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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 39, 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 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/39\hfil Existence and uniqueness]
{Existence and uniqueness for a dislocation model with short-range
 interactions and \\ varying stress field}

\author[H. Ibrahim \hfil EJDE-2015/39\hfilneg]
{Hassan Ibrahim}

\address{Hassan Ibrahim \newline
Lebanese University,
Faculty of Sciences I,
EDST, Equipe EDP-AN,
Hadath-Beirut, Lebanon}
\email{ibrahim@cermics.enpc.fr}

\thanks{Submitted January 9, 2015. Published February 12, 2015.}
\subjclass[2000]{35K50, 35K40, 35K55}
\keywords{Boundary value problem; parabolic system; comparison principle}

\begin{abstract}
 In this article, we consider a coupled singular parabolic system,
 describing the dynamics of dislocation densities in a bounded domain.
 The model takes into consideration the short-range interactions between
 dislocations, which causes the singularity that appears under the form
 of dividing by a gradient term. We prove a long time existence and uniqueness
 under the assumption that the applied stresses on the domain is bounded in
 space and time. The proof relies on a comparison principle to avoid singularity,
 and on exponential gradient estimates for the long time existence.
\end{abstract}

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

\section{Introduction}

\subsection{Setting of the problem and main result}

In this article, we study the singular parabolic system
\begin{equation}\label{pre_app_model}
\begin{gathered}
\kappa_{t}=\varepsilon\kappa_{xx}+\frac{\rho_{x}\rho_{xx}}{\kappa_{x}}+\sigma\rho_{x}\quad
\text{in }
(0,1)\times (0,\infty)\\
\rho_{t}=(1+\varepsilon)\rho_{xx}+\sigma\kappa_{x}\quad \text{in } (0,1)\times (0,\infty),
\end{gathered}
\end{equation}
with the initial conditions
\begin{equation}\label{ic}
\kappa(x,0)=\kappa^0(x),\quad \rho(x,0)=\rho^0(x),\quad x\in (0,1),
\end{equation}
and the Dirichlet boundary conditions
\begin{equation}\label{bc}
\rho(0,t)=\rho(1,t)=\kappa(0,t)=0,\quad \kappa(1,t)=1, \quad t>0.
\end{equation}
In what follows, we adopt the notation
$$
I_T = I \times (0,T)\quad \text{with $I=(0,1)$  and } T>0.
$$
System \eqref{pre_app_model} is an approximate model of the one introduced 
by Groma, Csikor and Zaiser \cite{GCZ} to describe the dynamics of dislocation
 densities in a constrained channel submitted to an applied stress 
(here represented by the function $\sigma = \sigma(x,t)$).
 A dislocation is a defect, or irregularity within a crystal structure that
 can be observed by electron microscopy. The theory was originally developed 
by Vito Volterra in 1905. Dislocations are
a non-stationary phenomena and their motion is the main explanation of 
the plastic deformation in metallic crystals (see \cite{nab, HL} and the 
references therein for a recent and mathematical presentation).

The approximation stated above is based on $\varepsilon$-$\Delta$ regularization of 
a spatially differentiated system of \eqref{pre_app_model}, 
which explains the presence of the different factors $\varepsilon$ and $(1+\varepsilon)$ 
in the two equations. The stress function
$$
\sigma: \overline{I}_{\infty} \longrightarrow \mathbb{R},
$$
is assumed bounded and regular, namely:
\begin{equation}\label{tau-regu}
\sigma \in C^{2}(\overline{I}_{\infty})\quad\text{with }
 b:=\|\sigma\|_{L^{\infty}(\overline{I}_{\infty})}<+\infty,
\end{equation}
while the initial data $\rho^0$ and $\kappa^0$ are assumed smooth over
$\overline{I}$.

The goal of the present paper is to show the long time existence and uniqueness 
of a smooth solution of \eqref{pre_app_model}, \eqref{ic} and \eqref{bc}. 
The same question has been raised and solved in \cite{IJMcpaa} 
(see also \cite{IJMcras} for a brief study focusing on the main ideas) 
but for a constant $\sigma\in \mathbb{R}$. The proof relied on a comparison principle 
on the gradient $\kappa_{x}$ leading to the inequality:
\begin{equation}\label{phys-r-cdn}
\kappa_{x} > |\rho_{x}|\quad \text{on }  \overline{I}_{\infty},
\end{equation}
that was used, first, to avoid the singularity in the first equation 
of \eqref{pre_app_model}, and second, to linearise it in order to 
obtain some \textit{a priori} estimates ensuring the long time existence. 
We show how to adapt this comparison principle to cover the case 
\eqref{tau-regu}, and how to obtain the \textit{a priori} estimates. 
This will finally lead to our main result.

\begin{theorem}\label{mainresult0}
Let $\rho^0,\kappa^0\in C^{\infty}(\overline{I})$ satisfy:
\begin{equation}\label{initial-compa}
\begin{gathered}
 \rho^0(0)=\rho^0(1) = 0,\\
 \kappa^0(0)=0,\quad k^0(1)=1,\\
(1+\varepsilon)\rho^0_{xx}+\sigma(\cdot, 0)\kappa^0_{x}=0\quad \text{on } \partial I,\\
(1+\varepsilon)\kappa^0_{xx}+\sigma(\cdot, 0)\rho^0_{x}=0\quad \text{on } \partial I,
\end{gathered}
\end{equation}
and
\begin{equation}\label{initial-singu}
\kappa^0_{x}>|\rho^0_{x}|\quad \text{in } \overline{I}.
\end{equation}
Then, for every $0<\alpha<1$, there exists a unique solution:
$$
\rho,\kappa\in C^{3+\alpha,\frac{3+\alpha}{2}}(\overline{I}_{\infty})\cap
C^{\infty}(\overline{I}\times (0,\infty)),
$$
of system \eqref{pre_app_model}, \eqref{ic} and \eqref{bc} satisfying
 \eqref{phys-r-cdn}.
\end{theorem}

It is worth mentioning that the boundary condition \eqref{initial-compa} 
on $\rho^0$ and $\kappa^0$ is natural. In fact, it appears from the H\"{o}lder 
regularity of the solution as well as \eqref{pre_app_model}, 
\eqref{ic} and \eqref{bc}. However, condition \eqref{phys-r-cdn} 
is of physical origin as it represents the positivity of the dislocation 
densities (see for example \cite{IJMcpde} for the derivation).

Let us briefly state the strategy of our proof. 
The existence and uniqueness is made by using a fixed point argument 
after an artificial modification of \eqref{pre_app_model} in order to avoid
dividing by zero.  We will first show the short time existence,
proving in particular that
$$
\kappa_{x}(x,t)\geq \sqrt{\gamma^{2}(t)+\rho_{x}^{2}(x,t)},
$$
for a well chosen function $\gamma(t)=ce^{-ct}$, where $c>0$ depends 
on $\sigma$ and the initial data. Here we obtain one of the key estimates
$|\frac{\rho_{x}}{\kappa_{x}}|\leq 1$ which, in addition to 
the boundedness of $\sigma$, somehow linearise the first equation of 
\eqref{pre_app_model}. Consequently, and due to some \textit{a priori} 
estimates, we can prove the global time existence.

To our knowledge, systems of equations involving the singularity in
$1/\kappa_{x}$ as in \eqref{pre_app_model} has not been directly handled
elsewhere in the literature. However, parabolic problems involving
singular terms have been widely studied in various aspects 
(see for instance 
\cite{vazquez_carrillo03, ChanKaper, vazquez_cha02, DiBook, Antonio, MerleZaag}).


\subsection{Organization of this article}
This paper is organized as follows. In Section \ref{sec2}, we show a comparison 
principle associated to \eqref{pre_app_model} that plays a crucial rule 
in the long time existence of the solution as well as the positivity of $\kappa_{x}$.
 In Section \ref{sec3}, we present a result of short time existence, 
uniqueness and regularity by using fixed point and bootstrap arguments. 
Finally, in Section \ref{sec4}, we provide some exponential bounds on the
 solution and we prove our main result: Theorem \ref{mainresult0}.

\section{A comparison principle}\label{sec2}

In this section, and for simplification reasons, we take $I = (-1,1)$ and we set
$$
G_{a}(x) := \sqrt{x^{2} + a^{2}},\quad x,a\in \mathbb{R}.
$$
We now state our comparison principle on the gradient of the solution.

\begin{proposition}[Comparison principle] \label{thm_cmp}
Let $(\rho, \kappa)$ be a regular solution of \eqref{pre_app_model} on the 
compact $\overline{I}_T$ with $\kappa_{x} > 0$, and the initial data
 $(\rho^0, \kappa^0)$ satisfying:
\begin{equation}\label{cmp_init}
\kappa^0_{x} \geq G_{\gamma_{0}}(\rho^0_{x}),\quad \gamma_{0}\in (0,1).
\end{equation}
Then there exists a positive function $\gamma:[0,T] \mapsto \mathbb{R}$ such that
\begin{equation}\label{cmp_t}
\kappa_{x} \geq G_{\gamma}(\rho_{x})\quad \text{on } \overline{I}_T.
\end{equation}
\end{proposition}

\begin{proof} 
We define
$M:= \kappa_{x} - G_{\gamma}(\rho_{x})$,
where the function $\gamma = \gamma(t)$, $t\in [0,T]$, is to be determined 
in a way that $M\geq 0$ on $\overline{I}_T$.
The proof is divided into three steps.
\smallskip

\noindent \textbf{Step 1. (Differential inequality satisfied by $M$)}
Assuming the regularity of $\rho, \kappa$ and $\gamma$, we compute
\begin{equation}\label{compu_1}
\begin{gathered}
 M_{t}=\kappa_{xt}-G'_{\gamma}(\rho_{x})\rho_{xt}-\Gamma,\\
 M_{x}=\kappa_{xx}-G'_{\gamma}(\rho_{x})\rho_{xx},\\
 M_{xx}=\kappa_{xxx}-G''_{\gamma}(\rho_{x})\rho^{2}_{xx}-G'_{\gamma}(\rho_{x})\rho_{xxx},
\end{gathered}
\end{equation}
where
$$
\Gamma = \frac{\gamma\gamma'}{\sqrt{\rho_{x}^{2} + \gamma^{2}}}.
$$
Differentiating \eqref{pre_app_model} with respect to $x$, we easily obtain
\begin{equation}\label{compu_2}
\begin{gathered}
\kappa_{xt}=\varepsilon\kappa_{xxx}+\frac{\rho_{xx}^{2}}{\kappa_{x}}+\frac{\rho_{x}\rho_{xxx}}{\kappa_{x}}-
\frac{\rho_{x}\rho_{xx}\kappa_{xx}}{\kappa_{x}^{2}} + \sigma_{x}\rho_{x} + \sigma\rho_{xx},\\
\rho_{xt}=(1+\varepsilon)\rho_{xxx} + \sigma_{x}\kappa_{x} + \sigma\kappa_{xx}.
\end{gathered}
\end{equation}
Using \eqref{compu_1} and \eqref{compu_2}, direct computations lead to
\begin{equation}\label{compu_3}
M_{t} = \varepsilon M_{xx} + A M_{x} + B M + C -\Gamma,
\end{equation}
where
\begin{equation}\label{compu_4}
\begin{gathered}
 A = -\frac{\rho_{x}\rho_{xx}}{\kappa^{2}_{x}} -\sigma G'(\rho_{x}),\\
 B = \frac{\rho^{2}_{xx}}{\kappa^{2}_{x}} -\sigma_{x}G'(\rho_{x})
- \frac{G'_{\gamma}(\rho_{x})\rho_{xxx}}{\kappa_{x}},\\
\begin{aligned}
 C &= G(\rho_{x})\Big(\frac{\rho^{2}_{xx}}{\kappa^{2}_{x}} 
 - \sigma_{x}G'(\rho_{x})\Big) - \frac{\rho_{x}\rho^{2}_{xx}}{\kappa^{2}_{x}} G'(\rho_{x})
  - \sigma(G'(\rho_{x}))^{2} \rho_{xx}\\
 &\quad+ \varepsilon G''(\rho_{x})\rho^{2}_{xx} + \sigma_{x}\rho_{x} + \sigma\rho_{xx}.
\end{aligned}
\end{gathered}
\end{equation}
We now estimate the term $C$ and we show, in particular, that we can 
eliminate $\sigma_{x}$. Thus we return back to a case similar to a constant
$\sigma$. In fact, since $G(\rho_{x})G'(\rho_{x})=\rho_{x}$, we obtain
\begin{align*}
C&=   \frac{\rho^{2}_{xx}}{\kappa^{2}_{x}}G(\rho_{x}) 
 - \sigma_{x}\rho_{x}- \frac{\rho_{x}\rho^{2}_{xx}}{\kappa^{2}_{x}} G'(\rho_{x}) 
 - \sigma(G'(\rho_{x}))^{2} \rho_{xx} + \varepsilon G''(\rho_{x})\rho^{2}_{xx} \\
&\quad + \sigma_{x}\rho_{x} + \sigma\rho_{xx}\\
&=  \overbrace{\frac{\rho^{2}_{xx}}{\kappa^{2}_{x}} \left(G(\rho_{x}) 
-  \rho_{x}G'(\rho_{x})\right)}^{I} 
 + \overbrace{\sigma\rho_{xx}\big(1 - (G'(\rho_{x}))^{2}\big)}^{II}
 + \overbrace{\varepsilon G''(\rho_{x})\rho^{2}_{xx}}^{III}.
\end{align*}
Notice that $ G(\rho_{x}) -  \rho_{x}G'(\rho_{x}) = \frac{\gamma^{2}}{G(\rho_{x})}$, 
thus $I\geq 0$. However, simple computations give:
$$
II = \frac{\sigma\gamma^{2}\rho_{xx}}{\rho^{2}_{x} + \gamma^{2}}\quad\text{and}\quad 
III = \frac{\varepsilon \gamma^{2} \rho^{2}_{xx}}{(\rho^{2}_{x} + \gamma^{2})^{3/2}}.
$$
Using \eqref{tau-regu}, and applying Young's inequality, we obtain:
$$
\big| \frac{\sigma\gamma^{2}\rho_{xx}}{\rho^{2}_{x} + \gamma^{2}}\big| 
\leq \frac{\gamma|\rho_{xx}|}{(\rho^{2}_{x} + \gamma^{2})^{3/4}}
 \frac{b\gamma}{(\rho^{2}_{x} + \gamma^{2})^{1/4}}
\leq \frac{\varepsilon \gamma^{2} \rho^{2}_{xx}}{(\rho^{2}_{x} + \gamma^{2})^{3/2}} 
+ \frac{b^{2}\gamma^{2}}{4\varepsilon(\rho^{2}_{x} + \gamma^{2})^{1/2}};
$$
therefore
$$
II \geq -III- \frac{b^{2}\gamma^{2}}{4\varepsilon(\rho^{2}_{x} + \gamma^{2})^{1/2}}.
$$
Consequently,
$$
C \geq - \frac{b^{2}\gamma^{2}}{4\varepsilon(\rho^{2}_{x} + \gamma^{2})^{1/2}}.
$$
This inequality, together with \eqref{compu_3} and \eqref{compu_4}, 
lead to the  differential inequality
$$
M_{t} \geq \varepsilon M_{xx} + A M_{x} + B M - \frac{b^{2}\gamma^{2}}{4\varepsilon(\rho^{2}_{x} 
+ \gamma^{2})^{1/2}} -\Gamma.
$$

\noindent \textbf{The choice of $\gamma$:}
We want to choose $\gamma$ such that the above differential inequality 
in $M$ is homogeneous. In fact, choosing
$$
 \gamma(t) = \gamma_{0} e^{- \frac{b^{2}}{4\varepsilon} t},
$$
we arrive at
\begin{equation}\label{compu_4_00}
M_{t} \geq \varepsilon M_{xx} + A M_{x} + B M.
\end{equation}
\smallskip

\noindent \textbf{Step 2. (Boundary analysis)}
Since $\rho$ and $\kappa$ are constants on the boundary 
$\partial I \times [0,T]$, we obtain
\begin{gather*}
\varepsilon\kappa_{xx}+\frac{\rho_{x}\rho_{xx}}{\kappa_{x}}+\sigma\rho_{x}=0\quad \text{on }
\partial I\times[0,T]\\
(1+\varepsilon)\rho_{xx}+\sigma\kappa_{x}=0\quad \text{on } \partial I\times[0,T];
\end{gather*}
therefore
\begin{equation}\label{bd-eq}
M_{x} = \frac{\sigma}{1 + \varepsilon} G'_{\gamma}(\rho_{x}) M\quad \text{on }
 \partial I\times [0,T].
\end{equation}
Here we would like to show that the minimum 
$m(t) = \min_{\overline{I}}M(\cdot,t)$ is not attained on the boundary,
 and then to use \eqref{compu_4_00} to show its positivity. 
However, this is not possible since the above boundary condition \eqref{bd-eq} 
carries no information that violates the presence of the minimal point 
on $\partial I$. For this reason, we carefully multiply $M$ by a suitable 
positive function having large values on the boundary. In particular, 
we consider $\overline{M}$ defined by:
$$
\overline{M}(x,t) = \cosh(\theta x) M(x,t),\quad\theta\in \mathbb{R},
$$
where $\theta$ is to be adjusted in a way that $\overline{M}(\cdot,t)$ 
attains its minimum in $I$. In fact, elementary computations show that
\begin{equation}\label{m-bar-bdry}
\overline{M}_{x} = \Theta\overline{M} \quad \text{on }  \partial I\times [0,T],
\end{equation}
where
$$
\Theta = \big(\theta \tanh(\theta x) +  \frac{\sigma}{1 + \varepsilon} G'_{\gamma}(\rho_{x}) \big).
$$
The boundedness of $ \frac{\sigma}{1+\varepsilon}G_{\gamma}'(\rho_{x})$ on 
$\overline{I}_T$ permits the existence of $\theta$ large enough so that
$\Theta(1,t)>0$ and $\Theta(-1,t)<0$ for all $t\in [0,T]$. 
Hence, by \eqref{m-bar-bdry}, the function $\overline{M}(\cdot, t)$ can not
 have a positive minimum on $\partial I$.
\smallskip

\noindent \textbf{Step 3. (Conclusion)}
We now write the partial differential inequality satisfied by $\overline{M}$:
\begin{equation}\label{pdi-M-bar}
\begin{aligned}
\overline{M}_{t} 
&\geq \varepsilon \overline{M}_{xx} 
+ \Big(-\frac{\rho_{x}\rho_{xx}}{\kappa^{2}_{x}} 
 -\sigma G'(\rho_{x})- 2\theta\varepsilon \tanh(\theta x)\Big)\overline{M}_{x}
 + \Big[\frac{\rho^{2}_{xx}}{\kappa^{2}_{x}} \\
&\quad -\sigma_{x}G'(\rho_{x})- \frac{G'_{\gamma}(\rho_{x})\rho_{xxx}}{\kappa_{x}}
 -\theta\tanh(\theta x) \Big(-\frac{\rho_{x}\rho_{xx}}{\kappa^{2}_{x}} 
 -\sigma G'(\rho_{x})\Big)\\
&\quad  + \varepsilon \theta^{2} \left( 2\tanh^{2}(\theta x)-1\right)\Big]\overline{M}.
\end{aligned}
\end{equation}
Due to the regularity of $\overline{M}$, we may find a curve 
$t\mapsto (x(t), t)$ such that
$$
\overline{m}(t) = \min_{x\in \overline{I}} \overline{M}(x, t) 
= \overline{M}(x(t),t)\quad t\in [0,T].
$$
Without loss of generality, we  assume that
$\kappa^0_{x} > \sqrt{(\rho^0_{x})^{2} + \gamma_{0}^{2}}$. 
In fact, it suffices to adjust $\gamma_{0}$ in \eqref{cmp_init}. 
Therefore (see Step 2):
$$
\overline{m}(0)>0\quad \text{and} \quad x(0)\in I.
$$
Again, the regularity of $\overline{M}$ ensures that
$x(t)\in I$ for all $ 0\leq t \leq t_{0}\leq T$.
We claim that $t_{0} = T$. Indeed, if not, we obtain $x(t_{0})\in \partial I$.
Let us show that this can not be true. Indeed, since $x(t)\in I$ for
 $0\leq t<t_{0}$ we directly obtain
$$
\overline{M}_{x}(x(t), t) = 0\quad \text{and}\quad \overline{M}_{xx}(x(t), t) \geq 0.
$$
Moreover, using \eqref{pdi-M-bar}, we obtain, for some constant $c\in \mathbb{R}$,
 the following ordinary differential inequality involving $\overline{m}$:
$$
\overline{m}' \geq c \overline{m}\quad \text{for } 0\leq t<t_{0},
$$
and therefore
\begin{equation}\label{odi-smb}
\overline{m}(t)\geq \overline{m}(0) e^{ct} \quad \text{for } 0\leq t<t_{0}.
\end{equation}
Since $\overline{m}(0)>0$, the above inequality gives $\overline{m}(t_{0})>0$. 
Consequently (see Step 2) $x(t_{0})\in I$ and the claim is true.

Now as we obtain $x(t)\in I$ for all $t\in [0,T]$, the inequality \eqref{odi-smb}
also holds true for all $t\in [0,T]$ with $\overline{m}(0)>0$. 
Hence, we can infer that $\overline{M}\geq 0$ on $I_T$ and therefore $M \geq 0$ 
on $I_T$.
\end{proof}

\section{Short time existence and uniqueness}\label{sec3}

In this section we prove the short time existence and uniqueness for 
\eqref{pre_app_model}, \eqref{ic} and \eqref{bc}. The main idea is to find 
a solution of a truncated system where we carefully truncate the 
gradients $\rho_{x}$ and $\kappa_{x}$ in order to make use of the fixed point theorem. 
After that, and due to the regularity of the obtained solution, we can eliminate 
the artificial modification and get back to our original system. 
Before stating the main result of this section, let us present some basic
 tools used in our analysis.

\noindent \textbf{Basic tools:}
 We first consider, for real values $a\geq 0$, the real valued function
 $\mathcal{I}_{a}$ defined by:
$$
\mathcal{I}_{a}(x) = x1\!\!1_{\{|x|\leq a\}} + a1\!\!1_{\{x\geq a\}} 
- a1\!\!1_{\{x\leq -a\}},
$$
where $1\!\!1_{A}$ is the indicator function of the set $A\subseteq \mathbb{R}$. 
It is easily seen that $\mathcal{I}_{a}$ is a truncation of the identity 
function which is  bounded and Lipschitz on $\mathbb{R}$, and this property will 
be repeatedly used hereafter in this section.

We now define the spaces and some fundamental estimates we are use. 
We may sometimes use the differentiation notation:
$$
D^{k}_{z}(u) = \frac{\partial^{k}u}{\partial z^{k}}.
$$
For $p>3$, we consider the parabolic Sobolev space:
$$
Y = W^{2,1}_{p}(I_T) = \{u\in L^{p}(I_T);\,
 D_{t}^{r}D_{x}^{s}u\in L^{p}(I_T)\,\text{ for } 2r+s\leq 2\},
$$
equipped with the norm
$$
\|u\|_{Y} = \sum_{2r+s\leq 2} \|D_{t}^{r}D_{x}^{s}u\|_{L^{p}(I_T)}.
$$
The value $p>3$ is taken to emphasis some regularity properties 
on the solution. Indeed, it is well known (see \cite{LSU} for the details) 
that if $p>3$ then $Y$ is continuously embedded in a parabolic H\"{o}lder space:
\begin{equation}\label{lem2.4i}
Y \hookrightarrow C^{1+\alpha,\frac{1+\alpha}{2}}(\overline{I}_T),\quad
\alpha = 1-\frac{3}{p},
\end{equation}
with the following fundamental estimate, valid for $u=0$ on the parabolic boundary,
$$
\partial^{p}(I_T) = (\partial I\times [0,T])\cup (I\times\{0\})
$$
that reads
\begin{equation}\label{est2.6}
\|u_{x}\|_{L^{\infty}(I_T)} \leq cT^{\frac{p-3}{2p}} \|u\|_{Y},
\end{equation}
where $c = c(p)>0$ is a constant depending only on $p$.

Recall that \cite[Section 1]{LSU}, for $l>0$, the parabolic H\"{o}lder 
space $C^{l,l/2}(\overline{I}_T)$ is the Banach space of functions $v$
that are continuous in $\overline{I}_T$, together with all derivatives
of the form
  $D^{r}_{t}D^{s}_{x}v$ for $2r+s<l$, and have a finite norm
$|v|_{I_T}^{(l)}=\langle v\rangle^{(l)}_{I_T}+\sum_{j=0}^{[l]}\langle v\rangle_{I_T}^{(j)}$,
where
\begin{gather*}
\langle v\rangle_{I_T}^{(0)}=|v|^{(0)}_{I_T}=\|v\|_{L^{\infty}(I_T)},\quad
\langle v\rangle_{I_T}^{(j)}=\sum_{2r+s=j}|D^{r}_{t}D^{s}_{x}v|^{(0)}_{I_T},\\
\langle v\rangle^{(l)}_{I_T}=\langle v\rangle_{x,I_T}^{(l)}+\langle v\rangle_{t,I_T}^{(l/2)}, \quad
\langle v\rangle_{x,I_T}^{(l)}=\sum_{2r+s=[l]}\langle D^{r}_{t}D^{s}_{x}v\rangle_{x,I_T}^{(l-[l])},\\
\langle v\rangle_{t,I_T}^{(l/2)}=\sum_{0<l-2r-s<2}\langle D^{r}_{t}D^{s}_{x}v
 \rangle_{t,I_T}^{(\frac{l-2r-s}{2})},
\end{gather*}
with
\begin{gather*}
\langle v\rangle_{x,I_T}^{(\alpha)}=\inf\{c;\; |v(x,t)-v(x',t)|\leq
c|x-x{'}|^{\alpha},\;(x,t), (x',t)\in \overline{I}_T\},\quad 0<\alpha<1,
\\
\langle v\rangle_{t,I_T}^{(\alpha)}=\inf\{c;\; |v(x,t)-v(x,t{'})|\leq
c|t-t{'}|^{\alpha},\;(x,t), (x,t')\in \overline{I}_T\},\quad 0<\alpha<1.
\end{gather*}
Another very useful inequality in our study is a Sobolev estimate for 
parabolic equations (see \cite[Lemma 2.3]{IJMcpaa}). 
To state this estimate, we consider solutions $u\in Y$, $u=0$ on 
$\partial^{p}(I_T)$, of
\begin{equation}\label{source}
u_{t} = \varepsilon u_{xx} + f,\quad f\in L^{p}(I_T)\text{ called the source term}.
\end{equation}
Then we have
\begin{equation}\label{cpaa18}
\frac{\|u\|_{L^{p}(I_T)}}{T}+\frac{\|u_{x}\|_{L^{p}(I_T)}}{\sqrt{T}}+
\|u_{xx}\|_{L^{p}(I_T)}+\|u_{t}\|_{L^{p}(I_T)}\leq {c}\|f\|_{L^{p}(I_T)},
\end{equation}
where $c = c(\varepsilon,p)>0$ is a constant depending only on $p$ and $\varepsilon$.
Now, we may state the main proposition of this section.

\begin{proposition}[Short time existence and uniqueness]
\label{short-exis-uni}
 Let $p>3$ and let
$$
\rho^0,\kappa^0\in C^{\infty}(\overline{I}),
$$
be two functions such that $\rho^0(0) = \rho^0(1) = \kappa^0(0) = 0$ and
 $\kappa^0(1) = 1$. Suppose furthermore that
\begin{equation}\label{initial-grad-est}
\begin{gathered}
 \kappa^0_{x} \geq \gamma_{0}\quad\text{on}\quad I,\\
 \|D^{s}_{x}\rho^0, D^{s}_{x}\kappa^0\|_{L^{\infty}(I)} \leq M_{0},\quad s=1,2,
\end{gathered}
\end{equation}
with $\gamma_{0}>0$ and $M_{0}>0$. Then there exists a unique solution 
$(\rho,\kappa)\in Y^{2}$ of \eqref{pre_app_model}, \eqref{ic} and \eqref{bc} where
\begin{equation}\label{lbnl}
T = T(M_{0}, \gamma_{0}, \varepsilon, b, p),\quad 0<T<1.
\end{equation}
Moreover, this solution satisfies
\begin{equation}\label{final-grad-est}
\begin{gathered}
 \kappa_{x} \geq \frac{\gamma_{0}}{2} \quad \text{on } \overline{I}_T,\\
 |\rho_{x}|\leq 2M_{0} \quad \text{on } \overline{I}_T.
\end{gathered}
\end{equation}
\end{proposition}

\begin{proof} 
Throughout the proof, and in various estimates we suppose that $0<T<1$. 
This is in no way a problem since we have to choose $T$ small enough to 
construct our solution. The proof uses a fixed point argument on a closed 
subspace of $Y$. Looking at \eqref{final-grad-est}, we can artificially
 modify \eqref{pre_app_model} using suitable truncations. 
To simplify our computations, we set
$\mathcal{I} := \mathcal{I}_{2M_{0}}$,
and  consider
\begin{equation}\label{r_k_app_trun}
\begin{gathered}
 \kappa_{t} = \varepsilon \kappa_{xx} + \frac{\rho_{xx}\mathcal{I}(\rho_{x})}{(\gamma_{0}/2) 
 + (\kappa_{x} - \gamma_{0}/2)^{+}} + \sigma\rho_{x}\quad \text{on } I_T\\
 \rho_{t}= (1 + \varepsilon) \rho_{xx} + \sigma\kappa_{x}\quad \text{on } I_T,
\end{gathered}
\end{equation}
with the same initial and boundary conditions \eqref{ic} and \eqref{bc}. 
It is worth noticing that when \eqref{final-grad-est} is satisfied then
 \eqref{r_k_app_trun} coincides with \eqref{pre_app_model}.
On the other hand, condition \eqref{final-grad-est} also suggests that we 
consider functions $u\in Y$ of bounded gradients, i.e.
 $\|u_{x}\|_{L^{p}(I_T)}\leq \lambda$ where $\lambda>0$ is a fixed sufficiently 
large constant. For this reason, define the spaces $Y^{\rho}_{\lambda}$ 
and $Y^{\kappa}_{\lambda}$ as follows:
\begin{gather*}
Y^{\rho}_{\lambda} = \{u\in Y;\, \|u_{x}\|_{L^{p}(I_T)}\leq \lambda,\, 
 u=\rho^0\,\text{ on }\, \partial^{p}(I_T)\},\\
Y^{\kappa}_{\lambda} = \{u\in Y;\, \|u_{x}\|_{L^{p}(I_T)}\leq \lambda,\,
 u=\kappa^0\,\text{ on }\, \partial^{p}(I_T)\}.
\end{gather*}
Define the application:
$\Psi:Y^{\rho}_{\lambda}\times Y^{\kappa}_{\lambda} \longmapsto Y^{\rho}_{\lambda}\times Y^{\kappa}_{\lambda}$ by
\[
    (\hat{\rho},\hat{\kappa}) \longmapsto
    \Psi(\hat{\rho},\hat{\kappa})=(\rho,\kappa)
\]
where $(\rho,\kappa)$ is the solution of
\begin{equation}\label{r_k_app_trun_fpt}
\begin{gathered}
 \kappa_{t} = \varepsilon \kappa_{xx} + \frac{\rho_{xx}\mathcal{I}(\hat{\rho}_{x})}{(\gamma_{0}/2) 
+ (\hat{\kappa}_{x} - \gamma_{0}/2)^{+}} + \sigma\hat{\rho}_{x}\quad \text{on } I_T\\
 \rho_{t}= (1 + \varepsilon) \rho_{xx} + \sigma\hat{\kappa}_{x}\quad \text{on } I_T.
\end{gathered}
\end{equation}

\noindent \textbf{The application $\Psi$ is well defined:}
The existence and uniqueness of $(\rho,\kappa)\in Y^{\rho}_{\lambda}\times Y^{\kappa}_{\lambda}$ 
solution of \eqref{r_k_app_trun_fpt} is obtained in two steps. 
In a first step, while having the initial and boundary conditions 
\eqref{ic} and \eqref{bc}, we find a solution $\rho\in Y$ of the second 
equation of \eqref{r_k_app_trun_fpt}, then we plug it into the first 
equation to get a solution $\kappa\in Y$. Here, the existence and uniqueness 
of both solutions are guaranteed by \cite[Theorem 9.1]{LSU}. 
It is worth mentioning that \cite[Theorem 9.1]{LSU} requires a compatibility 
condition of order $0$ on the initial and boundary data. Those conditions 
are satisfied by our boundary assumptions on $\rho^0$ and $\kappa^0$ 
(see Proposition \ref{short-exis-uni}).

In a second step, we use \eqref{cpaa18}, basically on the functions:
$$
\bar{\rho} = \rho - \rho^0\quad\text{and}\quad 
\bar{\kappa} = \kappa - \kappa^0,
$$
together with \eqref{initial-grad-est}, to gain the $L^p$ bounds on
 $\rho_{x}$ and $\kappa_{x}$ if we choose $T$ small enough. 
The above steps ensure that the application $\Psi$ is well defined at 
least for sufficiently small time.

\noindent \textbf{The application $\Psi$ is a contraction:}\\
\noindent We now show that $\Psi$ is a contraction. 
In fact, let $\Psi(\hat{\rho}, \hat{\kappa}) = (\rho,\kappa)$ and 
$\Psi(\hat{\rho}', \hat{\kappa}') = (\rho',\kappa')$. 
The couple $(\rho-\rho',\kappa-\kappa')$ is the solution of the  system
\begin{equation}\label{f5-thesis}
\begin{gathered}
(\kappa-\kappa')_{t}=\varepsilon(\kappa-\kappa')_{xx}+ \mathcal{F}_{1}\quad \text{on } I_T\\
(\rho-\rho')_{t}=(1+\varepsilon)(\rho-\rho')_{xx}+ \mathcal{F}_{2}\quad \text{on } I_T
\end{gathered}
\end{equation}
where
\begin{gather*}
\mathcal{F}_{1} = \frac{\rho_{xx}\mathcal{I}(\hat{\rho}_{x})}
{(\gamma_{0}/2)+(\hat{\kappa}_{x}-\gamma_{0}/2)^{+}}-\frac{\rho'_{xx}\mathcal{I}(\hat{\rho}'_{x})}
{(\gamma_{0}/2)+(\hat{\kappa}'_{x}-\gamma_{0}/2)^{+}}+ \sigma(\hat{\rho}_{x} - \hat{\rho}'_{x}),\\
\mathcal{F}_{2} = \sigma(\hat{\kappa}_{x}-\hat{\kappa}'_{x}),
\end{gather*}
with
$$
(\rho-\rho',\kappa-\kappa')=(0,0)\quad\text{on } \partial^{p}(I_T).
$$
In the remaining part of the proof, the variable $c>0$ is a generic 
constant that depends on all constants in Proposition \ref{short-exis-uni} 
but independent of $T$. Using \eqref{cpaa18} and \eqref{f5-thesis}, we deduce that
\begin{equation}\label{f5-thesis-b}
\begin{gathered}
 \|\kappa -\kappa'\|_{Y}\leq c \|\mathcal{F}_{1}\|_{L^{p}(I_T)},\\
 \|\rho -\rho'\|_{Y}\leq c \|\mathcal{F}_{2}\|_{L^{p}(I_T)}.
\end{gathered}
\end{equation}

\noindent \textbf{Estimate of $\|\rho -\rho'\|_{Y}$.}
 We note that if $v = \hat{\kappa} - \hat{\kappa}'$, then $v$ satisfies
\begin{gather*}
 v_{t} = v_{xx} + f\quad \text{on } I_T,\\
 v=0\quad \text{on } \partial^{p}(I_T),
\end{gather*}
with $f = (\hat{\kappa} - \hat{\kappa}')_{t} - (\hat{\kappa} - \hat{\kappa}')_{xx}$.
Then, by \eqref{cpaa18}, we obtain
$$
\|v_{x}\|_{L^{p}(I_T)} \leq c\sqrt{T} \big\|(\hat{\kappa} 
- \hat{\kappa}')_{t} - (\hat{\kappa} - \hat{\kappa}')_{xx}\big\|_{L^{p}(I_T)}
\leq c\sqrt{T} \|\hat{\kappa} - \hat{\kappa}'\|_{Y};
$$
therefore,
 \begin{equation}\label{estF_2}
\|\mathcal{F}_{2}\|_{L^{p}(I_T)} \leq cb\sqrt{T}\|\hat{\kappa} - \hat{\kappa}'\|_{Y}.
\end{equation}
Using the second equation of \eqref{f5-thesis-b} and \eqref{estF_2}, we finally obtain
\begin{equation}\label{estr-r'}
\|\rho -\rho'\|_{Y}\leq cb\sqrt{T}\|\hat{\kappa} - \hat{\kappa}'\|_{Y}.
 \end{equation}

\noindent \textbf{Estimate of $\|\kappa -\kappa'\|_{Y}$.} We write
$$
\mathcal{F}_{1} = \mathcal{A}_{1} + \mathcal{A}_{2} 
+ \mathcal{A}_{3} + \mathcal{A}_{4},
$$
with
\begin{equation}
\begin{gathered}
 \mathcal{A}_{1} = \frac{\mathcal{I}(\hat{\rho}_{x})}
{(\gamma_{0}/2)+(\hat{\kappa}_{x}-\gamma_{0}/2)^{+}}\left(\rho_{xx} - \rho'_{xx}\right),\\
 \mathcal{A}_{2} = \frac{\rho'_{xx}(\mathcal{I}(\hat{\rho}_{x})
 - \mathcal{I}(\hat{\rho}'_{x}))}{(\gamma_{0}/2)+(\hat{\kappa}_{x}-\gamma_{0}/2)^{+}},\\
 \mathcal{A}_{3} = \rho'_{xx}\mathcal{I}(\hat{\rho}'_{x})
\Big(\frac{1}{(\gamma_{0}/2)+(\hat{\kappa}_{x}-\gamma_{0}/2)^{+}} 
- \frac{1}{(\gamma_{0}/2)+(\hat{\kappa}'_{x}-\gamma_{0}/2)^{+}}\Big),\\
 \mathcal{A}_{4} = \sigma(\hat{\rho}_{x} - \hat{\rho}'_{x}).
\end{gathered}
\end{equation}
We estimate the $L^{p}$ norms of $\mathcal{A}_{i}, i=1,2,3,4$. 
First remark that the coefficient of $(\rho_{xx} - \rho'_{xx})$ 
in $\mathcal{A}_{1}$ is bounded, hence by \eqref{estr-r'} we deduce that
\begin{equation}\label{estA1}
\|\mathcal{A}_{1}\|_{L^{p}(I_T)}\leq cb\sqrt{T}\|\hat{\kappa} - \hat{\kappa}'\|_{Y}.
\end{equation}
For the term $\mathcal{A}_{2}$, we proceed as follows. 
We apply the $L^{\infty}$ control of the spatial derivative \eqref{est2.6} 
to the function $\hat{\rho} - \hat{\rho}'$, we obtain
\begin{equation}\label{t4.116}
\|(\hat{\rho} - \hat{\rho}')_{x}\|_{L^{\infty}(I_T)} 
\leq cT^{\frac{p-3}{2p}} \|\hat{\rho} - \hat{\rho}'\|_{Y}.
\end{equation}
For the term $\rho'_{xx}$ in $\mathcal{A}_{2}$, we first remark that if
 we let $\bar{\rho}' = \rho' - \rho^0$, this function satisfies
 (see the second equation of \eqref{r_k_app_trun_fpt}):
\begin{gather*}
\bar{\rho}'_{t}=(1+\varepsilon)\bar{\rho}'_{xx}+(1+\varepsilon)\rho^0_{xx}-\sigma\hat{\kappa}'_{x}\quad
\text{on } I_T,\\
\bar{\rho}'=0 \quad \text{on }\partial^{p}I_T,
\end{gather*}
and hence, by \eqref{cpaa18} and \eqref{initial-grad-est}, we deduce that
\begin{equation}\label{t4.117}
\|\rho'_{xx}\|_{L^{p}(I_T)} \leq c(M_{0} + \lambda).
\end{equation}
Knowing that $\mathcal{I}$ is Lipschitz, we use \eqref{t4.116} and 
\eqref{t4.117} to obtain
\begin{equation}\label{estA2}
\|\mathcal{A}_{2}\|_{L^{p}(I_T)}
\leq c\frac{(M_{0} + \lambda)}{\gamma_{0}}T^{\frac{p-3}{2p}}\|\hat{\rho} - \hat{\rho}'\|_{Y}.
\end{equation}
For $\mathcal{A}_{3}$, we use the same arguments as for $\mathcal{A}_{2}$,
 and we are lead to:
\begin{equation}\label{estA3}
\|\mathcal{A}_{3}\|_{L^{p}(I_T)}
\leq c\frac{M_{0}(M_{0} + \lambda)}{\gamma^{2}_{0}}T^{\frac{p-3}{2p}}\|\hat{\kappa}
 - \hat{\kappa}'\|_{Y}.
\end{equation}
Here we have used that the function
 $x\to \frac{1}{(\gamma_{0}/2)+(x-\gamma_{0}/2)^{+}}$ is Lipschitz continuous 
over $\mathbb{R}$. Finally,
\begin{equation}\label{estA4}
\|\mathcal{A}_{4}\|_{L^{p}(I_T)}\leq cb\sqrt{T}\|\hat{\rho} - \hat{\rho}'\|_{Y}.
\end{equation}
Using \eqref{estA1}, \eqref{estA2}, \eqref{estA3} and \eqref{estA4}, we finally obtain
\begin{equation}\label{estF_1}
\|\mathcal{F}_{1}\|_{L^{p}(I_T)} \leq cT^{\frac{p-3}{2p}}
\left(\|\hat{\kappa} - \hat{\kappa}'\|_{Y} + \|\hat{\rho} - \hat{\rho}'\|_{Y} \right),
\end{equation}
hence, by \eqref{estF_1}, and the first equation of \eqref{f5-thesis-b}, 
we arrive at
\begin{equation}\label{estk-k'}
\|\kappa -\kappa'\|_{Y}\leq cT^{\frac{p-3}{2p}}
\left(\|\hat{\kappa} - \hat{\kappa}'\|_{Y} + \|\hat{\rho} - \hat{\rho}'\|_{Y} \right).
 \end{equation}

\noindent\textbf{Conclusion:}
Equations \eqref{estr-r'} and \eqref{estk-k'} show that, for $T>0$ 
sufficiently small, the application $\Psi$ is a contraction, and eventually
 it has a unique fixed point $(\rho,\kappa)$ solution of system \eqref{r_k_app_trun}, 
\eqref{ic} and \eqref{bc}.
Lastly, to get rid of the artificial truncations ($\mathcal{I}$ and 
the positive part in the denominator in \eqref{r_k_app_trun}), 
and to show \eqref{final-grad-est}, we use the embedding \eqref{lem2.4i}
 and the initial conditions \eqref{initial-grad-est}. 
This is again with the possibility of reducing the time $T$.
\end{proof}

\begin{lemma}\label{lemma1}
If $\kappa$ is the solution obtained in Proposition \ref{short-exis-uni}, then
$$
\sigma\kappa_{x} \in C^{\alpha,\alpha/2}(\overline{I}_T).
$$
\end{lemma}

\begin{proof} 
The embedding \eqref{lem2.4i} infers that $\kappa_{x}\in C^{\alpha,\alpha/2}(\overline{I}_T)$ 
and hence $\sigma\kappa_{x}\in C(\overline{I}_T)$. We then compute, 
for $x,x'\in [0,1]$ and $t,t'\in [0,T]$, $0<T<1$:
\begin{align*}
&|\sigma(x,t)\kappa_{x}(x,t) - \sigma(x',t)\kappa_{x}(x',t)| \\
&\leq  b\langle\kappa_{x}\rangle_{x,I_T}^{(\alpha)}|x-x'|^{\alpha} 
 + \|\kappa_{x}\|_{L^{\infty}(I_T)}\|\sigma_{x}\|_{L^{\infty}(I_T)}|x-x'|\\
&\leq  c |x-x'|^{\alpha}
\end{align*}
and
\begin{align*}
&|\sigma(x,t)\kappa_{x}(x,t) - \sigma(x,t')\kappa_{x}(x,t')|\\
&\leq b\langle\kappa_{x}\rangle_{t,I_T}^{(\alpha/2)}|t-t'|^{\alpha/2} 
 + \|\kappa_{x}\|_{L^{\infty}(I_T)}\|\sigma_{t}\|_{L^{\infty}(I_T)}|t-t'|\\
&\leq  c |t-t'|^{\alpha/2}.
\end{align*}
Therefore, $\sigma\kappa_{x} \in C^{\alpha,\alpha/2}(\overline{I}_T)$.
\end{proof}

\begin{remark}\label{rem1} \rm
Lemma \ref{lemma1} suggests that we may have a better regularity for 
the solution $\rho$ obtained in Proposition \ref{short-exis-uni}, which in 
 turn, can also lead to a better regularity on $\kappa$ due to the coupling 
in \eqref{pre_app_model}. This is better illustrated by the next proposition.
\end{remark}

\begin{proposition}[Regularity by a bootstrap argument] \label{prop_reg_boot}
Under the same hypothesis of Proposition \ref{short-exis-uni} and if, 
in addition, the functions $\rho^0, \kappa^0$ satisfy the condition
\begin{equation}\label{compatibility-1}
\begin{gathered}
(1+\varepsilon)\rho^0_{xx}+\sigma(\cdot, 0)\kappa^0_{x}=0\quad \text{on } \partial I,\\
(1+\varepsilon)\kappa^0_{xx}+\sigma(\cdot, 0)\rho^0_{x}=0\quad \text{on } \partial I,
\end{gathered}
\end{equation}
then the solution $(\rho,\kappa)$ obtained in Proposition \ref{short-exis-uni} 
satisfies:
\begin{equation}\label{shrt-more-reg}
\rho,\kappa\in C^{3+\alpha,\frac{3+\alpha}{2}}(\overline{I}_T)
\cap C^{\infty}(\bar{I}\times (0,T)).
\end{equation}
\end{proposition}

\begin{remark}[Comments on condition \eqref{compatibility-1}] \label{rem2}\rm
The assumption \eqref{compatibility-1} on the initial data, together with 
the constant boundary values, define a compatibility condition of order $1$. 
In other words, we obtain
\begin{gather*}
 0 = \kappa_{t}(x,0) = \varepsilon \kappa_{xx}(x,0) 
 + \frac{\rho_{x}(x,0)\rho_{xx}(x,0)}{\kappa_{x}(x,0)} 
 + \sigma(x,0)\rho_{x}(x,0)\quad \text{for } x\in \partial I,\\
 0 = \rho_{t}(x,0) = (1 + \varepsilon) \rho_{xx}(x,0) 
 + \sigma(x,0)\kappa_{x}(x,0),\quad \text{for } x\in \partial I.
\end{gather*}
This boundary condition let us conclude that (see \cite[Theorem 5.2]{LSU}), 
if the source terms (see equation \eqref{source}) in \eqref{pre_app_model} 
is of class $C^{\beta,\beta/2}(\overline{I}_T)$ for $0<\beta<2$, then
 the solution $(\rho,\kappa)$ will be of class 
$C^{2+\beta,\frac{2+\beta}{2}}(\overline{I}_T)$.
\end{remark}

\begin{proof}[Proof of Proposition \ref{prop_reg_boot}]
The proof follows the idea of Remark \ref{rem1}. In fact, 
Lemma \ref{lemma1} show that the source term $\sigma\kappa_{x}$ of the second 
equation of \eqref{pre_app_model} is of class $C^{\alpha,\alpha/2}(\overline{I}_T)$. 
Remark \ref{rem2} with $0<\beta = \alpha <2$ and the compatibility conditions
 \eqref{compatibility-1}, ensures that 
$\rho\in C^{\alpha+2,\frac{\alpha + 2}{2}}(\overline{I}_T)$, which, by its turn 
(thanks to similar computations as those done in Lemma \ref{lemma1}), 
shows that the the source term $\frac{\rho_x\rho_{xx}}{\kappa_{x}} + \sigma\rho_{x}$ 
of the first equation of \eqref{pre_app_model} is of class 
$C^{\alpha,\alpha/2}(\overline{I}_T)$. Finally, we also deduce that 
$\kappa\in C^{\alpha+2,\frac{\alpha + 2}{2}}(\overline{I}_T)$ and then 
$\kappa_{x}\in C^{1+\alpha,\frac{1 + \alpha}{2}}(\overline{I}_T)$.

We now repeat exactly the same ideas but with this new regularity on
 $\kappa_{x}$, taking into consideration that still $0 < \beta = \alpha + 1 < 2$, 
and hence Remark \ref{rem2} is still applicable. In fact, since 
$\kappa_{x}\in C^{1+\alpha,\frac{1 + \alpha}{2}}(\overline{I}_T)$ then, by similar 
arguments as in Lemma \ref{lemma1}, we obtain
$\sigma\kappa_{x}\in C^{1+\alpha,\frac{1 + \alpha}{2}}(\overline{I}_T)$ 
and the whole process may be repeated. Therefore, we are finally lead to
 $\rho,\kappa \in C^{3+\alpha,\frac{3+\alpha}{2}}(\overline{I}_T)$. 
Note that this is the maximum H\"{o}lder regularity that could be obtained 
up to the boundary since the compatibility condition \eqref{compatibility-1}
is not sufficient when $\beta>2$. Indeed, higher order compatibility assumptions 
are required if we to achieve more regularity on the boundary.

To get the interior $C^{\infty}$ regularity, we need to carefully overcome 
the problem of compatibility at $t=0$, $x\in \partial I$. 
Indeed, let $0 < \delta < T$, and define any test function 
$\phi_{\delta}\in C^{\infty}[0,T]$ by
\begin{equation}\label{test-int-r}
\phi_{\delta}(t) =
\begin{cases}
 0 &\text{if }  0\leq t \leq \frac{\delta}{3},\\
 \phi_{\delta}(t)\in (0,1) &\text{if } 
 \frac{\delta}{3}\leq t \leq \frac{2\delta}{3},\\
 1 &\text{if }  \frac{2\delta}{3} \leq t \leq T.
\end{cases}
\end{equation}
By introducing
\begin{equation}\label{hafil}
\bar{\rho} = \rho\phi_{\delta}\quad \text{and} \quad \bar{\kappa} 
= \kappa\phi_{\delta},
\end{equation}
we can easily see that $(\bar{\rho}, \bar{\kappa})$ satisfy a parabolic 
system where higher order compatibility conditions on the initial data are 
satisfied. Hence, the parabolic H\"{o}lder regularity can be infinitely 
applied for $\bar\rho$ and $\bar\kappa$. Accordingly
$$
(\bar\rho, \bar\kappa)\in C^{\infty}(\overline{I}_T).
$$
From \eqref{test-int-r} and \eqref{hafil} we deduce that
$$
(\rho,\kappa) = (\bar\rho, \bar\kappa) \quad \text{in } [\frac{2\delta}{3}, T],\quad 
\forall\,0<\delta<T.
$$
Thus $\rho,\kappa \in C^{\infty}(\bar{I}\times (0,T))$ and the proof follows.
\end{proof}

\begin{remark}\label{rem3} \rm
It is worth noticing that the regularity \eqref{shrt-more-reg} of 
the solution $(\rho,\kappa)$ (obtained by Proposition \ref{short-exis-uni}) 
is sufficient to make use of the comparison principle (Proposition \ref{thm_cmp} 
in Section \ref{sec2}).
\end{remark}

\section{Exponential estimates and long time existence}\label{sec4}

In this section, we give some exponential bounds of the solution given 
by Proposition \ref{short-exis-uni}, and having the regularity shown
 by Proposition \ref{prop_reg_boot}. In this section, the generic 
constants $c>0$ and $c(T)>0$ may vary from line to line.

\begin{proposition}[Exponential bound in time for  $\rho_{x}$ and $\kappa_{x}$]
\label{prop:jin:1}
 Let
$$
\rho,\kappa\in C^{3+\alpha,\frac{3+\alpha}{2}}(\overline{I}_{\infty})\cap
C^{\infty}(\bar{I}\times(0,\infty)),
$$ 
be a solution of \eqref{pre_app_model}, \eqref{ic} and \eqref{bc}, with
$\rho^0(0)=\rho^0(1)=\kappa^0(0)=0$ and $\kappa^0(1)=1$. Suppose
furthermore that \eqref{phys-r-cdn} is satisfied. 
Then there exists a constant $c = c(\rho^0, \kappa^0, \varepsilon, b, p)>0$ such that
\begin{equation}\label{VVimp1}
\begin{gathered}
 \|\rho_{x}(\cdot, t)\|_{L^{\infty}(I)}\leq ce^{ct}\quad \text{for all } t\geq 0,\\
 \|\kappa_{x}(\cdot, t)\|_{L^{\infty}(I)}\leq ce^{ct}\quad \text{for all } t\geq 0.
\end{gathered}
\end{equation}
\end{proposition}

\begin{proof} 
The idea behind our exponential estimates is based upon linearising system 
\eqref{pre_app_model}. Indeed, condition \eqref{phys-r-cdn} implies the
 boundedness of the coefficient $\frac{\rho_{x}}{\kappa_{x}}$ of the term $\rho_{xx}$, 
whereas \eqref{tau-regu} implies the boundedness of the other coefficients.
 We begin by estimating $\|\kappa_{x}\|_{L^{p}(I_T)}$. 
By applying \eqref{cpaa18} to the function $\kappa-\kappa'$ where $\kappa'$ satisfies
\begin{equation}\label{cc:slav}
\begin{gathered}
\kappa_{t}'=\kappa_{xx}' \quad \text{on } I_T\\
\kappa'=\kappa \quad \text{on } \partial^{p} I_T,
\end{gathered}
\end{equation}
we obtain
\begin{equation}\label{ccc:aFi}
\|\kappa_{x}\|_{L^{p}(I_T)}\leq c(T)\left(\|\kappa^0\|_{W^{2-2/p}_{p}(I)}+1
\right) +cb\sqrt{T}\|\rho\|_{W^{2,1}_{p}(I_T)}.
\end{equation}
Here we have used the fact that
 $|\frac{\rho_{x}}{\kappa_{x}}|\leq 1$, and the solvability of parabolic 
equations in Sobolev spaces (see for instance \cite[Theorem 9.1]{LSU}). 
The same estimate now applied to $\rho-\rho'$ (where $\rho'$ satisfies an 
inequality similar to \eqref{cc:slav}) gives
\begin{equation}\label{jin:estbarr1}
\|\rho\|_{W^{2,1}_{p}(I_T)}\leq c(T)\|\rho^0\|_{W^{2-2/p}_{p}(I)}+
cb\|\kappa_{x}\|_{L^{p}(I_T)}.
\end{equation}
Combining \eqref{ccc:aFi} and \eqref{jin:estbarr1}, we conclude that for
$$
T^{*} = \frac{1}{2c^{4}b^{4}},
$$
we have
$$
\|\kappa_{x}\|_{L^{p}(I_{T^{*}})}
\leq c\Big(\|\kappa^0\|_{W^{2-2/p}_{p}(I)}+
\|\rho^0\|_{W^{2-2/p}_{p}(I)}+1\Big),\quad c=c(T^{*})>0.
$$
Having the special coupling of system
\eqref{pre_app_model}, together with the above estimate, we can
deduce that
$$
\|\rho\|_{W^{2,1}_{p}(I_{T^{*}})}, \|\kappa\|_{W^{2,1}_{p}(I_{T^{*}})}\leq
c\left(\|\kappa^0\|_{W^{2-2/p}_{p}(I)}+
\|\rho^0\|_{W^{2-2/p}_{p}(I)}+1\right),
$$
where $c=c(T^{*})>0$;
hence, by \eqref{lem2.4i}, we obtain
$$
|\rho|^{(1+\alpha)}_{I_{T^{*}}}, |\kappa|^{(1+\alpha)}_{I_{T^{*}}}\leq
c\left(\|\kappa^0\|_{W^{2-2/p}_{p}(I)}+
\|\rho^0\|_{W^{2-2/p}_{p}(I)}+1\right),\quad c=c(T^{*})>0.
$$
The trace theorem for parabolic Sobolev spaces \cite[Lemma 3.4]{LSU} give
$$
\|u(\cdot, t+T^{*})\|_{W^{2-2/p}_{p}(I)} 
\leq c\|u\|_{W^{2,1}_{p}(I\times (t,t+T^{*}))},\quad c=c(T^{*})>0;
$$
therefore, we can iterate backwards in time to finally obtain our result.
\end{proof}

\begin{proposition} \label{prop:jin:2}
Under the same hypothesis of Proposition \ref{prop:jin:1}, 
there exists a constant $c = c(\rho^0, \kappa^0, \varepsilon, b, p)>0$ such that
\begin{equation}\label{VVimp2}
\begin{gathered}
\|\rho_{xx}(\cdot, t)\|_{L^{\infty}(I)}\leq ce^{ct}\quad \text{for all } t\geq 0,\\
\|\kappa_{xx}(\cdot, t)\|_{L^{\infty}(I)}\leq ce^{ct}\quad \text{for all } t\geq 0.
\end{gathered}
\end{equation}
\end{proposition}

The proof of the above porposition can be found in 
\cite[Propositions 5 and 6]{IJMcpaa}.

\begin{proposition}\label{prop:jin:3}
Under the same hypothesis of Proposition \ref{prop:jin:1}, there exists a
 constant $c = c(\rho^0, \kappa^0, \varepsilon, b, p)>0$ such that
\begin{equation}\label{VVimp3}
\|\kappa_{x}(\cdot,t)\|_{L^{\infty}(I)}\geq ce^{-ct}\quad \text{for all }
0\leq t\leq T.
\end{equation}
\end{proposition}

The proof of the above proposition Follows immediately from \eqref{initial-singu} 
and \eqref{cmp_t}.
Now we are ready to show the main result of this paper, namely 
Theorem \ref{mainresult0}.

\begin{proof}[Proof of Theorem \ref{mainresult0}]
 Define the set $\mathcal{B}$ by
\begin{align*}
\mathcal{B}=Big\{& T>0; \;\exists \,! \;\text{ solution } (\rho,\kappa)
  \in C^{3+\alpha,\frac{3+\alpha}{2}}(\overline{I}_T) \text{ of }\\
&\text{\eqref{pre_app_model}, \eqref{ic} and  \eqref{bc},  satisfying 
 \eqref{phys-r-cdn}}
\end{align*}
This set is non empty by the short time existence result 
(Theorem \ref{short-exis-uni}). 
Set
$$
T_{\infty}=\sup \mathcal{B}.
$$
We claim that $T_{\infty}=\infty$. Assume, by contradiction that
$T_{\infty}<\infty$. In this case, let $\delta>0$ be an arbitrary
small positive constant, and apply the short time existence result
(Theorem \ref{short-exis-uni}) with $T_{0}=T_{\infty}-\delta$. Indeed, by
the exponential bounds \eqref{VVimp1}, \eqref{VVimp2} and \eqref{VVimp3}, we
deduce that the time of existence $T$ given by \eqref{lbnl} is in
fact independent of $\delta$. Hence, choosing $\delta$ small enough,
we obtain $T_{0}+T\in \mathcal{B}$ with $T_{0}+T>T_{\infty}$ and
hence a contradiction.
\end{proof}

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\end{document}