Omega|D^2(D^hu)|^2dx \\
&\leq C(\sigma_1)\Big(\epsilon\int_\Omega\big|D^2(D^hu)\big|^2
+\frac{1}{\epsilon}\int_\Omega\Big|\nabla\left(D^hf_u\right)
+\nabla\left(D^h\phi_0\right)\nabla u \\
&\quad +\nabla\phi_0\nabla(D^hu)\big|^2dx
 +\frac{1}{\epsilon}\int_\Omega\Big|D^hg_u\big|^2dx\Big)
\end{aligned}   \label{eqn:hephi}
\end{equation}
Similarly, we have
\begin{equation}
\begin{aligned}
&\frac{d}{dt}\Big(\int_\Omega\big|\nabla(D^hw)\big|^2dx\Big) +
\int_\Omega|D^2(D^hw)|^2dx \\
& \leq C(\sigma_1)\big(\epsilon\int_\Omega\big|D^2(D^hw)\big|^2
+\frac{1}{\epsilon}\int_\Omega\Big|\nabla\left(D^hf_w\right)
 +\nabla\left(D^h\psi_0\right)\nabla u \\
&\quad +\nabla\psi_0\nabla(D^hw)\Big|^2dx
 +\frac{1}{\epsilon}\int_\Omega\big|D^hg_w\big|^2dx\big)
\end{aligned}  \label{eqn:hepsi}
\end{equation}
Combining the above inequality with (\ref{eqn:hephi}), taking the limit
as $h\to 0$, choosing $\epsilon > 0$ sufficiently small, and simplifying,
we deduce
\begin{equation}
\begin{aligned}
&\frac{d}{dt}\Big(\int_\Omega|\nabla (D_lu)|^2dx+\int_\Omega|\nabla
(D_lw)|^2dx\Big)
+\int_\Omega|D^2(D_lu)|^2dx +\int_\Omega|D^2(D_lw)|^2dx \\
&\leq C(\sigma_1,\sigma_2)\Big[\int_\Omega|\nabla (D_lf_u) +
D^2\phi_0\nabla u + \nabla\phi_0\nabla(D_lu)|^2dx +
\int_\Omega|D_lg_u|^2dx\\
&\quad + \int_\Omega|\nabla (D_lf_w)
+ D^2\psi_0\nabla w + \nabla\psi_0\nabla(D_lw)|^2dx
+ \int_\Omega|D_lg_w|^2dx\Big]
\end{aligned}\label{eqn:chepsi}
\end{equation}
Note that this inequality is analogous to (\ref{eqn:cphipsi}).
Hence, by steps similar to steps (4) and (5), we obtain
\begin{equation}
\begin{aligned}
&\sup_{[0,T]}\|D^2u\|_{L^2(\Omega)}^2+\sup_{[0,T]}\|D^2w\|_{L^2(\Omega)}^2
+\int_0^T \int_\Omega|D^3u|^2\,dx\,dt
+\int_0^T \int_\Omega|D^3w|^2\,dx\,dt. \\
&\leq C\Big(\|D^2u_0\|_{L^2(\Omega)}^2+\|D^2w_0\|_{L^2(\Omega)}^2+C_1\Big)
=: C_2< \infty,
\end{aligned} \label{eqn:cheterms}
\end{equation}
where $C=C(\sigma_1,\sigma_2,\Omega,M,N,N_1,B,,B',Q,T)$ and
 $C_1$ is the bound in (\ref{eqn:heterms}).

 7. The last higher a priori estimates are obtained in this step. Define
\begin{equation}
\partial^r:=\frac{\theta(x,t)-\theta(x,t-r)}{r},\quad
\partial_l\theta:=\lim_{r\to 0}\partial^r \label{eqn:tquotient}
\end{equation}
for all $r>0$ such that $t-r\in [0,T)$.
Applying $\partial_l$ on the system \eqref{eqn:etemp}-\eqref{eqn:fw} yields
\begin{gather}
\frac{\partial}{\partial t}(\partial_lu)-\mathop{\rm div}
\left(\phi_0\nabla (\partial_lu)\right)
= \mathop{\rm div}(\partial_lf_u)+\partial_lg_u,\quad \text{in }\Omega \times [0,\infty),\label{eqn:letemp}\\
\frac{\partial }{\partial t}(\partial_lw)-\mathop{\rm div}
\left(\psi_0\nabla (\partial_lw)\right)
= \mathop{\rm div}(\partial_lf_w)+\partial_lg_w,\quad \text{in }
\Omega \times [0,\infty),\label{eqn:leconc}\\
\frac{\partial}{\partial n}(\partial_lu)
=\frac{\partial}{\partial n}(\partial_lw)=0\quad\text{on }\partial \Omega\times
[0,\infty),\label{eqn:leneum}\\
\partial_lu(x, 0)=\partial_lu_0(x),\quad
 \partial_lw(x, 0) = \partial_lw_0(x),\label{eqn:leinit}
\end{gather}
where
\begin{gather*}
\phi = h^1(x,t,u,w),\quad \psi=h^2(x,t,u,w), \label{eqn:lediff}\\
\quad \phi_0=h^1(x,t,u_0,w_0),\quad\psi_0=h^2(x,t,u_0,w_0), \label{eqn:loediff}\\
f_u = \int_0^t\partial_s\phi ds\nabla u,\quad g_u= Qwf(u),\label{eqn:flu}\\
f_w = \int_0^t\partial_s\psi ds\nabla w,\quad g_w= -wf(u).\label{eqn:flw}
\end{gather*}
Note that the system (\ref{eqn:letemp})-(\ref{eqn:flw}) is analogous to
\eqref{eqn:etemp}-\eqref{eqn:fw}. We therefore have an analogue to
\ref{eqn:cphipsi}, namely
\begin{equation}
\begin{aligned}
&\frac{d}{dt}\Big(\|\partial_lu\|_{H^1(\Omega)}^2
+\|\partial_lw\|_{H^1(\Omega)}^2\Big)
+\int_\Omega|D^2(\partial_lu)|^2dx
+\int_\Omega|D^2(\partial_lw)|^2dx \\
&\leq   C(\sigma_1,\sigma_2)\Big(\|\partial_lu\|_{H^1(\Omega)}^2
 +\|\partial_lw\|_{H^1(\Omega)}^2+\int_\Omega|\partial_lf_u|^2dx
 +\int_\Omega|\partial_lg_u|^2dx \\
&\quad +\int_\Omega|\partial_lf_w|^2dx+\int_\Omega|\partial_lg_w|^2dx +
  \int_\Omega|\nabla(\partial_lf_u)
  +\nabla\phi_0\nabla(\partial_lu)|^2dx \\
&\quad +\int_\Omega|\nabla(\partial_lf_w)
  +\nabla\psi_0\nabla(\partial_lw)|^2dx \Big)
\end{aligned} \label{eqn:chphipsi}
\end{equation}
Hence, by calculations and deductions similar to that of steps (4)-(5),
we obtain the estimates
\begin{equation}
\begin{aligned}
&\sup_{[0,T]}\|\partial_tu\|_{H^1(\Omega)}^2
+ \sup_{[0,T]}\|\partial_tw\|_{H^1(\Omega)}^2\\
& + \int_0^T \int_\Omega|D^2(\partial_tu)|^2\,dx\,dt
+ \int_0^T \int_\Omega|D^2(\partial_tu)|^2\,dx\,dt \\
& \leq C\big(\|u_0\|_{H^3(\Omega)}^2 +
\|w_0\|_{H^3(\Omega)}^2+\|u_0\|_{H^2(\Omega)}^2\|w_0\|_{H^1(\Omega)}^2
+ C_1+C_2\big) \\
&=: C_3 < \infty,
\end{aligned}\label{eqn:thphipsi}
\end{equation}
where $C=C(\sigma_1,\sigma_2,M,N,M_1,M_2,B,B',Q,\Omega,T)$,
$C_1$, and $C_2$ are the bounds in \eqref{eqn:ceterms} and
(\ref{eqn:cheterms}) respectively. Here we used the estimate (\ref{eqn:iestimate}) in Lemma \ref{eqn:lemma3}.

8. Combining estimates (\ref{eqn:heterms}), (\ref{eqn:cheterms}) and
   (\ref{eqn:thphipsi}) we complete the proof of Theorem \ref{thm:main}.
\end{proof}

\section{Existence of Solutions}

In this section, we shall prove the existence for a global unique strong
solution to \eqref{eqn:etemp}-\eqref{eqn:fw} in a subset of the space
$X\times X$, which is equipped with the norm
\begin{equation}
\begin{aligned}
\|(u,w)\|_{X\times X}
&= \Big[\sup_{[0,T]}\Big(\|u\|^2_{H^2(\Omega)}+\|w\|^2_{H^2(\Omega)}+\|\partial_t u\|^2_{H^1(\Omega)}+\|\partial_t
  w\|^2_{H^1(\Omega)}\Big)\\
&\quad +\int_0^T \int_\Omega|D^3u|^2\,dx\,dt
  +\int_0^T \int_\Omega|D^3w|^2\,dx\,dt\\
&\quad + \int_0^T \int_\Omega|D^2(\partial_t u)|^2\,dx\,dt
+ \int_0^T \int_\Omega|D^2(\partial_t w)|^2\,dx\,dt\Big]^{1/2}
\end{aligned}
\end{equation}

\begin{theorem} \label{thm3.1}
Let $u_0,w_0 \in H^3(\Omega)$. Then, there exists a global unique
strong solution to  \eqref{eqn:etemp}-\eqref{eqn:fw}.
\end{theorem}

\begin{proof}
1. The corresponding fixed point argument system to
\eqref{eqn:etemp}-\eqref{eqn:fw} is
\begin{gather}
\frac{\partial \chi}{\partial t}-\mathop{\rm div}(\phi_0\nabla \chi)
= \mathop{\rm div}(f_u)+g_u,\quad \text{in }\Omega \times [0,\infty),
 \label{eqn:aetemp}\\
\frac{\partial \tau}{\partial t}-\mathop{\rm div}(\psi_0\nabla \tau)
= \mathop{\rm div}(f_w)+g_w,\quad \text{in }
\Omega \times [0,\infty),\label{eqn:aeconc}\\
\frac{\partial \chi}{\partial n}
= \frac{\partial \chi}{\partial n}=0,\quad\text{on }\partial \Omega\times
[0,\infty),\label{eqn:aeneum}\\
\chi(x, 0) = u_0(x),\quad \tau(x, 0) = w_0(x),\label{eqn:aeinit}
\end{gather}
where
\begin{gather}
\phi = h^1(x,t,u,w),\quad \psi=h^2(x,t,u,w), \label{eqn:aediff}\\
\phi_0 = h^1(x,t,u_0,w_0),\quad \psi_0=h^2(x,t,u_0,w_0). \label{eqn:aoediff}\\
f_u=\int_0^t\partial_s\phi ds\nabla u,\quad g_u= Qwf(u),\label{eqn:afu}\\
f_w=\int_0^t\partial_s\psi ds\nabla w,\quad g_w= -wf(u).\label{eqn:afw}
\end{gather}

2. Define $A:X\times X\to X\times X$ by setting
   $A[(u,w)]=(\chi,\tau)$. We shall prove that if $T>0$ is small enough, then
   $A$ is a contraction mapping. We choose $(u,w), (\tilde u,\tilde w)\in
   X\times X$, and define $A[(u,w)]=(\chi,\tau),\
   A[(\tilde{u},\tilde{w})]=(\tilde{\chi},\tilde{\tau})$. For two solutions
   $(\chi,\tau)\ \text{and}\ (\tilde{\chi},\tilde{\tau})$ of
   (\ref{eqn:aetemp})- (\ref{eqn:afw}), we have
\begin{gather}
\frac{\partial }{\partial t}\left(\chi-\tilde{\chi}\right)
-\mathop{\rm div}\left(\phi_0\nabla\left(\chi-\tilde{\chi}\right)\right)
=\mathop{\rm div}\big(f_u-\tilde{f}_{\tilde{u}}\big)
+ g_u-\tilde{g}_{\tilde{u}},\quad \text{in }\Omega\times[0,\infty)
\label{eqn:detemp}\\
\frac{\partial }{\partial t} \left(\tau-\tilde{\tau}\right)
-\mathop{\rm div}\left(\phi_0\nabla\left(\tau-\tilde{\tau}\right)\right)
=\mathop{\rm div}\big(f_w-\tilde{f}_{\tilde{w}}\big)
+ g_w-\tilde{g}_{\tilde{w}}, \quad\text{in }\Omega\times[0,\infty)
\label{eqn:deconc}\\
\frac{\partial}{\partial  n}\left(\chi-\tilde{\chi}\right)
=\frac{\partial}{\partial  n}\left(\tau-\tilde{\tau}\right)
=0,\quad\text{on }\partial \Omega\times [0,\infty)\label{eqn:deneum},\\
\left(\chi-\tilde{\chi}\right)(x,0) = 0,\quad
\left(\tau-\tilde{\tau}\right)(x,0) = 0,\label{eqn:deinit}
\end{gather}
where
\begin{gather}
\phi=h^1(x,t,u,w),\quad \psi=h^2(x,t,u,w), \label{eqn:dediff}\\
\tilde{\phi}=h^1(x,t,\tilde{u},\tilde{w}),\quad
 \tilde{\psi}=h^2(x,t,\tilde{u},\tilde{w}), \label{eqn:tdediff}\\
\phi_0=h^1(x,t,u_0,w_0),\quad \psi_0=h^2(x,t,u_0,w_0), \label{eqn:doediff}\\
f_u=\int_0^t\partial_s\phi ds\nabla u,\quad g_u= Qwf(u),\label{eqn:dfu}\\
\tilde{f}_{\tilde{u}}=\int_0^t\partial_s\tilde{\phi} ds\nabla \tilde{u},
\quad \tilde{g}_{\tilde{u}}= Q\tilde{w}f(\tilde{u}),\label{eqn:tdfu}\\
f_w=\int_0^t\partial_s\psi ds\nabla w,\quad g_w= -wf(u),\label{eqn:dfw}\\
\tilde{f}_{\tilde{w}}=\int_0^t\partial_s\tilde{\psi} ds\nabla \tilde{w},
\quad \tilde{g}_{\tilde{w}}= -\tilde{w}f(\tilde{u}).\label{eqn:tdfw}
\end{gather}

3. Now, the system (\ref{eqn:detemp})-(\ref{eqn:tdfw}) is analogous to the
   system \eqref{eqn:etemp}-\eqref{eqn:fw}. Consequently, we have analogous
   estimates to a combination of estimates (\ref{eqn:cphipsi}),
   (\ref{eqn:chepsi}) and (\ref{eqn:chphipsi}). From the ensuing analogues,
   choosing $\epsilon > 0$, simplifying and integrating with respect to t over
   $[0,T]$ we obtain
\begin{equation}
\begin{aligned}
&\|\chi-\tilde{\chi}\|_{H^2(\Omega)}^2 +\|\tau-\tilde{\tau}\|_{H^2(\Omega)}^2
+ \|\partial_t\left(\chi-\tilde{\chi}\right)\|_{H^1(\Omega)}^2 \\
&+\|\partial_t\left(\tau-\tilde{\tau}\right)\|_{H^1(\Omega)}^2 +
\|D^3(\chi-\tilde{\chi})\|_{L^2[0,T; L^2(\Omega)]}^2  \\
&+ \|D^3(\tau-\tilde{\tau})\|_{L^2[0,T; L^2(\Omega)]}^2
+\|D^2\partial_t(\chi-\tilde{\chi})\|_{L^2[0,T; L^2(\Omega)]}^2\\
&+ \|D^2\partial_t(\tau-\tilde{\tau})\|_{L^2[0,T; L^2(\Omega)]}^2 \\
&\leq \int_0^T  \Big[\|\chi-\tilde{\chi}\|_{H^2(\Omega)}^2
 +\|\tau-\tilde{\tau}\|_{H^2(\Omega)}^2
 +\|\partial_t\left(\chi-\tilde{\chi}\right)\|_{H^1(\Omega)}^2 \\
&\quad + \|\partial_t\left(\tau-\tilde{\tau}\right)\|_{H^1(\Omega)}^2
 +\int_\Omega|f_u-\tilde{f}_{\tilde{u}}|^2dx
 + \int_\Omega|g_u-\tilde{g}_{\tilde{u}}|^2dx  \\
&\quad + \int_\Omega|f_w-\tilde{f}_{\tilde{w}}|^2dx
 + \int_\Omega|g_w-\tilde{g}_{\tilde{w}}|^2dx
 +\int_\Omega|\nabla\left(g_u-\tilde{g}_{\tilde{u}}\right)|^2dx \\
&\quad + \int_\Omega|\nabla\left(g_w-\tilde{g}_{\tilde{w}}\right)|^2dx
+ \int_\Omega|\nabla (f_u-\tilde{f}_{\tilde{u}})
 +\nabla\phi_0\nabla(u-\tilde{u})|^2dx \\
&\quad + \int_\Omega|\nabla (f_w-\tilde{f}_{\tilde{w}})
 +\nabla\psi_0\nabla(w-\tilde{w})|^2dx
 +\int_\Omega|\partial_t(f_u-\tilde{f}_{\tilde{u}})|^2dx+ \\
&\quad +\int_\Omega|D^2(f_u-\tilde{f}_{\tilde{u}})
 + D^2\phi_0\nabla(u-\tilde{u}) + \nabla\phi_0D^2(u-\tilde{u})|^2dx \\
&\quad +\int_\Omega|D^2(f_w-\tilde{f}_{\tilde{w}})
 +D^2\psi_0\nabla(w-\tilde{w}) + \nabla\psi_0D^2(w-\tilde{w})|^2dx  \\
&\quad +\int_\Omega|\nabla \partial_t(f_u-\tilde{f}_{\tilde{u}})
 +\nabla\phi_0\nabla\partial_t(u-\tilde{u})|^2dx
 + \int_\Omega|\partial_t(f_w-\tilde{f}_{\tilde{w}})|^2dx \\
&\quad +\int_\Omega|\nabla \partial_t(f_w-\tilde{f}_{\tilde{w}})
 +\nabla\psi_0\nabla\partial_t(w-\tilde{w})|^2dx
 +\int_\Omega|\partial_t(g_u-\tilde{g}_{\tilde{u}})|^2dx \\
&\quad + \int_\Omega|\partial_t(g_w-\tilde{g}_{\tilde{w}})|^2dx\Big]dt
\end{aligned} \label{eqn:dcombi}
\end{equation}
Calculations of the estimates of the necessary terms on the right side of
(\ref{eqn:dcombi}) are rather lengthy; and not rendered here for brevity.
 The estimates can be obtained by using the conditions
(\ref{eqn:func})-(\ref{eqn:phltlt}), Sobolev's embedding and Lemmas
\ref{eqn:first}-\ref{eqn:lemma}, after some suitable rearrangement.
If the estimates are substituted into (\ref{eqn:dcombi}), we deduce,
after an application of the integral form of the Gronwall's inequality,
the estimates
\begin{equation}
\begin{aligned}
&\|A[(u,w)] - A[(\tilde{u},\tilde{w})]\|_{X\times X}\\
&\leq C\Big(\sqrt{T}+T\sqrt{T}+T\Big)^{1/2}
\Big(1+\|u\|_{L^\infty[0,T; H^2(\Omega)]}^2   +\|\tilde{u}\|_{L^\infty[0,T; H^2(\Omega)]}^2\\
&\quad +\|w\|_{L^\infty[0,T; H^2(\Omega)]}^2+\|\tilde{w}|_{L^\infty[0,T; H^2(\Omega)]}^2\Big)^{1/2}
 \Big(1 + \|(\tilde{u},\tilde{w})\|_{X\times X}^2\\
&\quad + \|\partial_tu\|_{L^\infty[0,T; H^1(\Omega)]}^2 + \|\partial_tw\|_{L^\infty[0,T; H^1(\Omega)]}^2\Big)
 \|(u,w)-(\tilde{u},\tilde{w})\|_{X\times X},
\end{aligned}\label{eqn:mapping}
\end{equation}
where $C$ depends on $\Omega,Q$ and all the bounds in
(\ref{eqn:func})-(\ref{eqn:hslipschitz}).

4. We define a convex set
\begin{equation}
K = \big\{(u,w)|(u,w)-(u_0,w_0)\in X_0\times X_0,\;
\|(u,w)\|_{X\times X}\leq2\sqrt{\Lambda}\big\},\label{eqn:convex}
\end{equation}
where $X_0\times X_0$ is the set where the initial and boundary values are
zero; and $\Lambda=$ constant is the bound in (\ref{eqn:estimate}).
If $T>0$ is sufficiently small, we shall show that
\begin{equation}
A[K]\subseteq K,\ \|A[(u,w)] - A[(\tilde{u},\tilde{w})]\|_{X\times X}
\leq \frac{1}{2}\|(u,w)-(\tilde{u},\tilde{w})\|_{X\times X}\label{eqn:toshow}
\end{equation}
for all $(u,w), (\tilde{u},\tilde{w})\in K$.
Using (\ref{eqn:estimate}), we have
\begin{equation}
\|A[(u_0,w_0)]\|_{X\times X}=\|\left(\chi(x,0),\tau(x,0)\right)\|_{X\times X}=\|(u_0,w_0)\|_{X\times X}\leq \sqrt{\Lambda}\label{eqn:ainitial}.
\end{equation}
Thus for $(u,w)\in K$,
\begin{align*}
&\|A[(u,w)]\|_{X\times X}\\
&\leq \|A[(u_0,w_0)]\|_{X\times X} + \|A[(u,w)] -  A[(u_0,w_0)]\|_{X\times X} \\
&\leq \sqrt{\Lambda} + C\left(\sqrt{T}+T\sqrt{T}+T\right)^{1/2}\\
&\quad\times \Big(1+\|u\|_{L^\infty[0,T; H^2(\Omega)]}^2
 +\|w\|_{L^\infty[0,T; H^2(\Omega)]}^2+\|u_0\|_{H^2(\Omega)}^2
 +\|w_0|_{H^2(\Omega)}^2\Big)^{1/2} \\
&\quad\times \Big(1 + \|(u_0,w_0)\|_{X\times X}^2
 + \|\partial_tu\|_{L^\infty[0,T; H^1(\Omega)]}^2
 + \|\partial_tw\|_{L^\infty[0,T; H^1(\Omega)]}^2\Big)\\
&\quad\times \|(u,w)-(u_0,w_0)\|_{X\times X}
\quad \text{(using (\ref{eqn:mapping}), (\ref{eqn:ainitial}))}\\
&\leq \sqrt{\Lambda} + C\left(\sqrt{T}+T\sqrt{T}+T\right)^{1/2}
\left(1+4\Lambda+\|u_0\|_{H^2(\Omega)}^2+\|w_0|_{H^2(\Omega)}^2\right)^{1/2}\\
&\times \big(1+ 8\Lambda\big)\big(4\sqrt{\Lambda}\big)\\
&\leq 2\sqrt{\Lambda}
\end{align*}
%\label{eqn:emapping} \\
where we used (\ref{eqn:convex}),
for $T$ sufficiently small, such that
\[
4C\big(\sqrt{T}+T\sqrt{T}+T\big)^{1/2}
\big(1+4\Lambda+\|u_0\|_{H^2(\Omega)}^2+\|w_0|_{H^2(\Omega)}^2\big)^{1/2}
\big(1+8\Lambda\big)< 1.
\]
Thus $A[(u,w)]\in K$ for all $(u,w)\in K$. Furthermore, if $T$ is chosen
sufficiently small, such that
\[
C\big(\sqrt{T}+T\sqrt{T}+T\big)^{1/2}
\big(1+4\Lambda+\|u_0\|_{H^2(\Omega)}^2+\|w_0|_{H^2(\Omega)}^2\big)^{1/2}
\big(1+8\Lambda\big)< \frac{1}{2}.
\]
Then (\ref{eqn:mapping}) implies
\begin{equation}
\|A[(u,w)] - A[(\tilde{u},\tilde{w})]\|_{X\times X}\leq
\frac{1}{2}\|(u,w)-(\tilde{u},\tilde{w})\|_{X\times X},\label{eqn:showed}
\end{equation}
for all $(u,w),(\tilde{u},\tilde{w})\in K$.
 Thus the mapping $A$ is a strict contraction for sufficiently small $T>0$.
Hence, there exists a unique fixed point $(u,w)\in K$ such that
$A[(u,w)]=(u,w)$.

6. Given any $T>0$, we select $T_1>0$ small so that
\[
C\big(\sqrt{T_1}+T_1\sqrt{T_1}+T_1\big)^{1/2}
\big(1+4\Lambda+\|u_0\|_{H^2(\Omega)}^2+\|w_0|_{H^2(\Omega)}^2\big)^{1/2}
\big(1+8\Lambda\big)< \frac{1}{2}.
\]
We can thus apply Banach's fixed point theorem to find a unique strong
solution $(u,w)$ of  \eqref{eqn:etemp}-\eqref{eqn:fw} existing on
the interval $[0,T_1]$. Since $(u(x,t),w(x,t))\in K$ for almost everywhere
$0\leq t\leq T_1$, we can upon redefining $T_1$ if necessary, assume
$(u(x,T_1),w(x,T_1))\in K$. The argument above can then be repeated to extend
our unique strong solution to the interval $[T_1,2T_1]$. Continuing after
finitely many steps, we construct a unique strong solution of
\eqref{eqn:etemp}-\eqref{eqn:fw} existing on the full interval $[0,T]$.
\end{proof}

\subsection*{Acknowledgments}
The author would like to thank the anonymous referee whose
comments improved the original version of this manuscript.


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\end{document}