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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 95, pp. 1--11.\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-2009/95\hfil A parabolic-hyperbolic system]
{A parabolic-hyperbolic system modelling a moving cell}

\author[F. Cardetti, Y. S. Choi\hfil EJDE-2009/95\hfilneg]
{Fabiana Cardetti, Yung-Sze Choi}  % in alphabetical order

\address{Fabiana Cardetti \newline
Department of Mathematics\\
University of Connecticut\\
Ubox 3009, Storrs, CT 06269, USA}
\email{fabiana.cardetti@uconn.edu}

\address{Yung-Sze Choi \newline
Department of Mathematics\\
University of Connecticut\\
Ubox 3009, Storrs, CT 06269, USA}
\email{choi@math.uconn.edu}

\thanks{Submitted November 12, 2008. Published August 7, 2009.}
\subjclass[2000]{35Q80, 35R35, 35A07, 37N25, 92C17}
\keywords{Cell motility; moving boundary problems; coupled systems}

\begin{abstract}
 In this article, we study the existence and uniqueness
 of local solutions for a moving boundary problem governed
 by a coupled parabolic-hyperbolic system.
 The results can be applied to cell movement,
 extending a result obtained by Choi, Groulx, and Lui in 2005.
\end{abstract}

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

\section{Introduction} \label{sec:intro}

In this article, we consider the system of $n$ hyperbolic equations
coupled with a single parabolic equation
\begin{equation} \label{wus}
\begin{gathered}
 \mathbf{w}_t = -\sigma_x \mathbf{w}_x-\mathbf{F}(\mathbf{w},\sigma),
 \\
 \sigma_t =g(\mathbf {w},\sigma)\sigma_{xx} - \sigma_x^2
+h(\mathbf {w}, \sigma) \,,
 \end{gathered}
\end{equation}
where  $\mathbf{F}$, $g$, and $h$ are given $C^1$ functions in their
respective variables and $x \in [r(t),f(t)]$. The boundary conditions are
\begin{equation} \label{bc}
     \begin{gathered}
 \mathbf{ w}=\mathbf{w}_f (t), \quad \sigma=0, \quad \text{at } x=f(t),\\
 \sigma=0, \quad \text{at } x=r(t)
\end{gathered}
\end{equation}
 with $\mathbf{w}_f$ being a given $C^{1}$ function.
Motion of the boundaries are defined by
\begin{equation}\label{movingbdry}
    \begin{gathered}
    \frac{df}{dt}=V\vert_{f(t)-r(t)}+ \sigma_x\vert_{x=f(t)} \,, \\
    \frac{dr}{dt}=1+\sigma_x \vert_{x=r(t)} \,.
    \end{gathered}
\end{equation}
Here $V: (0, \infty) \to (0,\infty)$ is a given $C^1$ function with
$V(\ell)>0$ when $\ell>0$. We observe that $\ell(t)=f(t)-r(t)$
represents the instantaneous domain size.
The moving boundary problem consists of equations
\eqref{wus}, \eqref{bc},  \eqref{movingbdry},
 and initial conditions
 \begin{equation} \label{new_ic}
 r(0)=0, \quad
 f(0)=\ell_0>0, \quad
 \mathbf{w}(x,0)=\mathbf{w}_0(x), \quad
 \sigma(x,0)=\psi(x)
 \end{equation}
with  $\mathbf{w}_0 \in C^1[0,\ell_0]$ and
$\psi \in C^{2+\beta}[0,\ell_0]$
for some  $0<\beta<1\ $. In order for \eqref{wus}b) to be parabolic,
we impose
\begin{equation} \label{parabolicity}
g(\mathbf{w}_0, \psi) >0 \quad \mbox{for all } x \in [0, \ell_0] \,.
\end{equation}
We also require the initial conditions be compatible with the boundary
conditions and the moving conditions to first order
(see p.319, \cite{ladyzenskaja68} for the parabolic case).

The zeroth order compatibility is equivalent to
\begin{equation}
\label{zero_compat}
\psi(0)=\psi(\ell_0)=0 \,, \quad
\mathbf{w}_0(\ell_0)=\mathbf{w}_f(0) \,.
\end{equation}
For the first order compatibility one differentiates
$\sigma(f(t),t)=0$ which yields $\sigma_x \frac{df}{dt}+ \sigma_t=0$.
Therefore, at $(x,t)=(\ell_0,0)$, we have
\begin{equation}
\label{first_compat1}
\psi_x ( V+\psi_x)+ g(\mathbf{w}_0,\psi)
\psi_{xx}- \psi_x^2+h(\mathbf{w}_0,\psi)=0
\end{equation}
with $\psi$, $\psi_x$, $\psi_{xx}$, $\mathbf{w}_0$ and $V$ all
evaluated at $\ell_0$.
Similarly the first order compatibility for $\sigma$ at $(x,t)=(0,0)$
leads to
\begin{equation}
\label{first_compat2}
\psi_x ( 1+\psi_x)+ g(\mathbf{w}_0,\psi)
\psi_{xx}- \psi_x^2+h(\mathbf{w}_0,\psi)=0
\end{equation}
with all the terms evaluated at $x=0$.

One needs only the first order compatibility
for $\mathbf{w}_0$ at $(x,t)=(\ell_0,0)$. From $\mathbf{w}(f(t),t)=
\mathbf{w}_f(t)$, we obtain
\begin{equation} \label{first_compat3}
\mathbf{w}_{0,x} (V+\psi_x)- \psi_x \mathbf{w}_{0,x}
-\mathbf{F}(\mathbf{w}_0,\psi)
=\frac{d \mathbf{w}_f}{dt}(0)
\end{equation}
with $\psi$, $\psi_x$, $\mathbf{w}_0$, $\mathbf{w}_{0,x}$ and
$V$ all evaluated at $\ell_0$.

Under such conditions we will prove the following theorem by
mapping the changing domain $Q_{\epsilon} \equiv \{ (x,t): 0< t< \epsilon,
\ r(t) < x< f(t) \}$
to a rectangle with a unit length.

\begin{theorem} \label{main_theorem}
Consider the moving boundary problem
\eqref{wus}, \eqref{bc} and \eqref{movingbdry}
with initial conditions \eqref{new_ic} with
$\mathbf{w}_0 \in C^1[0,\ell_0]$ and $\psi \in C^{2+\beta}[0,\ell_0]$
for some $0<\beta<1$. Furthermore assume
\eqref{parabolicity} holds and
 the boundary and the initial conditions are
compatible to first order; i.e.,
\eqref{zero_compat} to \eqref{first_compat3} hold.
Then there exists an $\epsilon >0$ such that the moving boundary problem
has unique solutions $\mathbf{w} \in C^{1,1}$ and
$\sigma \in C^{2+\beta,(2+\beta)/2}$ in the domain
$\overline{Q_{\epsilon}}$.
\end{theorem}

We now explain the number of boundary conditions needed for $\mathbf{w}$.
Let the characteristics associated with hyperbolic equations \eqref{wus}a)
through a point $(x_0,t_0)$ be denoted by
$x=\tilde{x}(x_0,t_0,t)$.
Then $\partial \tilde{x}/\partial t=\sigma_x(\tilde{x},t)$.
From the moving conditions \eqref{movingbdry}, the front end is moving
at a positive speed $V(\ell(t))$ faster than
the characteristic. In other words the characteristics from the front
$x=f(t)$ are going
into the domain $[r(t),f(t)]$. Hence a boundary condition is needed for
$\mathbf{w}$ at the front. On the other hand the characteristic at
$x=r(t)$ is going outside the domain $[r(t),f(t)]$.
No boundary condition can be imposed at the rear end.


\section{motivation} \label{sec:motivate}

The study of the system \eqref{wus} is motivated by the one-dimensional
model for the movement of a nematode sperm cell proposed by Mogilner
 and Verzi \cite{move03}.  Based on the principles of mechanics,
they proposed
\begin{equation} \label{mogilner}
\begin{gathered}
\frac{\partial b}{\partial t} = -\frac{\partial}{\partial x}(bv)-\gamma_bb \,,\\
 \frac{\partial p}{\partial t} = -\frac{\partial}{\partial x}(pv)+\gamma_bb-\gamma_p p \,,\\
  \frac{\partial c}{\partial t} = -\frac{\partial}{\partial x}(cv) \,,
    \end{gathered}
\end{equation}
where $b,p$ denote the length densities of the bundled filaments and
free filaments, respectively, $c$ is the density of the cytoskeletal
nodes (responsible for cell adhesion),  $v$ is
the cytoskeletal velocity, $\gamma_b$ is the rate of unbundling of
the bundled filaments, and $\gamma_p$ is the rate of disassembly
of the free filaments.
System \eqref{mogilner} is derived from mass balance and is assumed
to hold for  $x \in [r(t),f(t)]$, which denotes the spatial
interval defined by the rear and front
ends of the cell at time $t$.

A balance between frictional and elastic filament forces
leads Mogilner and Verzi to assume
\begin{equation} \label{velocity}
v(x,t)= \frac1\xi \frac{\partial \sigma}{\partial x},
\end{equation}
where $\xi$  is the effective drag coefficient between the cell and the substratum,
and $\sigma$ is the total filament stress with a constitutive
law defined by
 \begin{equation} \label{stress}
    \sigma=Kb ( \frac1c - \rho ) + \kappa \frac pc \,,
\end{equation}
where $K$ and $\kappa$ are the effective spring constants for the
bundled and free filaments, respectively, and $\rho$ is the rest
length of the bundled filament while the free filament is assumed
to have natural length $0$. This formula for stress is based on
Hooke's law with the average distance between two cytoskeletal
nodes being $1/c$.

Motion of the boundaries are defined by
\begin{equation} \label{movingbdryOLD}
    \begin{gathered}
    \frac{df}{dt}=V_p \vert_{f(t)-r(t)}+v \vert_{f(t)} \,, \\
    \frac{dr}{dt}=V_d+v \vert_{r(t)}
    \end{gathered}
\end{equation}
with initial conditions  $f(0)=\ell_0$ and $r(0)=0$, respectively.
Here $V_d$ is a given positive constant
representing the rate of disassembly at the rear, and
$V_p$  is a given function, depending on the instantaneous cell
length, which represents
the rate of filament polymerization at the front.

In \cite{choi05}, Choi, Groulx, and Lui proved the local existence
of solutions assuming that $\gamma_p=0$ and $K=\kappa$ so that
$(b+p)/c$ is a conserved quantity as time evolves.

It can be shown that system~\eqref{wus}  is a generalization of
 \eqref{mogilner}. In fact, let $w= \frac {b}{c}$.
Using \eqref{mogilner}a) and \eqref{mogilner}c), we
then obtain the following equation for $w$:
 \begin{align*}
w_t &=\frac{b_tc-bc_t}{c^2}\\
    &=\frac{(-b_xv-bv_x- \gamma_b b)c-b(-c_xv-cv_x)}{c^2}\\
%&=\frac{(-b_xvc-bv_xc- \gamma_b bc+bc_xv+bcv_x)}{c^2}\\
    &=-v [  \frac{b_xc-bc_x}{c^2} ] - \gamma_b \frac bc\\
    &=-vw_x-\gamma_b w \,.
\end{align*}
Let $u=\frac {p}{b}$. Using \eqref{mogilner}b) and
\eqref{mogilner}c),
we arrive at
\[
u_t = -vu_x- (\gamma_p-\gamma_b)u+\gamma_b \,.
\]
With $K=1$ and $\rho=1$, (\ref{stress}) can be recast
in terms of $w$ and $u$ as
\begin{equation} \label{stressnew}
\sigma= w(1+ \kappa u) -b \,.
\end{equation}
On taking the time derivative and
substituting $w_t$ and $u_t$  from the above calculations,
\[
\sigma_t = bv_x -v\sigma_x-\gamma_b w-\kappa  \gamma_p u w
+\kappa w \gamma_b+\gamma_bb \,.
\]
Replacing $v$ by $\frac{\sigma_x}{\xi}$ and rearranging
and using \eqref{stressnew}, system \eqref{mogilner} is equivalent to
\begin{equation} \label{wusOLD}
    \begin{gathered}
    w_t= -\big(\frac{\sigma_x}{\xi} \big)w_x-\gamma_bw,\\
    u_t=-\big(\frac{\sigma_x}{\xi} \big)u_x - (\gamma_p-\gamma_b) u + \gamma_b,\\
\sigma_t =b \big(\frac{\sigma_x}{\xi} \big)_x - \frac{\sigma_x^2}{\xi} - \gamma_b \sigma+ \kappa \gamma_bw(1+u) -\kappa \gamma_puw \,.
    \end{gathered}
\end{equation}
With $b=w(1+ \kappa u) - \sigma$, (\ref{wusOLD}) can be cast
totally in term of $w$, $u$ and $\sigma$.

 It is now clear that the system consists of a second-order equation and
a pair of first-order hyperbolic equations that can be rewritten
as system \eqref{wus}.

\section{Preliminary lemma}

Throughout this article, we will use
$u \in C^{1,0}$ to denote $u$ and $u_x$ are continuous functions in
a $(x,t)$ domain, while
$u \in C^{1,1}$ means $u$, $u_x$ and $u_t$ are continuous.
The following lemma  will be needed in Section \ref{localexistence}
to prove the main theorem.

\begin{lemma} \label{lemma2.1}
Let $\mathcal{R}_{\delta}=\{(x,t) : 0<x<1, \, 0<t< \delta\}$.
Let $a \in C^{1,0}(\overline{\mathcal{R}_{\delta}})$,
$\mathbf{f}:{\mathbf R}^n \times \overline{\mathcal{R}_{\delta}} $
be a continuous function of $(\mathbf{u},x,t)$
with $\mathbf{f}_\mathbf{u}$ and $\mathbf{f}_x$ being continuous,
$\mathbf{g} \in C^1[0,\delta]$, and $\mathbf{u}_0 \in C^1[0,1]$.
 Consider the system
\begin{equation} \label{eq_lemma2.1}
\begin{gathered}
 \mathbf{u}_t+a(x,t)\mathbf{u}_x={\mathbf f}(\mathbf{u},x,t)\,, \\
 \mathbf{u}(1,t)=\mathbf{g}(t) \quad \text{for }   0\leq t \leq \delta, \\
 \mathbf{u}(x,0)=\mathbf{u}_0(x) \quad \text{for }  0 \leq x \leq 1 .
\end{gathered}
\end{equation}
 Suppose $a(0,t)<0$ and $a(1,t)<0$ for $0\leq t \leq \delta$, and the compatibility conditions
 $\mathbf{g}(0)=\mathbf{u}_0(1)$ and
 $\mathbf{g}_t(0)+a(1,0)\mathbf{u}'_0(1)=\mathbf{f}(\mathbf{u}_0(1),1,0)$ holds. Then there exists a unique solution
$\mathbf{u} \in C^{1,1}(\overline{\mathcal{R}_{\delta_1}})$ for some positive
$\delta_1 \leq \delta$ to the above system. Furthermore,

(a) there exists a constant $M_1>0$, depending on the $L^{\infty}$-norm of $a, \mathbf{g}, \mathbf{u}_0$
and $\mathbf{f}$ in the compact set $[- \| \mathbf{u}_0 \|_{\infty}-\| \mathbf{g} \|_{\infty}-1, \| \mathbf{u}_0 \|_{\infty}+\| \mathbf{g} \|_{\infty}+1] \times
\overline{\mathcal{R}_{\delta}}$,
such that $\|\mathbf{u}\|_{\infty} \leq M_1$,

(b) there exists a constant $M_2>0$, depending on the $C^1$ norm of $\mathbf{g}$ and
$\mathbf{u}_0$,  the $C^{1,0}$-norm of coefficient
 $\ a$,
and the $L^{\infty}$-norm of $\mathbf{f}, \mathbf{f}_\mathbf{u},
\mathbf{f}_x$ in the compact set
$[- \| \mathbf{u}_0 \|_{\infty}-\| \mathbf{g} \|_{\infty}-1, \| \mathbf{u}_0 \|_{\infty}+\| \mathbf{g} \|_{\infty}+1] \times
\overline{\mathcal{R}_{\delta}}$,
 such that $\|\mathbf{u}\|_{C^{1,1}} \leq M_2$. The time interval of existence
 $[0,\delta_1]$ also depends on the same norms.
\end{lemma}

\begin{proof}
Let $x=\tilde{x}(t, \tau)$ be the  characteristic curve coming out from
$x=1$ at time $\tau$ into the domain $\mathcal{R}_{\delta}$.
 In other words, $\tilde{x}$ satisfies
\begin{equation} \label{one}
 \begin{gathered}
\frac{\partial \tilde{x}}{\partial t}=a(\tilde{x}(t,\tau),t),\\
\tilde{x}(\tau,\tau)=1.
\end{gathered}
\end{equation}
Now define $\mathbf{v}(t,\tau)=\mathbf{u}(\tilde{x}(t,\tau),t)$.
Then using \eqref{one}a) and the governing equation on $\mathbf{u}$,
we obtain
\begin{equation} \label{two}
 \begin{gathered}
\frac{\partial \mathbf{v}}{\partial t}
=\mathbf{f}(\mathbf{v},\tilde{x}(t,\tau),t),\\
\mathbf{v}(\tau,\tau)=\mathbf{g}(\tau).
  \end{gathered}
\end{equation}
By ODE theory there exists a unique solution $\mathbf{v}$ for a small
time interval $[\tau,\tau+\delta_1]$
with $\delta_1$ being uniform with respect to initial conditions
$\mathbf{g}(\tau)$ for all $\tau \in [0,\delta]$.
One can perform  similar calculations for characteristics starting
from initial point $(x,0)$ for $x \in [0,1]$ by shrinking the time
interval of existence $[0,\delta_1]$ if necessary.
These lead to a $L^{\infty}$-norm bound on $\mathbf{v}$, which leads
to statement (a) in the theorem.
Continuity of solutions across the characteristic $\Gamma_0$
coming out from $(x,t)=(1,0)$ is an easy consequence of the above
analysis and the compatibility
condition $\mathbf{g}(0)=\mathbf{u}_0(1)$.

Now differentiating the initial condition \eqref{one}b) with respect to
$\tau$ yields $\tilde{x}_t(\tau,\tau)+\tilde{x}_{\tau}(\tau,\tau)=0$,
which simplifies to $\tilde{x}_{\tau}(\tau,\tau)=-a(1,\tau)>0$.
 Hence from \eqref{one},
\begin{gather*}
\frac{\partial}{\partial t}
\big(\frac{\partial \tilde{x}}{\partial \tau}\big)
=a_x(\tilde{x}(t,\tau),t)\frac{\partial \tilde{x}}{\partial \tau},\\
\frac{\partial \tilde{x}}{\partial \tau}(\tau,\tau)=-a(1,\tau).
  \end{gather*}
Therefore, if $\|a_x\|_{\infty} \leq M$, then
\begin{equation} \label{dxdtau}
|a(1,\tau)| e^{-M(t-\tau)} \leq
\frac{\partial \tilde{x}}{\partial \tau}
\leq |a(1,\tau)| e^{M(t-\tau)} \,.
\end{equation}
Similarly we differentiate the initial condition \eqref{two}b)
obtaining $\mathbf{v}_t(\tau,\tau)+\mathbf{v}_{\tau}(\tau,\tau)=\mathbf{g}_t(\tau)$,
which simplifies
to $\mathbf{v}_{\tau}(\tau,\tau)= \mathbf{g}_t(\tau)
- \mathbf{f}(\mathbf{g}(\tau),1,\tau)$.
Therefore (\ref{two}) yields the following governing equation
for $\frac{\partial \mathbf{v}}{\partial \tau}$,
\begin{equation}\label{three}
 \begin{gathered}
\frac{\partial}{\partial t}
\big(\frac{\partial \mathbf{v}}{\partial \tau}\big)=
\mathbf{f}_{\mathbf{u}}(\mathbf{v},\tilde{x}(t,\tau),t)
 \frac{\partial \mathbf{v}}{\partial \tau}
 + \mathbf{f}_x (\mathbf{v},\tilde{x}(t,\tau),t)
 \frac{\partial\tilde{x}}{\partial \tau},
\\
\frac{\partial \mathbf{v}}{\partial \tau}(\tau,\tau)=\mathbf{g}_t(\tau)
- \mathbf{f}(\mathbf{g}(\tau),1,\tau) \,.
  \end{gathered}
\end{equation}
Thus we get the linear system
$\frac{\partial}{\partial t}
\big(\frac{\partial \mathbf{v}}{\partial \tau}\big)
=A(t)\frac{\partial \mathbf{v}}{\partial \tau}+\mathbf{b}(t)$
with matrix $A$, vector $\mathbf{b}$ and initial condition all with
$L^{\infty}$-norm bounds.
Using $< \cdot,\cdot>$ to denote scalar product in $\mathbf{R}^n$,
which is related to $\ell_2$ norm.
Then there exist positive constants $c_1$ and $c_2$ such that
\begin{align*}
\frac{\partial}{\partial t}
\big(\|\frac{\partial \mathbf{v}}{\partial \tau}\|^2_{\ell^2}\big)
&= 2 \big\langle \frac{\partial \mathbf{v}}{\partial \tau} ,
\frac{\partial}{\partial t}
\big(\frac{\partial \mathbf{v}}{\partial \tau}\big)\big\rangle\\
&=2 \big\langle \frac{\partial \mathbf{v}}{\partial \tau} , A(t)
\frac{\partial \mathbf{v}}{\partial \tau}+b(t)\big\rangle\\
& \leq c_1\|\frac{\partial \mathbf{v}}{\partial \tau}\|^2_{\ell^2}+c_2 \,,
\end{align*}
which leads to boundedness of
$\|\frac{\partial \mathbf{v}}{\partial \tau}\|_{\infty}$.

From the definition of $\mathbf{v}$ we  have that
$\frac{\partial \mathbf{v}}{\partial \tau}
=\frac{\partial \mathbf{u}}{\partial x}
\frac{\partial \tilde{x}}{\partial \tau}$.
With $\partial \tilde{x}/\partial \tau$ in (\ref{dxdtau}) having a
positive lower bound, it is immediate that there exists a positive
constant $m$ such that
$\|\frac{\partial \mathbf{u}}{\partial x}\|_{\infty} \leq m$.
Next with $\frac{\partial \mathbf{v}}{\partial t}
=a\frac{\partial \mathbf{u}}{\partial x}
+ \frac{\partial \mathbf{u}}{\partial t}$,
we can also bound $\frac{\partial \mathbf{u}}{\partial t}$.
Therefore, there exists a positive constant $M_2$ such that
$\|\mathbf{u}\|_{C^{1,1}} \leq M_2$ for those solutions whose
characteristics originate from $x=1$.

 A similar analysis can be performed with solutions whose characteristics
originate from $t=0$. To complete the proof of statement (b), we
need $\mathbf{u}_t$ and $\mathbf{u}_x$ to be continuous across
$\Gamma_0$.

Assuming the solution is smooth in these two regions for the time being,
we have
$$
(\mathbf{u}_x)_t+a(\mathbf{u}_x)_x+a_x \mathbf{u}_x
=\mathbf{f}_{\mathbf{u}}\mathbf{u}_x +\mathbf{f}_{x} \,.
$$
Let $[ \cdot ]$ to denote the jump across $\Gamma_0$. Due to continuity
of $\mathbf{u}$ across $\Gamma_0$, on subtraction of the above equations
in the two regions we obtain
\begin{equation} \label{four}
 \begin{gathered}
 \left([\mathbf{u}_x]\right)_t+a([\mathbf{u}_x])_x
+(a_x-\mathbf{f}_{\mathbf{u}})[\mathbf{u}_x]=0,\\
[\mathbf{u}_x](0,0)=0,
\end{gathered}
\end{equation}
where the zero initial condition is a consequence of the second
compatibility condition.
Hence $[\mathbf{u}_x]=0$ at all subsequent time by integrating along
the characteristic $\Gamma_0$.

Since the coefficients $a$ and $a_x-\mathbf{f}_{\mathbf{u}}$ in
(\ref{four}) do not depend on higher smoothness
of solution $\mathbf{u}$, one can obtain (\ref{four}) by using
approximations of $a$ and $\mathbf{f}$
by smoother functions and take the limit.
The proof of the Lemma is now complete.
\end{proof}


\section{Fixing the domain}   \label{sec:fix}


To facilitate our discussion, we let
$ Q_{\epsilon}=\{(x,t) \mid r(t)<x<f(t), 0<t< \epsilon\}$.
It is  convenient to work on a fixed domain so we first straighten
out the moving boundaries.
Let  $\ell(t)=f(t)-r(t)$ and $x=r(t)+ \bar{x} \ell(t)$.
The region $Q_{\epsilon}$ is mapped onto the region
$ \mathcal{R}_{\epsilon}=\{(\bar{x},t) \mid 0<\bar{x}<1, 0<t< \epsilon\}$.
Define $\tilde{\mathbf{w}}(\bar{x},t)=\mathbf{w}(r(t)+ \bar{x} \ell(t),t)$
and $\tilde{\sigma}(\bar{x},t)=\sigma(r(t)+ \bar{x} \ell(t),t)$.
Then $\tilde{\mathbf{w}}_{\bar{x}}=\mathbf{w}_x \ell$ and
 \begin{align*}
\tilde{\mathbf{w}}_{t}&=(r'+\bar{x}\ell')\mathbf{w}_x+\mathbf{w}_t \\
&=\frac{(r'+\bar{x}\ell')}{\ell} \tilde{\mathbf{w}}_{\bar{x}}
 - \frac{\sigma_x}{\ell} \tilde{\mathbf{w}}_{\bar{x}}
 -\mathbf{F}(\tilde{\mathbf{w}})\\
&=- \Big(\frac{\tilde{\sigma}_{\bar{x}}}{\ell^2}
 -\frac{(r'+\bar{x}\ell')}{\ell}\Big) \tilde{\mathbf{w}}_{\bar{x}}
 -\mathbf{F}(\tilde{\mathbf{w}})
\end{align*}
  with boundary condition $\tilde{\mathbf{w}}(1,t)=\mathbf{w}_f(t)$.
Similarly we can obtain the governing equation for $\tilde{\sigma}$.
System~\eqref{wus}  is then transformed into the system
  \begin{equation}   \label{3.4}
  \begin{gathered}
 \tilde{\mathbf{w}}_{t}
=- \Big( \frac{\tilde{\sigma}_{\bar{x}}}{\ell^2}
 -\frac{(r'+\bar{x}\ell')}{\ell}\Big) \tilde{\mathbf{w}}_{\bar{x}}
 -\mathbf{F}(\tilde{\mathbf{w}}) \,, \\
\tilde{\sigma_t}
= \frac{g(\tilde{\mathbf{w}},\tilde{\sigma})}{\ell^2}
 \tilde{\sigma}_{\bar{x}\bar{x}}
 - \Big(\frac  {\tilde{\sigma}_{\bar{x}}}{\ell^2}
  - \frac{(r'+\bar{x}\ell')}{\ell} \Big)
  \tilde{\sigma}_{\bar{x}} +h(\tilde{\mathbf{w}},\tilde{\sigma}),
 \end{gathered}
\end{equation}
which holds in   $\mathcal{R}_{\epsilon}$.  The boundary
conditions \eqref{bc} become
\begin{equation} \label{3.2}
\begin{gathered}
\tilde{\sigma}(0,t)=0, \\
\tilde{\sigma}(1,t)=0,\quad \tilde{\mathbf{w}}(1,t)=\mathbf{w}_f(t) ,
\end{gathered}
\end{equation}
and the equations for the moving boundaries~\eqref{movingbdry} become
 \begin{equation} \label{3.3}
 \begin{gathered}
  \frac{df}{dt}=V(\ell(t))+ \frac{\tilde{\sigma}_{\bar{x}}(1,t)}{\ell} \,, \ \ f(0)=\ell_0,\\
  \frac{dr}{dt}=1+\frac{\tilde{\sigma}_{\bar{x}}(0,t)}{\ell}, \quad
  r(0)=0.
 \end{gathered}
 \end{equation}
We observe that the first order compatibility conditions
\eqref{zero_compat} to \eqref{first_compat3} between the initial
and the boundary conditions in the domain
$ Q_{\epsilon}$ give rise to the corresponding compatibility conditions
of $\tilde{\sigma}$ at $(\bar{x},t)=(0,0), (1,0)$ and
$\tilde{\mathbf{w}}$ at
$(\bar{x},t)=(1,0)$ in the domain $\mathcal{R}_{\epsilon}$.
Hence one can establish Theorem \ref{main_theorem} by considering
\eqref{3.4}-\eqref{3.3}
with corresponding initial conditions which are compatible to the boundary
conditions to first order.

   The idea of our existence proof is to make a guess for
$\tilde{\sigma}$. Next using such a guess and
\eqref{3.3} we find the moving boundaries  $f$, $r$ and $\ell=f-r$.
Via Lemma \ref{lemma2.1},
we have enough a priori bounds for the solution $\tilde{\mathbf{w}}$
of  the hyperbolic equations \eqref{3.4}a).
 The final step is to solve \eqref{3.4}b) for a new $\tilde \sigma$.
If we can find a fixed point of such
an iterative procedure, this will be a solution we are looking for.

 \section{Proof of Theorem \ref{main_theorem}}
\label{localexistence}

 Define $g_0=g(\tilde{\mathbf{w}},\tilde{\sigma})\vert_{t=0}$
and $h_0=h(\tilde{\mathbf{w}},\tilde{\sigma})\vert_{t=0}$.
Let $z$ be the solution of the following initial-boundary
value problem
 \begin{equation} \label{z}
    \begin{gathered}
z_t=\frac{g_0(\bar{x}l_0)}{\ell_0^2} z_{\bar{x}\bar{x}}
 - \Big(\psi'(\bar{x}\ell_0)
 - \frac{[r'(0)+\bar{x}\ell'(0)] } { \ell_0 }  \Big)
  {\psi}'(\bar{x} \ell_0) +h_0(\bar{x}l_0),
    \\
    z(0,t)=z(1,t)=0,\\
    z(\bar{x},0)=\psi(\bar{x}l_0) \,,
    \end{gathered}
 \end{equation}
which is a linear second order parabolic equation.
It is derived from \eqref{3.4}b) with all
the terms, except for the
ones involving the time derivative and the second
spatial derivative, evaluated using the initial conditions.
By hypothesis the coefficients and the non-homogeneous terms in the
above equation are  time-independent and in $C^1([0,1])$.
Since the initial conditions satisfy the first order compatibility
conditions at $(0,0)$ and $(1,0)$, respectively,  a unique solution
$z$ exists and is in
$C^{2+\beta,(2+\beta)/2}(\overline{\mathcal{R}}_{\epsilon})$
for some   $0<\beta<1$ as defined in the hypothesis. It is also clear
that if solution $\tilde{\sigma}$ to \eqref{3.4}b) exists, then
 $\tilde{\sigma}(\bar{x},0)=z(\bar{x},0)$ and
$\tilde{\sigma}_t(\bar{x},0)=z_t(\bar{x},0)$.
 Let $0<\epsilon  \leq 1$ and let
\begin{align*}
\mathcal{S}_{\epsilon}
= \big\{& \sigma \in C^{2,1}(\overline{\mathcal{R}}_{\epsilon}) :
 \| \sigma - z\|_{C^{2,1}(\overline{\mathcal{R}}_{\epsilon})} \leq 1,\;
 \sigma(\bar{x},0)=z(\bar{x},0), \\
&\sigma_t(\bar{x},0)=z_t(\bar{x},0), \;
 \sigma(0, \cdot)=\sigma(1, \cdot)=0  \big\}.
\end{align*}
The goal is to define, for sufficiently small $\epsilon$,
a compact continuous map
$T: \mathcal{S}_{\epsilon}  \to \mathcal{S}_{\epsilon}$
and then apply the Schauder fixed-point Theorem.

Recall $[0,\delta_1]$ is the interval of existence in
Lemma \ref{lemma2.1}.
Let $\sigma \in \mathcal{S}_{\epsilon_1}$ where
$\epsilon_1 \leq \delta_1$ will be determined later and let $\ell$
be the solution to the equation $\ell'=L(\ell,t)$ with initial
condition $\ell(0)=\ell_0$, where
$$
L(\ell,t)= V(\ell) -1 + \frac{\sigma_{\bar{x}}(1,t)
-\sigma_{\bar{x}}(0,t)}{\ell}.
$$
 Since $L$ is $C^1$ in $\ell$, $C^{1/2}$ in $t$,
there exists an $\epsilon_1>0$, uniform with respect to
$\sigma \in \mathcal{S}_1$,
such that the solution $\ell$ exists, belongs to
$C^{1+1/2}([0,\epsilon_1])$
and satisfies $2 \ell_0 \geq \ell \geq \ell_0/2$.
Now solve~\eqref{3.3} separately for $f$ and $r$.
It is clear that $f,r \in C^{1+1/2}[0,\epsilon_1]$ and $f-r=\ell$.

 Using $f,r$, and $\sigma$, the next step is to solve the hyperbolic
equations \eqref{3.4}a)  for $\tilde{\mathbf{w}}$.
By Lemma \ref{lemma2.1}, the solution $\mathbf{w}$ exists and has
uniform $C^{1,1}$ bound which is independent of the choice of
$\sigma \in \mathcal{S}_{\epsilon_1}$.

Now let $\hat{\sigma}$ be the solution to the linear initial-boundary
value problem
\begin{equation} \label{sigma_hat}
\begin{gathered}
\hat{\sigma}_t=\frac{g(\mathbf{w}(\bar{x},t),
{\sigma}(\bar{x},t))}{\ell^2(t)}  \hat{\sigma}_{\bar{x} \bar{x}}
+G(\bar{x},t),\\
    \hat{\sigma}(0,t)=\hat{\sigma}(1,t)=0,\\
    \hat{\sigma}(\bar{x},0)=\psi(\bar{x}l_0)
    \end{gathered}
\end{equation}
in $\overline{\mathcal{R}_{\epsilon_1}}$, where
\begin{equation} \label{F}
G(\bar{x},t)= - \Big(\frac  {{\sigma}_{\bar{x}}}
 { \ell^2}   - \frac{(r'+\bar{x}\ell')}{\ell} \Big)
  {\sigma}_{\bar{x}} + h(\mathbf{w},\sigma)\,.
\end{equation}
Having uniform time derivative bounds on $\mathbf{w}$ and $\sigma$,
they stay close to $\mathbf{w}_0$ and $\psi$ by reducing
$\epsilon_1$ if necessary. Hence we have the parabolicity
$g(\mathbf{w},\sigma)>0$ because of \eqref{parabolicity}.

With  $\sigma \in \mathcal{S}_{\epsilon_1}$ and our established
estimates on $h$ and $\ell$, we have
$\| G\|_{C^{1,1/2}(\overline{\mathcal{R}}_{\epsilon_1})} $
being uniformly  bounded independently of the choice of
$\sigma \in \mathcal{S}_{\epsilon_1}$.
Define the operator
$T: \mathcal{S}_{\epsilon_1} \to \mathcal{S}_{\epsilon_1}$
by $T\sigma=\hat{\sigma}$.

  To show that   $\hat{\sigma} \in \mathcal{S}_{\epsilon_1}$,
let $\tilde{g}(\bar{x},t)=g(\mathbf{w}(\bar{x},t),\sigma(\bar{x},t))$ and
$u=\hat{\sigma}-z$.
Observe that (\ref{z}) is the same as
\[
z_t= \frac{\tilde{g}(\bar{x},0)}{\ell_0^2} z_{\bar{x}\bar{x}}
+G(\bar{x},0) \,.
\]
Then it can readily be checked that $u$ satisfies the equation
 \begin{equation} \label{reqn}
 u_t=\frac{\tilde{g}(\bar{x},t)}{\ell^2} u_{\bar{x} \bar x}
+H(\bar{x},t) ,
\end{equation}
where
\begin{equation}
H(\bar{x},t)= \Big( \frac{\tilde{g}(\bar{x},t)}{\ell^2}
- \frac{\tilde{g}(\bar{x},0)}{\ell_0^2} \Big)  z_{\bar{x}\bar{x}}
+ G(\bar{x},t)-G(\bar{x},0) \,.
\end{equation}
The established estimates allow us to conclude that there is a uniform
bound on  $\| H\|_{C^{\beta,\beta/2}(\overline{\mathcal{R}}_{\epsilon_1})}$,
which is independent of the choice of
$\sigma \in \mathcal{S}_{\epsilon_1}$. Observe that $u$ has zero initial
and boundary conditions and $H(\cdot,0)=0$.
Hence by \cite[ch.4, Thm. 5.4]{ladyzenskaja68},
\begin{equation} \label{thm5.4}
 \| u\|_{C^{2+\beta,(2+\beta)/2}(\overline{\mathcal{R}}_{\epsilon_1})} \leq M_1 \| H\|_{C^{\beta,\beta/2}(\overline{\mathcal{R}}_{\epsilon_1})} \,,
 \end{equation}
where $M_1$ is independent of the choice of $\sigma\in \mathcal{S}_{\epsilon_1}$ and remains bounded as $\epsilon  \downarrow 0$. Since $u(\cdot, 0)=u_t(\cdot,0)=0$,
by choosing $\epsilon_1$ smaller if necessary,
(\ref{thm5.4}) allows us to conclude
$\| u\|_{C^{2,1}(\overline{\mathcal{R}}_{\epsilon_1})} \leq 1$
so that $\hat{\sigma} \in \mathcal{S}_{\epsilon_1}$.
Inequality (\ref{thm5.4}) also implies that
$\| \hat{\sigma}\|_{C^{2+\beta,(2+\beta)/2}
(\overline{\mathcal{R}}_{\epsilon_1})}$
is bounded independently of the choice of  $\sigma$ in
$\mathcal{S}_{\epsilon_1}$. Thus $T$ is a compact operator.

As $\sigma \in \mathcal{S}_{\epsilon_1}$ varies continuously in
$C^{2,1}(\overline{\mathcal{R}}_{\epsilon_1})$ norm,
it is readily checked that $r, f, \ell$ varies continuously in
$C^{1+1/2}[0,\epsilon_1]$ norm,
which leads to a corresponding variation of $\mathbf{w}(\bar{x},t)$
in $C^{1,1}(\overline{\mathcal{R}}_{\epsilon_1})$ norm.
Standard parabolic estimate then requires
$\hat{\sigma}$ to vary continuously in
$C^{2,1}(\overline{\mathcal{R}}_{\epsilon_1})$ norm.
Hence $T$ is continuous on $\mathcal{S}_{\epsilon_1}$.
Schauder fixed point Theorem implies that $T$ has a fixed point
and the proof of the existence of solution is complete.


\subsection*{Proof of uniqueness}
Now we turn our attention to the uniqueness of smooth solutions.
 Let $(\tilde{\sigma}_i,\tilde{\mathbf{w}}_i, f_i,r_i), i=1,2$,
be two solutions of the moving boundary problem with the same
initial conditions. Let $g_i=g(\tilde{\mathbf{w}}_i,\tilde{\sigma}_i)$
and $h_i=h(\tilde{\mathbf{w}}_i,\tilde{\sigma}_i)$ for $i=1,2$.
Define $\hat{\sigma}=\tilde{\sigma}_{1}-\tilde{\sigma}_2$,
$\hat{g}=g_1-g_2$, $\hat{h}=h_1-h_2$,
$\hat{\mathbf{w}}=\tilde{\mathbf{w}}_1-\tilde{\mathbf{w}}_2$,
$\hat{\ell}={\ell}_1-{\ell}_2$, and $\hat{r}={r}_1-{r}_2$.
Then from \eqref{3.4}b),  $\hat{\sigma}$  satisfies
 \begin{equation} \label{sigmahat}
 \hat{\sigma}_{t}=\frac{g_1}{\ell_{1}^2}\hat{\sigma}_{\bar{x}\bar{x}}
+\frac{(r'_1-\bar{x}\ell'_1)}{\ell_1}\hat{\sigma}_{\bar{x}}
-\frac{(\tilde{\sigma}_{1\bar{x}}+\tilde{\sigma}_{2\bar{x}})}
{\ell_{1}^2}\hat{\sigma}_{\bar{x}}
 +G_{\sigma}(\bar{x}, t)
  \end{equation}
where the nonhomogeneous term is
\[
 G_{\sigma}=\Big(\frac{g_1}{\ell_{1}^2}- \frac{g_2}{\ell_{2}^2}\Big)
\tilde{\sigma}_{2\bar{x}\bar{x}}
+\Big(\frac{(r'_1-\bar{x}\ell'_1)}{\ell_1}
 - \frac{(r'_2-\bar{x}\ell'_2)}{\ell_2}\Big)\tilde{\sigma}_{2\bar{x}}
 -\Big(\frac{1}{\ell_{1}^2}-\frac{1}{\ell_{2}^2}\Big)
\tilde{\sigma}^2_{2\bar{x}} +(h_1-h_2)\,.
\]

Since $G_{\sigma}(\bar{x},0)=0$ and $\hat{\sigma}$ has zero
boundary and initial conditions in $\mathcal{R}_{\tau}$,
by \cite[ch.4, Thm. 9.2]{ladyzenskaja68},  for any $q>1$,
there exists a constant $K_{q,\tau}>0$, which remains bounded as
$\tau \to 0$, such that
 \[
 \| \hat{\sigma}\|_{W^{2,1}_{q}(\mathcal{R}_{\tau})} \leq K_{q,\tau}
 \left( \| \hat{g}\|_{C(\overline{\mathcal{R}_{\tau}})}
+\|\hat{r}\|_{C^1{([0,\tau])}}+\|\hat{\ell}\|_{C^1([0,\tau])}
+\|\hat{h}\|_{C(\overline{\mathcal{R}_{\tau}})} \right) \,,
  \]
where the right hand side is obtained by estimating
$L^{\infty}$-norm of $G_\sigma$.

By increasing the constant $K_{q,\tau}$ if necessary,
we can recast the above estimate as
 \begin{equation}
 \label{6.12}
 \| \hat{\sigma}\|_{W^{2,1}_{q}(\mathcal{R}_{\tau})}
\leq K_{q,\tau}
 \left( \| \hat{\sigma}\|_{C(\overline{\mathcal{R}_{\tau}})}
+\|\hat{r}\|_{C^1{([0,\tau])}}+\|\hat{\ell}\|_{C^1([0,\tau])}
+\|\hat{\mathbf{w}}\|_{C(\overline{\mathcal{R}_{\tau}})}\right) \,.
  \end{equation}
A similar calculation for $\hat{\mathbf{w}}$ using \eqref{3.4}a) yields
\[
\hat{\mathbf{w}}_t+\big[  \frac{\tilde{\sigma}_{1\bar{x}}}{\ell_1^2 }
-\frac{(r'_1-\bar{x}\ell'_1)}{\ell_1} \big] \hat{\mathbf{w}}_{\bar{x}}=
 G_w(\bar{x},t) \,,
 \]
 where
\[
 G_w(\bar{x},t)={-\big[  \frac{\tilde{\sigma}_{1\bar{x}}}{\ell_1^2 }
- \frac{\tilde{\sigma}_{2\bar{x}}}{\ell_2^2 }
-\frac{(r'_1-\bar{x}\ell'_1)}{\ell_1}
+ \frac{(r'_2-\bar{x}\ell'_2)}{\ell_2}\big]
\tilde{\mathbf{w}}_{2\bar{x}}-(\mathbf{F}( \tilde{\mathbf{w}}_1)
-\mathbf{F}( \tilde{\mathbf{w}}_2)) } \,.
\]
With $G_w(\bar{x},0)=0$ and $\hat{\mathbf{w}}$ vanishing at $t=0$
and on the right boundary of $\mathcal{R}_\tau$,
the compatibility conditions
at $(x,t)=(1,0)$ are satisfied.
By integrating along the characteristics,
there exists a constant $K_1>0$ such that
\begin{equation} \label{6.14}
\|\hat{\mathbf{w}}\|_{C(\overline{\mathcal{R}_{\tau}})}
\leq K_1\|G_w\|_{C(\overline{\mathcal{R}_{\tau}})}
\leq K_1(\|\hat{\sigma}\|_{C^{1,0}(\overline{\mathcal{R}_{\tau}})}
+\|\hat{r}\|_{C^1([0,\tau])}+\|\hat{\ell}\|_{C^1([0,\tau])}) \,.
\end{equation}
Next we estimate $\hat{\ell}$ and $\hat{r}$. By subtracting \eqref{3.3}b)
from \eqref{3.3}a), we obtain a governing equation for $\ell$.
Thus $\hat{\ell}$ satisfies an equation of the form
\begin{equation}\label{6.16}
\hat{\ell}'=m(t)\hat{\ell}+n(t)
\end{equation}
 for some functions $m$ and $n$ with initial condition $\hat{\ell}(0)=0$.
We note that $\|m\|_{C([0,\tau])}$ is bounded and
$\|n\|_{C([0,\tau])}\leq K_3 \|\hat{\sigma}
\|_{C^{1,0}(\overline{\mathcal{R}_{\tau}})} $
for some constant $K_3>0$. From (\ref{6.16}),
\begin{equation}  \label{6.17}
 \|\hat{\ell}\|_{C^1([0,\tau])}\leq K_4\|n\|_{C(0,\tau])}\leq K_5\|\hat{\sigma}\|_{C^{1,0}(\overline{\mathcal{R}_{\tau}})} \end{equation}
  for some constant $K_5>0$. A similar calculation gives
 \begin{equation}
 \label{6.18}
\|\hat{r}\|_{C^1([0,\tau])}\leq K_6(\|\hat{\sigma}\|_{C^{1,0}(\overline{\mathcal{R}_{\tau}})}+\|\hat{\ell}\|_{C([0,\tau])}) \leq K_7\|\hat{\sigma}\|_{C^{1,0}(\overline{\mathcal{R}_{\tau}})}
\end{equation}
 for some positive constants $K_6, \, K_7$.

Substituting (\ref{6.14}), (\ref{6.17}), (\ref{6.18}) in (\ref{6.12}),
we have
 $ \| \hat{\sigma}\|_{W^{2,1}_{q}(\mathcal{R}_{\tau})}
\leq K_ 8 \| \hat{\sigma}\|_{C^{1,0}(\overline{\mathcal{R}_{\tau}})}$
for some $K_8>0$. Note that the constants $K_1$ to $K_8$ remain
bounded as $\tau \downarrow 0$. Lemma 3.3 in
\cite[ch.4]{ladyzenskaja68}, with $\ell=1$,  $r=0$,  $s=1$,  and $q=6$
implies that
$ \| \hat{\sigma}\|_{C^{1+\lambda,
\frac{1+\lambda}{2}}(\overline{\mathcal{R}_{\tau}})}
\leq K_ {9} \| \hat{\sigma}\|_{W^{2,1}_{6}(\mathcal{R}_{\tau})}$
where $\lambda=\frac12$ and $K_{9}$ is independent of $\tau$.
This means that $\hat{\sigma}_{\bar{x}}$ is H\"older continuous
in $t$ with exponent $1/4$. Since $\hat{\sigma}(\cdot, 0)=0$,
combining the above inequalities, we have
\begin{equation}
\|\hat{\sigma}\|_{C^{1+\frac12,\frac34}(\overline{\mathcal{R}_{\tau}})}
 \leq K_{10} \tau^{1/4} \|\hat{\sigma}\|_{C^{1+\frac12,\frac34}
(\overline{\mathcal{R}_{\tau}})}
 \end{equation}
 for some constant $K_{10}>0$. By choosing $\tau$ small enough that
$K_{10}\tau^{1/4}<1$, we have $\hat{\sigma}=0$; i.e.,
 $\tilde{\sigma}_1=\tilde{\sigma}_2$. That
$\tilde{\mathbf{w}}_1=\tilde{\mathbf{w}}_2$,
$\tilde{\ell}_1=\tilde{\ell}_2$, and $\tilde{r}_1=\tilde{r}_2$
follow immediately
from (\ref{6.14}), (\ref{6.17}), (\ref{6.18}).
The uniqueness part of the proof is complete.

\section{Conclusion and open questions}

In \cite{choi05}, the  existence and uniqueness of local solutions
to a moving boundary problem \eqref{mogilner} modelling
cell motility is established when
$\gamma_p=0$ and $K =\kappa$ in (\ref{stress}). Such
assumptions are discarded in this paper so that some conservation
relation becomes unavailable.

If one puts (\ref{velocity}) and (\ref{stress}) into \eqref{mogilner},
at first glance the governing equations  look like a strongly
coupled parabolic system. It is, however, a system of first order
hyperbolic equations coupled with a parabolic equation and requires a
careful reformulation to make such an issue clear.
Through a clever choice of new independent
variables in section~\ref{sec:motivate}, the transformed equations
(\ref{wusOLD}) are weakly coupled, allowing a simpler analysis to
establish a priori bounds. The corresponding generalized problem
presented in section~\ref{sec:intro} is then transformed into the
problem \eqref{3.4}-\eqref{3.3} with a fixed domain.
A fixed point iterative scheme leads to the existence of solutions
to this moving boundary problem. Uniqueness then follows by
applying a priori estimates on the difference of two solutions.
We now cite some open problems
associated with this model:

(a) Having proved the local existence and uniqueness of the solution, a
natural step is the study of global existence of solutions.
Besides $\gamma_p=0$ and $K = \kappa$,
some special initial conditions  are needed in \cite{choi05}
to prove the global existence of solution.
Such simplifications allow the reduction of the model to
 a scalar parabolic equation with some
non-local moving boundary conditions. This reduction allows certain
techniques which are not possible for a system of equations.
There is some progress in the global existence  for
 a single simple hyperbolic equation coupled with a parabolic equation
 (Choi and Miller, in preparation).
For the system \eqref{wus} with appropriate restrictions on $\mathbf{F}$,
it will be interesting to see if a modification of such
ideas will work or some totally different tricks are necessary in the
study of its global existence of solution.

(b) The constitutive law (\ref{stress})  proposed by Mogilner
and Verzi in \cite{move03} is based on the assumption that the stress
can be modelled as the sum of two linear spring forces.
The actual stress-strain relationship inside a cell may be more
complicated.
For example one may just require that stress increases with extension
beyond its natural length.
Under such more general conditions,  the local  and global existence
of solutions can be studied.

 (c) A two-dimensional model has been proposed by Choi and Lui
in \cite{choiLui2D}.
The model was shown to admit a travelling domain solution, in the
sense that both the shape of the domain and the steady travelling speed
are parts of the solution.
Both the local and the global existence of solution to such a 2D model
has not been  established.

 \begin{thebibliography}{0}

\bibitem{choi05} Y. S. Choi, P. Groulx, and R. Lui;
 \emph{Moving boundary problem for a one-dimensional crawling nematode
sperm cell model}. Nonlinear Anal.ÊReal World Appl.
{\bf 6} (2005), no. 5, 874--898.

\bibitem{choileelui04} Y. S. Choi, J. Lee, and R. Lui;
 \emph{Traveling wave solutions for a one-dimensional crawling nematode
sperm cel model}. J. Math. Biol. {\bf 49} (2004), no. 3, 310--328.

\bibitem{choiLui2D} Y. S. Choi and R. Lui;
\emph{Existence of traveling domain solutions for a
two-dimensional moving boundary problem}. Transactions of AMS, to appear.

\bibitem{ladyzenskaja68}
O. A. Lady{\v{z}}enskaja, V. A. Solonnikov, and N. N. Ural'ceva;
\emph{Linear and quasilinear equations of parabolic type}
Translations of Mathematical Monographs,  Vol. 23. American
Mathematical Society, Providence, RI 1967.

\bibitem{move03} A. Mogilner and D. W. Verzi;
\emph{A simple 1-D physical model for the crawling nematode sperm cell}.
J. Stat. Phys. {\bf 110} (2003), 1169--1189.

\end{thebibliography}

\end{document}