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\markboth{\hfil A Stefan problem with kinetics  \hfil EJDE--2002/15}
{EJDE--2002/15\hfil Michael L. Frankel \& Victor Roytburd \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 2002}(2002), No. 15, pp. 1--27. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu  (login: ftp)}
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  Compact attractors for a Stefan problem with kinetics
 %
\thanks{ {\em Mathematics Subject Classifications:} 35R35, 74N20, 80A25.
\hfil\break\indent
{\em Key words:} Stefan problem, compact attractors, kinetic condition.
\hfil\break\indent
\copyright 2002 Southwest Texas State University. \hfil\break\indent
Submitted January 25, 2002. Published February 12, 2002.} }
\date{}
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\author{Michael L. Frankel \& Victor Roytburd}
\maketitle

\begin{abstract} 
 We prove existence of a unique bounded classical
 solution for a one-phase free-boundary problem with kinetics
 for continuous initial conditions.
 The main result of this paper establishes existence of a
 compact attractor for classical solutions of the problem.
\end{abstract}


\newtheorem{theorem}{Theorem}[section]
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\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{proposition}[theorem]{Proposition}

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\section{Introduction}

In this paper we study the asymptotic behavior of solutions of the modified
one-phase Stefan problem in one spatial dimension:
\begin{gather}
u_{t} = u_{xx}-\gamma u,\quad -\infty <x<s(t),  \label{prob1} \\
(\partial u/\partial x)|_{x=s(t)} = -V(t),\quad g(u|_{x=s(t)})=V(t),
\label{prob2} \\
u(x,0) = u^{0}(x).  \label{prob3}
\end{gather}
Here $u(x,t)$ is the temperature, the damping term is due to the volumetric
heat losses, $\gamma \geq 0$. The two boundary conditions overdetermine the
problem and allow us to find the free boundary whose position is denoted by
$s(t)$, and $V(t)=\dot{s}(t)$ is the velocity.

The free-boundary problem (\ref{prob1}-\ref{prob3}) arises naturally as a
mathematical model of a variety of exothermic phase transition type
processes, such as condensed phase combustion \cite{matsiv-solid} (also
known as self-sustained high-temperature synthesis or SHS \cite{munir}),
solidification with undercooling \cite{langer}, laser induced evaporation
\cite{gt}, rapid crystallization in thin films \cite{saarloos} etc. These
processes are characterized by production of heat at the interface, and
their dynamics is determined by the feedback mechanism between the heat
release due to the kinetics $g(u|_{x=s(t)})$\ and the heat dissipation by
the medium. The first boundary condition in (\ref{prob2}) (the Stefan
boundary condition) expresses the balance between the heat produced at the
free boundary and the heat diffusion through the adjacent medium. As the
problem (\ref{prob1})-(\ref{prob3}) describes propagation of the phase
transition front, the second boundary condition in (\ref{prob2}) is a
manifestation of the non-equilibrium nature of the transition; its analog for
the classical Stefan problem is just $u|_{x=s(t)}=0.$\ In the context of
condensed phase combustion the kinetic boundary condition expresses the
dependence of the propagation velocity on the flame front temperature.

\emph{The principal objective of the paper is to prove existence of a
compact attractor for the infinite-dimensional dynamical system generated by
the one-phase problem (\ref{prob1}-\ref{prob3}). }In other words, if the
problem is viewed as evolution in a functional space then for all the
initial conditions, solutions approach a compact set in the functional
space. This result is motivated by results of DNS and in particular by
numerical experiments described in \cite{port} and \cite{fin-let} (see also
\cite{physd}). In \cite{port} we performed DNS of dynamics of the system for
a variety of kinetic functions (actually, for a parametric family of kinetic
functions). We demonstrated that for a wide range of initial conditions the
system develops complex thermo-kinetic oscillations. The resulting asymptotic
regimes are attained very fast and do not depend on the initial conditions,
thus indicating existence of an attractor for the dynamics. As the parameter
governing kinetics is varied, the dynamical patterns exhibited by the system
include a Hopf bifurcation, period doubling cascades leading to chaotic
pulsations, a Shilnikov-Hopf bifurcation etc. Most of these patterns appear
to be of a finite-dimensional nature and are well-known for the
finite-dimensional dynamical systems.

On the other hand, in \cite{fin-let} we derived and studied a $3\times 3$\
system of ODEs which is a three-mode pseudo-spectral approximation to the
free-boundary problem: the dynamics of the finite-dimensional dynamical
system mimic those of the infinite-dimensional system to an amazing degree.
These observations led to the conjecture that the asymptotic dynamics of (
\ref{prob1}-\ref{prob3}) should be finite-dimensional. The result of this
paper is therefore a first step in proving this conjecture. In the sequel to
this paper \cite{fr-hausdorff} (to be published elsewhere), based on the
compactness results we demonstrate that the attractor actually has a finite
Hausdorff dimension.

In a nutshell, the idea of the proof of existence of a compact attractor is
as follows: we start with initial data inside a ball in a functional space
and establish uniform a priori estimates on the solutions and their first
spatial derivative that secure the existence of an absorbing compact set.
Then we use the simple abstract result from dynamical systems that the
$\omega $-limit set of a compact absorbing set is a compact attractor. It
should be noted however that the actual proof is not as straightforward as
just indicated. If the solution is viewed as a sum of contributions from the
free boundary and from initial data, it is easy to see that while the former
does compactify with time, the latter cannot compactify. Fortunately the
damping guarantees this contribution to decay exponentially. Also, since we
are working on an infinite spatial interval, the compactness is predicated
on a uniform spatial decay, which \textit{can} be justified for the
contribution from the free boundary.

The rest of the paper is organized as follows. In Sec.~2 we present some
minimal background information that places the model in the context of
condensed phase combustion. A local existence result is obtained in Sec.~3.
We are not interested in local existence per se since it is well established
in the literature for various versions of the problem, see for example \cite
{radkevich, yin, frsima}.The principal value of this result for our purposes
is in the direct construction for a solution that allows us to derive
necessary estimates for the solution and its spatial derivative. A solution
is found in the form of a single layer potential. We demonstrate that the
potential density necessarily should have a one over the square root
singularity. We develop a suitable version of potential theory and design a
convergent iteration scheme that produces the density and the free boundary
velocity.

The single-layer representation for the solution is employed in Sec.~4 to
obtain an estimate for the spatial derivative of the solution. This estimate
can be proved uniform with respect to the sup norm of initial data, provided
the solution itself is uniformly bounded. Such a uniform bound is also
presented in Sec.~4: namely, assuming that the kinetic function $g$
satisfies some natural requirements, we demonstrate an a priori estimate on
the solutions uniformly with respect to the initial conditions. The estimate
is based on a representation of the solution in terms of a combination of
single and double layer potentials. The latter representation is
instrumental for the argument of Sec.~5,  since it gives a natural
decomposition of the solution into the two contributions, one from the
initial conditions, and another one from the free boundary.

We show that the contributions from the free boundary are uniformly bounded
and decaying at infinity. Together with the uniform bound on the spatial
derivative, this guarantees that contributions from the free boundary for
different initial data from a fixed ball form a precompact set. At this
point the presence of the heat loss becomes crucial. Although the
contribution from initial data is not compactified by the evolution, it
decays exponentially with time due to the damping. Thus the evolution is a
combination of a compactifying part and a decaying part. We complete the
proof of existence of a compact attractor by incurring an appropriate
abstract result from dynamical systems.

\section{Motivation of the free boundary model}

In this section we sketch a derivation of the free
boundary model (\ref{prob1})-(\ref{prob3}). We note that although the
derivation is purely heuristic, the model itself is well accepted in the
literature. The interpretation of the free boundary problem
(\ref{prob1})-(\ref{prob3}) in terms of nonequilibrium phase transition
with the interface
attachment kinetics $g$ is rather transparent (see e.g. \cite{saarloos}),
here we would like to demonstrate its relevance for the condensed phase
combustion. For this type of combustion the solid fuel mixture is
transformed directly into a solid product. In addition to its theoretical
interest, gasless combustion currently finds technological applications as a
method of synthesizing certain technologically advanced materials for
high-temperature semiconductors, nuclear safety devices, fuel cells etc.,
see \cite{munir}, \cite{var2} and also \cite{var1} for a popular exposition.
The process is characterized by highly exothermic reactions propagating
through mixtures of fine elemental reactant powders (e.g., Ti + C, Ti +2B),
resulting in the synthesis of compounds.

The most primitive model of gasless combustion involves a system of
differential equations for the temperature $u$ and the concentration of the
fuel $C$ (see Shkadinsky \textit{et al.}, \cite{ShkaKhaMer}). For
appropriately nondimensionalized variables the one-dimensional formulation
of the model takes the form:
\begin{gather}
u_{t}=(\kappa u_{x})_{x}+qW(C,u)-\gamma u  \label{et} \\
C_{t}=-W(C,u)  \label{ec}
\end{gather}
where $\kappa $ is the thermal diffusivity, $W$ is the chemical reaction
rate, $q$ is the heat release and $\gamma u$, the volumetric heat loss.

For physically relevant values of parameters, the system is characterized by
the strong temperature sensitivity of the rate and by rather sharply defined
regions of dramatic change in the field variables that are usually
associated with propagating fronts. This suggests an alternative to the
model with distributed kinetics (the so-called flame sheet approximation,
see \cite{z}): the distributed reaction rate in (\ref{et})--(\ref{ec}) is
replaced by the $\delta $-function,
\begin{equation*}
W=g(u)\delta (x-s(t))
\end{equation*}
with an appropriate rate $g(u)$ supported at the interface $x=s(t)$ between
the fresh $(C=1)$ and burnt $(C=0)$ material (see, Matkowsky \& Sivashinsky,
\cite{matsiv-solid}). In the case of gaseous combustion with Arrhenius
kinetics, the sharp interface model can be obtained as an asymptotic
approximation of the distributed kinetics model in the large activation
energy limit. In this case the strength of the $\delta $-function $g(u)$ is
determined through an asymptotic analysis by matching relevant inner and
outer solutions. Of course, all the intricacies of the behavior in the
reaction zone are lost in this approximation.

The system (\ref{et})--(\ref{ec}) with the $\delta $-function source is
understood in the sense of distributions. This leads to the system of two
heat equations coupled at the interface
\begin{equation}\begin{gathered}
u_{t}^{-} =(\kappa u_{x}^{-})_{x}-\gamma u,\quad u_{t}^{+}=(\kappa
u_{x}^{+})_{x}-\gamma u   \\
u^{-}|_{x=s(t)} = u^{+}|_{x=s(t)},\quad (\kappa u_{x}^{+}-\kappa
u_{x}^{-})_{x=s(t)}=-w(u)_{x=s(t)}   \\
\frac{ds}{dt} = -g(u)|_{x=s(t)}
\end{gathered} \label{plusmin}
\end{equation}
where
\begin{gather*}
u^{-}(x,t) = u(x,t)\quad \mbox{for}\quad x<s(t),  \\
u^{+}(x,t) = u(x,t)\quad \mbox{for}\quad x>s(t).
\end{gather*}
This is the free interface \emph{two-phase} problem of condensed phase
combustion. The physical properties of the material such as the heat
diffusion coefficient $\kappa $ may differ substantially ahead and behind
the interface. If, for instance, the product is a foam-like substance then
$\kappa _{product}\ll \kappa _{fuel}$. By setting $\kappa _{product}=0$ in
the equation and the boundary condition for $u^{+}$ in (\ref{plusmin}), we
arrive at the \emph{one-phase} model problem in (\ref{prob1})-(\ref{prob2})
for $u\equiv u^{+}$.

We note that in the context of solidification of overcooled liquids or the
amorphous to crystalline transition the kinetic boundary condition
corresponds to the so-called interface attachment kinetics, which are
determined by various microscopic mechanisms of incorporating the matter
into the crystalline lattice at the interface. Concerning the choice of the
kinetic function we remark that this issue is far from settled either
theoretically or experimentally. For example, for solid combustion the
widely used exponential approximation of Arrhenius kinetics has not been
obtained from an analysis of molecular collisions in the spirit of the
kinetic theory of gases and, consequentally, asymptotic expansion in
transition to the $\delta $-function approximation, but rather
``transplanted'' from the sharp interface model of gas combustion. There are
several types of functions that were suggested for a more realistic
description of kinetics in specific chemical and physical settings.

We will assume that $g(u)$ is a monotonically decreasing differentiable
function on $[0,\infty ]$ with $|g'|\leq C$ and satisfying
\begin{equation}
-V_{0}\leq g(u)\leq -v_{0}\;\mathrm{{for}\;{some}\;}V_{0},v_{0}>0.
\label{kinetics}
\end{equation}
The lower bound is satisfied for the standard Arrhenius kinetics where
$V=ce^{-A/u}$ while the upper bound $v_{0}$ corresponds to the ignition
temperature (in our case, ``ignition velocity'') kinetics: the model is
valid only for moving fronts.

\section{Existence of local classical solutions}

 In order not to clutter formulas with factors of
the type $e^{-\gamma t}$, from now on, until Sec.~\ref{absorb-sec} we set
the damping coefficient $\gamma =0$. The modifications to the $\gamma >0$
case are trivial and will be indicated when needed. A short-time solution of
the free boundary problem (\ref{prob1}-\ref{prob3}) will be sought in the
form of a superposition of heat potentials,
\begin{equation}
u(x,t)=\int_{0}^{t}G(x,s(\tau ),t-\tau )\varphi (\tau )d\tau +\int_{-\infty
}^{0}G(x,\xi ,t)u^{0}(\xi )d\xi ,  \label{ro}
\end{equation}
where $G$ is the fundamental solution of the heat equation,
\begin{equation*}
G(x,\xi ,t-\tau )=\exp \left\{ -\frac{(x-\xi )^{2}}{4(t-\tau )}\right\}
\left[ 4\pi (t-\tau )\right] ^{-1/2}
\end{equation*}
The density of the single layer potential $\varphi $ and the front position
$s(t)$ are to be determined.

We will demonstrate a little later that the single-layer potential is
continuous up to the boundary and its derivative possesses the standard jump
property:
\begin{equation}
\lim_{x\to s(t)-}\frac{\partial }{\partial x}\int_{0}^{t}G(x,s(\tau
),t-\tau )\varphi (\tau )d\tau =\frac{\varphi (t)}{2}
+\int_{0}^{t}G_{x}(s(t),s(\tau ),t-\tau )\varphi (\tau )d\tau  \label{jump}
\end{equation}
This result is, of course, well-known if $\varphi $ is continuous. It turns
out however, that by the nature of the free-boundary problem at hand,
$\varphi $ must have a $1/\sqrt{t}$ singularity at 0. Thus a justification of
(\ref{jump}) will require an extra effort. If the jump property in (\ref
{jump}) holds then for the solution represented by (\ref{ro}), the boundary
conditions in (\ref{prob2}) yield the following equations
\begin{eqnarray}
u(s(t),t)&=&g^{-1}(V(t)) \label{ro1}\\
&=&\int_{0}^{t}G(s(t),s(\tau ),t-\tau )\varphi (\tau
)d\tau +\int_{-\infty }^{0}G(s(t),\xi ,t)u^{0}(\xi )d\xi  \nonumber \\
u_{x}(s(t),t)&=&-V(t) \label{ro2} \\
&=&\frac{\varphi }{2}-\int_{0}^{t}G_{\xi }(s(t),s(\tau
),t-\tau )\varphi (\tau )d\tau -\int_{-\infty }^{0}G_{\xi }(s(t),\xi
,t)u^{0}(\xi )d\tau  \nonumber
\end{eqnarray}

We will choose the density of the form $\varphi (t)=\psi (t)/\sqrt{t},$where
$\psi (t)$ is continuous on $[0,T]$. To motivate this choice, let us
consider asymptotics of (\ref{ro2}) as $t\to 0$. Let us assume for
simplicity of the argument that $u^{0}\in C^{1}$ and $V$ is continuous on
$[0,T]$. First we integrate by parts the second integral in (\ref{ro2}) and
note that it has a $1/\sqrt{t}$ singularity:
\begin{eqnarray}
\lefteqn{-\int_{-\infty }^{0}G_{\xi }(s(t),\xi ,t)u^{0}(\xi )d\xi }\nonumber\\
&=&
-u^{0}(0)\frac{\exp \{-s(t)^{2}/4t\}}{\sqrt{4\pi t}} +\int_{-\infty
}^{0}G(s(t),\xi ,t)u_{\xi }^{0}(\xi )d\xi \label{sing2}\\
&\sim& -u^{0}(0)\frac{\exp
\{-V(0)^{2}t/4\}}{\sqrt{4\pi t}}+\frac{u_{\xi }^{0}(0)}{2}  \nonumber
\end{eqnarray}
As to the first integral in (\ref{ro2}), for continuous $\varphi $ it
converges to $0$ as $t\to 0$ :
\begin{eqnarray}
|\int_{0}^{t}G_{\xi }(s(t),s(\tau ),t-\tau )\varphi (\tau )d\tau | &=&\frac{1
}{2}|\int_{0}^{t}\frac{s(t)-s(\tau )}{t-\tau }G\varphi (\tau )d\tau |  \notag
\\
&\sim &\frac{1}{2}|V(0)|\sup |\varphi |\sqrt{t},  \label{tzero1}
\end{eqnarray}
since $|G(\cdot ,\cdot ,t-\tau )|\leq 1/\sqrt{t-\tau }$. Thus, for a
continuous $\varphi $ the singularities in (\ref{ro2}) cannot balance.

If $\varphi $ has a singularity of the type $b/\sqrt{t}$ then the estimate
in (\ref{tzero1}) should be augmented by the term
\begin{eqnarray*}
\lefteqn{\int_{0}^{t}G_{\xi }(s(t),s(\tau ),t-\tau )\frac{b}{\sqrt{\tau }}
d\tau }\\
&=&\frac{b}{2}\int_{0}^{t}\frac{s(t)-s(\tau )}{t-\tau }\frac{\exp
\{-(s(t)-s(\tau ))^{2}/4(t-\tau )\}}{\sqrt{4\pi (t-\tau )\tau }}d\tau \\
&\sim& \frac{b}{2}V(0)\exp \{-V(0)^{2}t/4\}\int_{0}^{t}\frac{d\tau }{\sqrt{
4\pi (t-\tau )\tau }} \\
&=&\frac{b}{4}V(0)\sqrt{\pi }\exp \{-V(0)^{2}t/4\}
\end{eqnarray*}
which converges to the finite value. Thus, the only way to balance the
singularity (\ref{sing2}) in the boundary condition in (\ref{ro2}) is for
$\varphi $ itself to have a singularity. The balance condition then reads:
\begin{equation}
\lim\limits_{t\to 0}\sqrt{t}\varphi (t)=u^{0}(0)/\sqrt{\pi }
\label{bdef}
\end{equation}
A similar limit obtained from the first integral equation (\ref{ro1}) leads
to the initial condition for $V$:
\begin{equation}
V(0)=g(u^{0}(0))
\end{equation}

Next we rewrite the integral equations in (\ref{ro1})-(\ref{ro2}) in terms
of $\varphi $ and $V$:
\begin{gather}
V=K_{1}(V,\varphi )  \label{i-eq1}\\
\varphi =-2K_{1}(V,\varphi )+K_{2}(V,\varphi )  \label{i-eq2}
\end{gather}
where the nonlinear operators $K_{1},K_{2}$ are defined as follows
\begin{gather}
K_{1}(V,\varphi )=g\{\int_{0}^{t}G(s(t),s(\tau ),t-\tau )\varphi (\tau
)d\tau +\int_{-\infty }^{0}G(s(t),\xi ,t)u^{0}(\xi )d\xi \}  \label{i-eq3}\\
K_{2}(V,\varphi )=2\int_{0}^{t}G_{\xi }(s(t),s(\tau ),t-\tau )\varphi (\tau
)d\tau +2\int_{-\infty }^{0}G_{\xi }(s(t),\xi ,t)u^{0}(\xi )d\xi
\label{i-eq4}
\end{gather}
Here as usual,
\begin{equation}
s(t)=\int_{0}^{t}V(\tau )d\tau .  \label{i-eq-s}
\end{equation}
The equations are supplemented by the initial conditions:
\begin{equation}
V(0)=g(u^{0}(0));\qquad \lim\limits_{t\to 0}\sqrt{t}\varphi
(t)=u^{0}(0)/\sqrt{\pi }  \label{i-eq-ic}
\end{equation}

The principal goal of the present section is the proof of the following
local existence result:

\begin{theorem}\label{local}
Let $g<0$ be continuously differentiable, monotone
decreasing function, $u^{0}\in C(-\infty ,0]$, $\ u^{0}>0$. Then the problem
in (\ref{i-eq1})-(\ref{i-eq2}) has a unique solution $V,\varphi $ such that
$V$ and $\sqrt{t}\varphi (t)$ are continuous on $[0,\sigma ]$ for some
$\sigma >0$, where $\sigma $ depends only on $\sup u^{0}$.  The solution to
the free boundary problem is determined by $V,\varphi $ via the
representation (\ref{ro}) with $s(t)=\int_{0}^{t}V(\tau )d\tau$.
\end{theorem}

The proof of the theorem is given in the next subsections. Its outline is as
follows. First of all, we justify the integral equations by establishing the
single-layer potential jump property (\ref{jump}) for densities with the $1/
\sqrt{t}$ singularity. Then we demonstrate that the solution of the system
of integral equations (\ref{i-eq1})-(\ref{i-eq2}) generates a solution to
the free boundary problem via the representation (\ref{ro}). After that we
concentrate on existence for the system of integral equations. We show that,
if $\sigma >0$ is small enough, the integral operator is a contraction.

It should be noted that the singularity in the potential density precludes a
simple-minded iteration scheme from being a contraction. Roughly speaking,
the contraction rate for nonsingular densities is on the order of
$\sqrt{\sigma }$. The $1/\sqrt{t}$ singularity leads to a ``cancelation''
(the rate of order one) and prevents us from making the rate coefficient
smaller than one. To overcome this difficulty we introduce a two-step
iteration scheme.

Another standard precaution should be taken for the proof to proceed.
Because of the nonlinearity of the problem the contraction rate depends on
the size of $\{V,\varphi \}$ . Thus to guarantee that the iteration sequence
does not deteriorate the contraction rate and therefore requires smaller and
smaller $\sigma $, we need to secure the existence of a ball in the
functional space which is mapped by the operator into itself.

All the results of the section hold without the basic assumption on the
kinetic function in (\ref{kinetics}). Nonetheless, we do not hesitate to
assume it whenever it leads to a substantial simplification of the
presentation.

\subsection{Two lemmas on the single-layer potential\label{loc-ex}}

In this section we study properties of the single-layer potential whose
density has a one over square root singularity. For our purposes it is
convenient to introduce a norm which is appropriate for functions with this
singularity:
\begin{equation}
\left\| \varphi \right\| _{\sigma }=\sup_{0\leq \tau \leq \sigma }\sqrt{\tau
}|\varphi (\tau )|  \label{norm}
\end{equation}
Obviously, if $\varphi (t)=\psi (t)/\sqrt{t}$, where $\psi (t)$ is
continuous , then $\left\| \varphi \right\| _{\sigma }=\|\psi \|_{C[0,\sigma
]}$.

Specifically we are interested in the behavior of the spatial derivative of
the potential and its limit at the boundary.

\begin{lemma}
\label{lemma1} Let $\varphi (t)=\psi (t)/\sqrt{t}$, where $\psi (t)$ is a
continuous function on $[0,T]$ and let $s(t)$ be Lipschitz continuous on
$[0,T]$ and non-increasing. Then for every $0<t\leq T$, and $x<s(t)$
\begin{equation}
|\Phi (x,t)|=|\frac{\partial }{\partial x}\int_{0}^{t}G(x,s(\tau ),t-\tau
)\varphi (\tau )d\tau |\leq \text{\rm const}  \label{lem1}
\end{equation}
\end{lemma}

\paragraph{Proof}
The lemma holds if monotonicity condition for $s$ is dropped, but in our
case \ $s$ is monotone which simplifies the proof. It is convenient to
consider separately the two cases: $|s(t)-x|>1$ and $|s(t)-x|<1$.

For the case $|s(t)-x|>1$
\begin{equation*}\begin{aligned}
|\Phi (x,t)|=&\Big|\int_{0}^{t}\frac{x-s(\tau )}{2(t-\tau )}\frac{e^{-(x-s(\tau
))^{2}/4(t-\tau )}}{\sqrt{4\pi (t-\tau )}}\varphi (\tau )d\tau \Big| \\
=&\Big|\int_{0}^{t}\frac{(x-s(\tau ))^{2}}{2(t-\tau )(x-s(\tau ))}e^{-(x-s(\tau
))^{2}/8(t-\tau )}   \\
&\times \exp \{-\frac{(x-s(t))^{2}+2(x-s(t))(s(t)-s(\tau ))+(s(t)-s(\tau ))^{2}}{
8(t-\tau )}\} \\
&\times \frac{\psi (\tau ) }{\sqrt{4\pi \tau (t-\tau )}} \,d\tau \Big|  \\
\leq &\frac{C\left\| \varphi \right\| _{t}}{|s(t)-x|}e^{-v_{0}|x-s(t)|/4}
\int_{0}^{t}\frac{e^{-v_{0}^{2}(t-\tau )/8}}{\sqrt{4\pi \tau (t-\tau )}}
d\tau \leq \frac{Ce^{-v_{0}|x-s(t)|/4}}{|s(t)-x|}\left\| \varphi \right\|
_{t}
\end{aligned}
\end{equation*}
In the last estimate we used the following simple observations: $\eta
e^{-\eta }\leq $const, for $\eta =\dfrac{(x-s(\tau ))^{2}}{4(t-\tau )}>0$,
$|s(\tau )-x|>|s(t)-x|$ and
\begin{equation}
\int_{0}^{t}1/\sqrt{\tau (t-\tau )}d\tau =\allowbreak \pi .  \label{g-sqrt}
\end{equation}

\begin{remark}
Thus the proof above shows that if $|s(t)-s(\tau )|\geq v_{0}|t-\tau |$
which holds if the basic assumption on the kinetics in (\ref{kinetics}) is
satisfied, then the derivative decays exponentially
\begin{equation}
|\Phi (x,t)|\leq \frac{Ce^{-v_{0}|x-s(t)|/4}}{|s(t)-x|}\left\| \varphi
\right\| _{t}  \label{expdec}
\end{equation}
The exponent $-v_{0}/4$ can be improved to $-v_{0}/(2+\varepsilon )$ (at the
price of increasing $C$).
\end{remark}

For the case $|s(t)-x|<1$ we split the integral into the two parts
\begin{multline*}
\frac{\partial }{\partial x}\int_{0}^{t}G(x,s(\tau ),t-\tau )\varphi (\tau
)d\tau \\
=-\big[ \int_{0}^{t-\delta }+\int_{t-\delta }^{t}\big] \frac{
x-s(\tau )}{2(t-\tau )}G(x,s(\tau ),t-\tau )\varphi (\tau )d\tau ,
\end{multline*}
where $0<\delta <t$ to be chosen later on. In the estimates below we follow
rather closely the argument from Friedman \cite{friedman} (inequality
(1.18), p.219):
\begin{eqnarray*}
\lefteqn{
\int_{t-\delta }^{t}\frac{x-s(\tau )}{2(t-\tau )}G(x,s(\tau ),t-\tau
)\varphi (\tau )d\tau }\\
&=&\int_{t-\delta }^{t}\frac{x-s(t)}{2(t-\tau )}G(x,s(\tau ),t-\tau )\varphi
(\tau )d\tau \\
&&+\int_{t-\delta }^{t}\frac{s(t)-s(\tau )}{2(t-\tau )}G(x,s(\tau
),t-\tau )\varphi (\tau )d\tau \\
&=&I_{1}+I_{2}
\end{eqnarray*}
We shall estimate the difference between $I_{1}$ and
\begin{equation*}
J_{1}=\int_{t-\delta }^{t}\frac{x-s(t)}{2(t-\tau )}G(x,s(t),t-\tau )\varphi
(\tau )d\tau
\end{equation*}
as follows
\begin{equation}
|I_{1}-J_{1}|=\Big| \int_{t-\delta }^{t}\frac{x-s(t)}{2(t-\tau )}
G(x,s(t),t-\tau )\Big\{ 1-e^{\frac{(x-s(t))^{2}-(x-s(\tau ))^{2}}{4(t-\tau )
}}\Big\} \varphi (\tau )d\tau \Big|  \label{i1j1}
\end{equation}
Since obviously $1-\exp (-\eta )<\eta $ for any $\eta >0$, the expression in
the braces is estimated:
\begin{eqnarray*}
0 &<&1-\exp [\frac{(x-s(t))^{2}-(x-s(\tau ))^{2}}{4(t-\tau )}]<\frac{s(\tau
)-s(t)}{4(t-\tau )}\left[ s(t)-x+s(\tau )-x\right] \\
&=&\frac{s(\tau )-s(t)}{4(t-\tau )}[2(s(t)-x)+s(\tau )-s(t)]\leq \frac{V_{0}
}{4}[2(s(t)-x)+s(\tau )-s(t)]
\end{eqnarray*}
here $V_{0}$ is the Lipschitz constant for $s(t)$ (the maximal velocity). We
note now that
\begin{equation*}
\sup_{t-\delta \leq \tau \leq t}|\varphi (\tau )|=\sup_{t-\delta \leq \tau
\leq t}(|\varphi (\tau )|\sqrt{\tau })|/\sqrt{\tau }\leq \left\| \varphi
\right\| _{t}/\sqrt{t-\delta }
\end{equation*}
and continue (\ref{i1j1}):
\begin{align*}
&|I_{1}-J_{1}| \\
&\leq \int_{t-\delta }^{t}\frac{s(t)-x}{2(t-\tau )}G(x,s(t),t-\tau )\frac{
V_{0}}{4}[2(s(t)-x)+s(\tau )-s(t)]\frac{\left\| \varphi \right\| _{t}}{\sqrt{
t-\delta }}d\tau \\
&=\frac{V_{0}}{4}\frac{\left\| \varphi \right\| _{t}}{\sqrt{t-\delta }}
\int_{t-\delta }^{t}\Big\{ \frac{[s(t)-x]^{2}}{(t-\tau )}+\frac{s(\tau
)-s(t)}{2(t-\tau )}[s(t)-x]\Big\} e^{-\frac{(x-s(t))^{2}}{4(t-\tau )}}
\frac{d\tau }{\sqrt{4\pi (t-\tau )}} \\
&\leq \frac{V_{0}}{4}\frac{\left\| \varphi \right\| _{t}}{\sqrt{t-\delta }}
\sqrt{\delta /\pi }\Big\{ C+\sqrt{\delta }\frac{V_{0}}{2}[s(t)-x]\Big\} .
\end{align*}
In the last inequality we have used $\eta ^{p}\exp (-\eta )\leq C$, for any
$p>0$.

The integral $J_{1}$ can be reduced via a substitution $4(t-\tau
)/[s(t)-x]^{2}=z$ as follows
\begin{eqnarray*}
|J_{1}| &=&\int_{t-\delta }^{t}\frac{s(t)-x}{4\sqrt{\pi }(t-\tau )^{3/2}}e^{-
\frac{(x-s(t))^{2}}{4(t-\tau )}}|\varphi (\tau )|d\tau \\
&\leq &\frac{\left\| \varphi \right\| _{t}}{\sqrt{t-\delta }}\frac{1}{2\sqrt{
\pi }}\int\nolimits_{0}^{\delta /[s(t)-x]^{2}}z^{-3/2}e^{-1/z}dz
\end{eqnarray*}
Since $\frac{1}{\sqrt{\pi }}\int\nolimits_{0}^{\infty
}z^{-3/2}e^{-1/z}dz=1/2 $ and the integrand is positive we have
\begin{equation*}
|J_{1}|<\frac{1}{2}\frac{\left\| \varphi \right\| _{t}}{\sqrt{t-\delta }}.
\end{equation*}
Now we need to estimate $I_{2}$.
\begin{eqnarray*}
|I_{2}| &=&\Big| \int_{t-\delta }^{t}\frac{s(t)-s(\tau )}{2(t-\tau )}
G(x,s(\tau ),t-\tau )\varphi (\tau )d\tau \Big| \\
&\leq &\frac{V_{0}}{2}\frac{\left\| \varphi \right\| _{t}}{\sqrt{t-\delta }}
\int_{t-\delta }^{t}G(x,s(\tau ),t-\tau )d\tau \leq \frac{V_{0}}{2}\frac{
\left\| \varphi \right\| _{t}}{\sqrt{t-\delta }}\sqrt{\delta /\pi }
\end{eqnarray*}
Finally, for $\left| x-s(t)\right| <1$ we get that on the interval
$t-\delta \leq \tau \leq t
$\begin{align*}
&\Big| \int_{t-\delta }^{t}\frac{x-s(\tau )}{2(t-\tau )}G(x,s(\tau ),t-\tau
)\varphi (\tau )d\tau \Big|\\
&\leq |I_{1}|+|I_{2}| \leq |I_{1}-J_{1}|+|J_{1}|+|I_{2}| \\
&\leq \frac{V_{0}}{4}\frac{\left\| \varphi \right\| _{t}}{\sqrt{t-\delta }}
\sqrt{\delta /\pi }\Big\{ C+\sqrt{\delta }\frac{V_{0}}{2}[s(t)-x]\Big\} +
\frac{1}{2}\frac{\left\| \varphi \right\| _{t}}{\sqrt{t-\delta }}+\frac{V_{0}
}{2}\frac{\left\| \varphi \right\| _{t}}{\sqrt{t-\delta }}\sqrt{\delta /\pi }
\\
&=\frac{\left\| \varphi \right\| _{t}}{\sqrt{t-\delta }}\Big[ \frac{1}{2}
+C_{1}\sqrt{\delta /\left( t-\delta \right) }\Big] .
\end{align*}
As for the estimate on the interval $0\leq \tau \leq t-\delta $ for $\left|
x-s(t)\right| <1$ we get
\begin{equation*}
\Big|\int_{0}^{t-\delta }\frac{x-s(\tau )}{2(t-\tau )}\frac{e^{-(x-s(\tau
))^{2}/4(t-\tau )}}{\sqrt{4\pi (t-\tau )}}\varphi (\tau )d\tau \Big|
\leq C_{2}\frac{\left\| \varphi \right\| _{t}}{\delta }\sqrt{t-\delta }
\end{equation*}
Now, by combining the estimates above
\begin{equation*}
|\Phi (x,t)|\leq C_{2}\frac{\left\| \varphi \right\| _{t}}{\delta }\sqrt{
t-\delta }+\frac{\left\| \varphi \right\| _{t}}{\sqrt{t-\delta }}\Big[
\frac{1}{2}+C_{1}\sqrt{\delta /\left( t-\delta \right) }\Big]
\end{equation*}
we conclude the proof of the lemma for $\left| x-s(t)\right| <1$. It is
possible to optimize the above estimate by choosing an appropriate $\delta $.
However for our purposes it will suffice to set $\delta =ct$ that results
in
\begin{equation}
|\Phi (x,t)|\leq C\left\| \varphi \right\| _{t}/\sqrt{t}  \label{putnumb}
\end{equation}
\quad\hfill$\diamondsuit$

\begin{remark} \rm
The above estimate for the derivative $\Phi $ is obtained for the density
$\varphi =\psi (t)/\sqrt{t}$. If $\varphi $ itself is a continuous function
then the above estimate becomes
\begin{equation}
|\Phi (x,t)|\leq C\left\| \varphi \right\| _{t}/\sqrt{t}=C\sup_{0\leq \tau
\leq t}|\varphi (\tau )\sqrt{\tau }|/\sqrt{t}\leq C\sup_{0\leq \tau \leq
t}|\varphi (\tau )|  \label{rem}
\end{equation}
\end{remark}

The next lemma presents a version of the classical jump property for the
single-layer potential with singularity.

\begin{lemma}
\label{dpotential/dt}Let $\varphi (t)=\psi (t)/\sqrt{t}$, where $\psi (t)$
is a continuous function on $[0,T]$ and let $s(t)$ be Lipschitz continuous
on $[0,T]$ and non-increasing. Then for every $0<t\leq T$
\begin{equation}
\lim_{x\to s(t)-}\Phi (x,t)=\frac{1}{2}\varphi
(t)+\int_{0}^{t}G_{x}(s(t),s(\tau ),t-\tau )\varphi (\tau )d\tau  \notag
\end{equation}
\end{lemma}

\paragraph{Proof}
As in the previous lemma the result holds if the \ monotonicity condition
for $s$ is dropped. For $\varphi (t)$ continuous the result is contained in
Friedman \cite{friedman}. The proof for our case follows the continuous case
with relatively minor modifications.

It is easy to see that for any fixed $\delta >0$
\begin{equation*}
\int_{0}^{t-\delta }\frac{x-s(\tau )}{2(t-\tau )}\frac{e^{-\frac{(x-s(\tau
))^{2}}{4(t-\tau )}}\varphi (\tau )d\tau }{\sqrt{4\pi (t-\tau )}}\to
\int_{0}^{t-\delta }\frac{s(t)-s(\tau )}{2(t-\tau )}\frac{e^{-\frac{
(s(t)-s(\tau ))^{2}}{4(t-\tau )}}\varphi (\tau )d\tau }{\sqrt{4\pi (t-\tau )}
}
\end{equation*}
as $x\to s(t)_{-}$ since the singularity at $\tau =0$ is integrable.
On the other hand, on the interval $[t-\delta ,t]$ the density $\varphi
(\tau )$ is nonsingular and the classical argument shows that
\begin{equation*}
\int_{t-\delta }^{t}\frac{x-s(\tau )}{2(t-\tau )}\frac{e^{-\frac{(x-s(\tau
))^{2}}{4(t-\tau )}}\varphi (\tau )d\tau }{\sqrt{4\pi (t-\tau )}}\to
\int_{t-\delta }^{t}\frac{s(t)-s(\tau )}{2(t-\tau )}\frac{e^{-\frac{
(s(t)-s(\tau ))^{2}}{4(t-\tau )}}\varphi (\tau )d\tau }{\sqrt{4\pi (t-\tau )}
}+\frac{\varphi (\xi )}{2}
\end{equation*}
where $t-\delta $ $\leq \xi \leq t$. By passing to the limit $\delta
\to 0$ one obtains the result of the lemma.
\hfill$\diamondsuit$ \smallskip

Now consider the integral representation (\ref{ro}) with $\varphi ,V$ (
$s(t)=\int_{0}^{t}V(\tau )d\tau $) being a solution of the system of integral
equations (\ref{i-eq1})-(\ref{i-eq2}). Since $G$ is a fundamental solution
of the heat equation, for $x<s(t)$, $t>0$, $u(x,t)$ solves the heat
equation. Similar to the argument in the proof of the lemma, it is easy to
show that $\lim_{x\to s(t)-}u(x,t)$ exists and is equal to the right
hand side of the integral equation (\ref{ro1}). Thus, by the virtue of the
integral equation the kinetic boundary conditions is satisfied. The Stefan
boundary conditions is nothing else than the integral equation in (\ref{ro2})
which is justified through the lemma. Finally, for $x<0$ it is easily seen
that $\lim_{t\to 0}u(x,t)=u^{0}(x)$.

\subsection{Iteration scheme}

The system of integral equations in (\ref{i-eq1})-(\ref{i-eq2}) will be
solved iteratively. Given $\phi =$ $(\varphi ,V)$ we define the operator
$K:(\varphi ,V)\to \omega =(\chi ,v)$ through the following two-stage
procedure. First we define
\begin{eqnarray*}
\chi &=&-2V+K_{2}(V,\varphi )\\
&=& -2V+2\int_{0}^{t}G_{\xi }(S(t),S(\tau ),t-\tau )\varphi (\tau )d\tau
+2\int_{-\infty }^{0}G_{\xi }(S(t),\xi ,t)u^{0}(\xi )d\xi
\end{eqnarray*}
where
$S(t)=\int_{0}^{t}V(\tau )d\tau $.
Then, on the base of the just found $\chi $ we compute $v$:
\begin{equation*}
v=K_{1}(V,\chi )=g\{\int_{0}^{t}G(S(t),S(\tau ),t-\tau )\chi (\tau )d\tau
+\int_{-\infty }^{0}G(S(t),\xi ,t)u^{0}(\xi )d\xi \}
\end{equation*}
We will show that $K$ has a fixed point, which obviously provides a solution
to the original integral equations (\ref{i-eq1})-(\ref{i-eq2}),
(\ref{i-eq-s})-(\ref{i-eq-ic}).

\subsection{Invariant ball}

We start with the following remark. Based on its physical interpretation,
the kinetic function $g(u)$ is defined for $0<u<\infty $ and varies in the
interval $-V_{0}<V<-v_{0}=g(0)$. It is not clear a priori, whether the
integral operator $K$ preserves the ``physical'' cone of positive
temperatures. To avoid complications caused by using an iteration scheme in
the set $\{g(K_{1}(V,\varphi ))>0\}$, we extend the function $g$ to the
interval $(-\infty ,0)$ as $g(u)\equiv -v_{0}$. We abuse the notation
slightly using the same letter for the extension (which has the same
Lipschitz constant as the original $g$).

In the space of pairs $\Xi =\{\phi =(\varphi ,V):\varphi (.)\sqrt{.},V\in
C[0,\sigma ]\}$ we define the norm
\begin{equation*}
\|\phi \|=\max \{\|\varphi \|_{\sigma },\|V\|_{C[0,\sigma ]}\}=\max
\{\|\varphi (.)\sqrt{.}\|_{C[0,\sigma ]},\|V\|_{C[0,\sigma ]}\},
\end{equation*}
that makes $\Xi $ a Banach space. The fixed point will be sought in the
closed set $B_{M,\sigma }=\{\phi =(\varphi ,V):-V_{0}\leq V\leq -v_{0},\,\|\varphi \|_{\sigma }\leq M\}$ with $M$ and $\sigma $ to be determined.

First we note that the velocity component of the operator automatically
remains in $B_{M,\sigma }$ by virtue of the definition of $g$:
\begin{equation}
-V_{0}\leq g\{\int_{0}^{t}G(s(t),s(\tau ),t-\tau )\varphi (\tau )d\tau \
+\int_{-\infty }^{0}G(s(t),\xi ,t)u^{0}(\xi )d\tau \}\leq -v_{0}  \notag
\end{equation}
In a similar fashion, for the $\varphi $-component of $K\phi $ we obtain:
\begin{eqnarray*}
\|\chi \|_{\sigma } &=&\sup_{0\leq t\leq \sigma }\sqrt{t}
\Big(2V+2|
\int_{0}^{t}G_{\xi }(S(t),S(\tau ),t-\tau )\varphi (\tau )d\tau | \\
&&+2|\int_{-\infty }^{0}G_{\xi }(S(t),\xi ,t)u^{0}(\xi )d\xi |\Big)
\end{eqnarray*}
To estimate the first integral we again use (\ref{g-sqrt}),
\begin{eqnarray*}
\lefteqn{\Big|\int_{0}^{t}G_{\xi }(S(t),S(\tau ),t-\tau )\varphi (\tau )d\tau
\Big|}\\
&=&\Big| \int_{0}^{t}\frac{S(t)-S(\tau )}{2(t-\tau )}G(S(t),S(\tau ),t-\tau
)\varphi (\tau )d\tau \Big|\\
&\leq&
\int_{0}^{t}\frac{1}{2}|V(\theta )|\frac{1}{\sqrt{4\pi (t-\tau )}\sqrt{\tau }
}|\varphi (\tau )|\sqrt{\tau }d\tau \leq \frac{\sqrt{\pi }}{2}V_{0}M
\end{eqnarray*}
The second integral is treated as follows:
\begin{eqnarray*}
\lefteqn{\Big|\int_{-\infty }^{0}\frac{\xi -S(t)}{2t}\frac{e^{-(\xi -S(t))
^{2}/4t}}{\sqrt{4\pi t}}u^{0}(\xi )d\xi \Big| }\\
&=&\frac{1}{\sqrt{t}}|\int_{-\infty }^{0}\sqrt{8}\frac{\xi -S(t)}{\sqrt{8t}}
e^{-(\xi -S(t))^{2}/8t}\sqrt{2}\frac{e^{-(\xi -S(t))^{2}/8t}}{\sqrt{8\pi t}}
u^{0}(\xi )d\xi | \\
&\leq &\frac{4}{e\sqrt{t}}\|u^{0}\|.
\end{eqnarray*}
Thus
\begin{equation}
\|\chi \|_{\sigma }\leq \frac{\sqrt{\pi }}{2}V_{0}M\sqrt{\sigma }+\frac{4}{e}
\|u^{0}\|  \notag
\end{equation}
If the right side of the above inequality is less or equal than $M$ then $K$
will map $B_{M,\sigma }$ into itself. This is insured by choosing $\sqrt{
\sigma }<2/(V_{0}\sqrt{\pi })$ and consequently
\begin{equation*}
M\geq \frac{8\|u^{0}\|}{e(2-V_{0}\sqrt{\pi \sigma })}
\end{equation*}

\subsection{Iteration for density}

Now we will prove that for a sufficiently small $\sigma $, $K$ is a
contraction in the density component. Let $\omega =K\phi $ , $\omega
'=K\phi '$. For the $\chi $-component of $\omega -$
$\omega '$ the estimates are as follows,
\begin{eqnarray}
\lefteqn{|\chi -\chi '|} \nonumber \\
&\leq& 2\| V-V'\|  \label{density}\\
&&+2\Big|\int_{-\infty }^{0}G_{\xi }(S(t),\xi ,t)u^{0}(\xi )d\xi -\int_{-\infty
}^{0}G_{\xi }(S'(t),\xi ,t)u^{0}(\xi )d\xi \Big|  \notag \\
&&+2\Big|\int_{0}^{t}G_{\xi }(S(t),S(\tau ),t-\tau )\varphi (\tau )d\tau
-\int_{0}^{t}G_{\xi }(S'(t),S'(\tau ),t-\tau )\varphi
'(\tau )d\tau \Big|  \nonumber \\
&=&2\| V-V'\| +2\ |w_{1}|+2|w_{2}|  \nonumber
\end{eqnarray}
First we estimate $w_{1}$. Suppose $S(t)<S'(t)<0$ and split the
integral for $w_{1}$ into three integrals:
\begin{equation}
w_{1}=\int_{-\infty }^{S}\delta G_{\xi }u^{0}d\xi +\int_{S}^{S'}\delta G_{\xi }u^{0}d\xi +\int_{S'}^{0}\delta G_{\xi }u^{0}d\xi
\label{lc-w1}
\end{equation}
where
\begin{equation*}
\delta G_{\xi }=G_{\xi }(S(t),\xi .t)-G_{\xi }(S'(t),\xi ,t).
\end{equation*}
By the mean value theorem,
\begin{eqnarray*}
|\delta G_{\xi }| &=&\Big| (S-S')\frac{\partial ^{2}G}{\partial
x^{2}}(\tilde{s}-\xi ,0,t)\Big| \\
&=&\frac{1}{\sqrt{4\pi }}\left| S(t)-S'(t)\right| \Big| \frac{(
\tilde{s}-\xi )^{2}}{4t^{5/2}}-\frac{1}{2t^{3/2}}\Big| \ e^{-\frac{(\tilde{
s}-\xi )^{2}}{4t}}
\end{eqnarray*}
where $\tilde{s}$ is an intermediate value:
\begin{equation*}
\tilde{s}=\tilde{s}(t,\xi ),\quad S(t)\leq \tilde{s}\leq S'(t).
\end{equation*}
Thus for the first integral in (\ref{lc-w1}) we have
\begin{eqnarray*}
\lefteqn{\big|\int_{-\infty }^{S}\delta G_{\xi }u^{0}d\xi \big|}\\
&\leq&
\frac{\left| S(t)-S'(t)\right| }{t}\int_{-\infty }^{S}
\Big| \frac{(\tilde{s}-\xi )^{2}}{4t}-\frac{1}{2}\Big|
e^{-\frac{(\tilde{s}-\xi )^{2}}{8t}}\frac{1}{\sqrt{4\pi t}}
 e^{-\frac{(\tilde{s}-\xi )^{2}}{8t}}|u^{0}(\xi)|d\xi \\
&\leq& \| V-V'\| \int_{-\infty }^{0}C\frac{1}{\sqrt{4\pi t}}\
e^{-\frac{(\tilde{s}-\xi )^{2}}{8t}}|u^{0}(\xi )|d\xi \leq C\sup |u^{0}(\xi
)|\,\| V-V'\| .
\end{eqnarray*}
The integral over $(S',0)$ in (\ref{lc-w1}) is estimated similarly.
As to the second integral in (\ref{lc-w1}) , it is even simpler:
\begin{eqnarray*}
\lefteqn{\big|\int_{S}^{S'}\delta G_{\xi }u^{0}d\xi \big|}\\
&=&\big|\int_{S}^{S'}\frac{1}{\sqrt{4\pi t}}
\big( \frac{\xi -S}{2t}e^{-\frac{(S-\xi )^{2}}{4t}}-
\frac{\xi -S'}{2t}e^{-\frac{(S'-\xi )^{2}}{4t}}\big)
u^{0}(\xi )d\xi \big| \\
&\leq& \int_{S}^{S'}\frac{1}{\sqrt{4\pi t}}\big( \frac{\xi -S}{2t}+
\frac{S'-\xi }{2t}\big) u^{0}(\xi )d\xi
 \frac{(S'-S)}{2t}\frac{1}{\sqrt{4\pi t}}2(S'-S)\|u^{0}\| \\
&\leq& C\ t^{1/2}\| V-V'\| \, \| u^{0}\|
\end{eqnarray*}
Finally, by combining the preceding estimates we get:
\begin{equation}
w_{1}\leq (C+C_{3}\ t^{1/2})\| V-V'\| \,\,\| u^{0}\| .
\end{equation}
Next we estimate the free boundary contribution $w_{2}$:
\begin{equation}
w_{2}\leq \int_{0}^{t}|\Delta G_{\xi }|\ |\varphi |\ d\tau
+\int_{0}^{t}|G_{\xi }(S'(t),S'(\tau ),t-\tau )|\,\|\varphi -\varphi '|\ d\tau  \label{w21}
\end{equation}
For $|\Delta G_{\xi }|$ we get:
\begin{align*}
&|\Delta G_{\xi }|\equiv |G_{\xi }(S(t),S(\tau ),t-\tau )-G_{\xi }(S'(t),S'
(\tau ),t-\tau )| \\
&=|S(t)-S(\tau )-(S'(t)-S'(\tau ))|\,|\frac{\partial ^{2}G
}{\partial x^{2}}(\tilde{s},0,t-\tau )| \\
&=\frac{1}{\sqrt{4\pi }}\left| S(t)-S'(t)-(S(\tau )-S'(\tau
))\right| \,|\frac{\tilde{s}^{2}}{4(t-\tau )^{5/2}}-\frac{1}{2(t-\tau )^{3/2}
}|\ e^{-\frac{\tilde{s}^{2}}{4(t-\tau )}} \\
&\leq \frac{1}{(t-\tau )^{1/2}}\big( \frac{1}{e}+\frac{1}{2}\big) \frac{1}{
\sqrt{4\pi }}\big| \frac{dS}{dt}(\tilde{\tau})-\frac{dS'}{dt}(
\tilde{\tau})\big| \leq C\frac{\|V-V'\|}{(t-\tau )^{1/2}}
\end{align*}
where $\tilde{s}$ is between $S(t)-S(\tau )$ and $S'(t)-S'(\tau )$.
Therefore
\begin{eqnarray*}
\int_{0}^{t}|\Delta G_{\xi }|\ |\varphi |\ d\tau
&\leq& C\|V-V'\|\int_{0}^{t}\frac{\sqrt{\tau }|\varphi |}{\sqrt{\tau }
\sqrt{t-\tau }}\ d\tau \\
&\leq &C\pi \|V-V'\|\,\|\varphi\|_{t}=C_{1}\|V-V'\|\,\|\varphi \|_{t}\,.
\end{eqnarray*}
Meanwhile,
\begin{eqnarray}
\lefteqn{\int_{0}^{t}|G_{\xi }(S'(t),S'(\tau ),t-\tau )|\,
|\varphi -\varphi '|\ d\tau }\nonumber\\
&=&  \int_{0}^{t}\big| \frac{S'(t)-S'(\tau )}{2(t-\tau )}
\big| |G(S'(t),S'(\tau ),t-\tau )|\ \frac{1}{\sqrt{\tau }
}\sqrt{\tau }\ |\varphi -\varphi '|\ d\tau  \notag\\
&\leq& C_{2}\|V'\|\,\|\varphi -\varphi '\|_{t}\   \label{w22}
\end{eqnarray}
Together (\ref{w21})-(\ref{w22}) yield
\begin{equation*}
w_{2}\leq C_{1}\|V-V'\|\,\|\varphi \|_{t}+C_{2\ }\|V'\|\,\|\varphi -\varphi '\|_{t}\
\end{equation*}
Thus
\begin{eqnarray*}
|\chi -\chi '|&\leq& 2\| V-V'\| +2\ |w_{1}|+2|w_{2}| \\
&=&\big( 2+(C+C_{3}\ t^{1/2})\| u^{0}\| +C_{1}\|\varphi \|_{t}\big)
\| V-V'\| +C_{2\ }\|V'\|\,\|\varphi -\varphi
'\|_{t}
\end{eqnarray*}
and we observe that although the densities $\chi $ and $\chi '$
both have singularities at zero, their difference is bounded. This rather
remarkable result demonstrates that the integral operator is a contraction.
Indeed,
\begin{equation}
\|\chi -\chi '\|_{\sigma }=\sup_{0\leq t\leq \sigma }\sqrt{t}|\chi
(t)-\chi '(t)|\leq \sqrt{\sigma }[c_{1}\| V-V'\|
+c_{2\ }\|\varphi -\varphi '\|_{\sigma }]  \label{esthi}
\end{equation}
where the constants $c_{1}$ and $c_{2}$ depend explicitly on $M$ and $\|
u^{0}\| $. By taking $\sigma $ sufficiently small, one can make $K$ a
contraction (in the $\varphi $-component).

\subsection{Iteration for velocity}

For the $V$-component of $\omega -$ $\omega '$ we have
\begin{eqnarray}
|v-v'| &\leq &L\Big\{\big|\int_{-\infty }^{0}[G(S(t),\xi ,t)u^{0}(\xi
)-G(S'(t),\xi ,t)]u^{0}(\xi )d\xi \big|  \notag \\
&&+\big|\int_{0}^{t}[G(S(t),S(\tau ),t-\tau )\chi -G(S'(t),S'(\tau ),t-\tau )
\chi ']d\tau \big|\Big\}  \notag \\
&=&L\{W_{1}+W_{2}\}  \label{contr-v}
\end{eqnarray}
where $L$ is the Lipschitz constant for $g$. This estimate is very similar
to (\ref{density}), the principal difference being that the integrand there
contains $G_{\xi }$ instead of $G$. The estimates for separate terms are
quite elementary and are based on the mean value theorem. To estimate $W_{2}$
we will need a bound for the difference,
\begin{eqnarray*}
|\Delta G|&\equiv& |G(S(t)-S(\tau ),t-\tau ,0)-G(S'(t)-S'(\tau ),t-\tau ,0)| \\
&=&|S(t)-S(\tau )-(S'(t)-S'(\tau ))|\,|\frac{\partial G}{
\partial x}(\tilde{s},t-\tau ,0)| \\
&=&\big| \frac{S(t)-S'(t)-(S(\tau )-S'(\tau ))}{2(t-\tau )}
\big| \,|\tilde{s}G(\tilde{s},t-\tau ,0)|\  \\
&=&\frac{1}{2}\big| \frac{dS}{dt}(\tilde{\tau})-\frac{dS'}{dt}(
\tilde{\tau})\big| \,|\tilde{s}G(\tilde{s},t-\tau ,0)|\\
&\leq& \frac{1}{2}\|V-V'\| \,|\tilde{s}G(\tilde{s},t-\tau ,0)|
\end{eqnarray*}
where $\tau \leq \tilde{\tau}\leq t$ and $\tilde{s}$ are intermediate
values. Since $\tilde{s}$ is between $S'(t)-S'(\tau )$ and
$S(t)-S(\tau )$,
\begin{equation*}
|\tilde{s}|\leq \max \left\{ |S'(t)-S'(\tau
)|,|S(t)-S(\tau )|\right\} \leq V_{0}(t-\tau )
\end{equation*}
Taking into account $|G|\leq \lbrack 4\pi (t-\tau )]^{-1/2}$ we get the
estimate
\begin{equation}
|\Delta G|\leq C_{1}\|V-V'\|\ (t-\tau )^{1/2}.  \label{contr-del}
\end{equation}
where $C_{1}=V_{0}/4\sqrt{\pi }$ is an absolute constant.
For the term $W_{2}$ we obtain
\begin{eqnarray}
W_{2}&\leq& \int_{0}^{t}|\Delta G|\ \chi \ d\tau
+\int_{0}^{t}G(S'(t),S'(\tau ),t-\tau )\ |\chi -\chi '|\ d\tau  \notag \\
&\leq& C_{1}\|V-V'\|\int_{0}^{t}\frac{\sqrt{(t-\tau )}}{\sqrt{\tau }}
\sqrt{\tau }\ \chi \ d\tau  \nonumber \\
&&+\int_{0}^{t}\ \frac{e^{-(S'(t)-,S'(\tau ))^{2}/4(t-\tau )}
}{\sqrt{4\pi (t-\tau )\tau }}\sqrt{\tau }|\chi -\chi '|\ d\tau \label{w2}\\
&\leq& \ C_{1}\frac{t\pi }{2}\|V-V'\|\ M+\|\chi -\chi '\|_{t}
\frac{\sqrt{\pi }}{2}  \notag \\
&\leq& \ (C_{1}\frac{t\pi }{2}+C\sqrt{t})\|V-V'\|\ M+C_{0}\sqrt{t}
\|\varphi -\varphi '\|_{t}\frac{\sqrt{\pi }}{2}  \notag
\end{eqnarray}

The estimation for the $W_{1}$ in (\ref{contr-v}) is a little different.
Suppose $S(t)<S'(t)<0$ and split the integral for $W_{1}$ into
three integrals:
\begin{equation}
W_{1}=\int_{-\infty }^{S}\delta Gu^{0}d\xi +\int_{S}^{S'}\delta
Gu^{0}d\xi +\int_{S'}^{0}\delta Gu^{0}d\xi  \label{contr-ic}
\end{equation}
where
$\delta G=G(S(t),t,\xi )-G(S'(t),t,\xi )$.
By the mean value theorem,
\begin{equation*}
\delta G=(S-S')\frac{\partial G}{\partial x}(\tilde{s}-\xi
,t,0)=(S-S')\frac{\tilde{s}-\xi }{2t}G(\tilde{s}-\xi ,t,0)
\end{equation*}
where $\tilde{s}$ is an intermediate value:
$\tilde{s}=\tilde{s}(t,\xi )$, $S(t)\leq \tilde{s}\leq S'(t)$.

If $\xi <S<\tilde{s}<S'$ then
\begin{eqnarray*}
\lefteqn{(\tilde{s}-\xi )G(\tilde{s}-\xi ,t,0)}\\
 &=&(4\pi )^{-1/2}(\tilde{s}-\xi
)t^{-1/2}e^{-(\tilde{s}-\xi )^{2}/4t} \\
&=&(4\pi )^{-1/2}(8t)^{1/2}\frac{\tilde{s}-\xi }{(8t)^{1/2}}e^{-(\tilde{s}
-\xi )^{2}/8t}2^{1/2}(2t)^{-1/2}e^{-(\tilde{s}-\xi )^{2}/8t} \\
&\leq &4t^{1/2}c_{1}G(\tilde{s}-\xi ,2t,0)\leq 4c_{1}t^{1/2}G(\tilde{s}-\xi
,2t,0)
\end{eqnarray*}
where $c_{1}=\max (xe^{-x^{2}})$. Thus for the first integral in (\ref
{contr-ic}) we have
\begin{eqnarray*}
\big|\int_{-\infty }^{S}\delta Gu^{0}d\xi \big|
 &\leq &\frac{S-S'}{2t} 4c_{1}t^{1/2}
 \int_{-\infty }^{S}G(S-\xi ,2t,0)|u^{0}(\xi )|d\xi \\
&\leq &2c_{1}t^{1/2}\frac{1}{t}\int_{0}^{t}\left[ V(\tau )-V'(\tau )
\right] d\tau \int_{-\infty }^{0}G(S-\xi ,2t,0)|u^{0}(\xi )|d\xi \\
&\leq &2c_{1}t^{1/2}\sup |u^{0}(\xi )|\,\| V-V'\| .
\end{eqnarray*}
The integral over $(S',0)$ in (\ref{contr-ic}) is estimated
similarly. As to the second integral in (\ref{contr-ic}), it is even
simpler:
\begin{eqnarray*}
\big|\int_{S}^{S'}\delta Gu^{0}d\xi \big|
&=&\big|\int_{S}^{S'}[G(S(t),t,\xi )-G(S'(t),t,\xi )]\ d\xi \big|\\
&\leq& (S'-S)2\sup G\cdot \| u^{0}\|  \\
&=&(S'-S)2(4\pi )^{-1/2}t^{-1/2}\| u^{0}\|  \\
&\leq &2(4\pi )^{-1/2}\ t^{1/2}\| V-V'\| \,\| u^{0}\|
\end{eqnarray*}
Finally, by combining the preceding estimates we get:
\begin{equation}
W_{1}\leq C_{3}\ t^{1/2}\| V-V'\| \,\,\| u^{0}\| .
\label{contr-w1}
\end{equation}
Thus, the estimates in (\ref{w2}), (\ref{contr-w1}) yield
\begin{eqnarray}
\|v-v'\|&=&\sup_{0\leq t\leq \sigma }|v-v'|  \notag \\
&\leq& C_{3}\ \sigma ^{1/2}\| V-V'\| \,\,\| u^{0}\| +\
C_{1}\frac{\sigma \pi }{2}\|V-V'\|\, M \notag\\
&&+C_{0}\sqrt{\sigma }\|\varphi-\varphi '\|_{\sigma }\frac{\sqrt{\pi }}{2}
 \label{estv}\\
&\leq& (A_{1}\sigma ^{1/2}+A_{2}\sigma )\| V-V'\| +A_{3}\sigma
^{1/2}\,\|\varphi -\varphi '\|_{\sigma }, \nonumber
\end{eqnarray}
where the constants depend only on $M$ and $\| u^{0}\| $. It is clear
from (\ref{esthi}) and (\ref{estv}) that the map $K$ is a contraction for a
sufficiently small $\sigma $.

This completes the proof of Theorem \ref{local} since by the contraction
mapping principle, the system of integral equations has a unique solution.
As we saw at the end of Sec.~\ref{loc-ex} it produces the solution for the
problem (\ref{prob1})-(\ref{prob3}).

\section{A priori bounds and global existence}

\setcounter{equation}{0}

\setcounter{equation}{0}In this section we present a priori bounds for
solutions of the free-boundary problem and their first spatial derivatives.
The a priori bound for the solutions was established in our prior work \cite
{all-amer} (where a more involved case of $g$ sublinear is also considered);
its proof is included here for reader's convenience.

\subsection{A priori estimate for the solution}

The derivation of the bound is based upon an integral representation of the
solution that we will obtain next. If $G$ is the fundamental solution of the
heat equation and $u(x,t),\;s(t)$ is a classical solution of (\ref{prob1})-(
\ref{prob3}), then by integrating Green's identity
\begin{equation}
\frac{\partial }{\partial \xi }\big( G\frac{\partial u}{\partial \xi }-u
\frac{\partial G}{\partial \xi }\big) -\frac{\partial }{\partial \tau }
(Gu)=0
\end{equation}
over the domain $\xi <s(\tau ),\,0<\tau <t$ and using the Stokes formula, we
obtain the integral representation for the solution:
\begin{eqnarray}
u(x,t)&=&\int_{0}^{t}G(x,s(\tau ),t-\tau )\left[ -V(\tau )+U(\tau )V(\tau )
\right] d\tau  \label{mp} \\
&&-\int_{0}^{t}\frac{\partial G}{\partial \xi }(x,s(\tau ),t-\tau )U(\tau
)d\tau +\int_{-\infty }^{0}G(x,\xi ,t)u^{0}(\xi )d\xi  \notag
\end{eqnarray}
which employs both the single-layer and double-layer potentials. In the
limit $x\to s(t)-$ the integral representation yields the following
equation:
\begin{gather}
\frac{1}{2}U(t)=\int_{0}^{t}G(s(t),s(\tau ),t-\tau )[-V(\tau )]d\tau
+\int_{0}^{t}G(s(t),s(\tau ),t-\tau )U(\tau )V(\tau )d\tau  \notag \\
-\int_{0}^{t}\frac{\partial G}{\partial \xi }(s(t),s(\tau ),t-\tau )U(\tau
)d\tau +\int_{-\infty }^{0}G(x,\xi ,t)u^{0}(\xi )d\xi  \label{inteq1}
\end{gather}
where
\begin{equation}
U(t)=g^{-1}(V(t)),\quad s(t)=\int_{0}^{t}V(\tau )d\tau .
\end{equation}
Note that the factor $\frac{1}{2}$ arises from the jump relation for the
second integral in (\ref{mp}).

We will show first that the temperature at the front, $U(t)$, is uniformly
bounded. The proof consists of separate estimates for different terms in (
\ref{inteq1}). For the first integral we obtain:
\begin{eqnarray}
\int_{0}^{t}G(s(t),s(\tau ),t-\tau )[-V(\tau )]d\tau
&\leq& V_{0}\int_{0}^{t}{
\frac{e^{-v_{0}^{2}(t-\tau )/4}}{\sqrt{4\pi (t-\tau )}}}d\tau \\
&\leq& V_{0}\int_{0}^{\infty }{\frac{e^{-v_{0}^{2}\tau /4}}{\sqrt{4\pi \tau }}}
d\tau =V_{0}/v_{0}.  \notag
\end{eqnarray}
In the above estimate we have used the obvious bound:
\begin{equation*}
\frac{(s(t)-s(\tau ))^{2}}{t-\tau }=\big[ \frac{s(t)-s(\tau )}{t-\tau }
\big] ^{2}(t-\tau )\geq v_{0}^{2}(t-\tau ).
\end{equation*}
We combine together the two subsequent integrals with respect to $\tau $
from (\ref{inteq1}):
\begin{equation}
\Phi (t)=\int_{0}^{t}G(s(t),s(\tau ),t-\tau )U(\tau )\big[ V(\tau )-\frac{1
}{2}\frac{s(t)-s(\tau )}{t-\tau }\big] d\tau ,  \label{heat2}
\end{equation}
note that the term with the average velocity,
\begin{equation*}
\frac{s(t)-s(\tau )}{t-\tau },
\end{equation*}
arises from the explicit differentiation in $\partial G/\partial \xi $.
\noindent The estimate is based on the observation that the integrand in (
\ref{heat2}) is \emph{negative } for the velocity of large magnitude.

\begin{lemma}
\label{l2} Let $g(U)\geq -V_{0}$ for $U\geq 0$ then the function $\Phi (t)$
defined by (\ref{heat2}) is bounded uniformly in $t$.
\end{lemma}


\paragraph{Proof}
We split the domain of integration in (\ref{heat2}) into the two sets:
\begin{equation}
\lbrack 0,t]=\{\tau :V(\tau )<-V_{0}/2\}\cup \{\tau :V(\tau )\geq
-V_{0}/2\}=B_{-}\cup B_{+}.  \label{split}
\end{equation}
Note that on $B_{-}$
\begin{equation}
V(\tau )-\frac{1}{2}\frac{s(t)-s(\tau )}{t-\tau }=V(\tau )-\frac{1}{2}
V(\sigma )\leq -V_{0}/2+V_{0}/2=0,  \label{dif1}
\end{equation}
where $\tau \leq \sigma \leq t$. On $B_{+}$ the absolute value of the
difference in (\ref{dif1}) is bounded by $\sup_{B_{+}}|V|=V_{0}/2$,
therefore, since $U\geq 0$ and $G>0$, we obtain the estimate:
\begin{eqnarray}
\Phi (t) &\leq &\int_{B_{+}}G(s(t),s(\tau ),t-\tau )U(\tau )\left[ V(\tau )-
\frac{1}{2}\frac{s(t)-s(\tau )}{t-\tau }\right] d\tau ,  \notag  \label{est2}
\\
&\leq &\frac{g^{-1}(-V_{0}/2)V_{0}}{2}\int_{B_{+}}\frac{e^{-v_{0}^{2}(t-\tau
)/4}}{\sqrt{4\pi (t-\tau )}}d\tau \leq \frac{g^{-1}(-V_{0}/2)V_{0}}{2v_{0}}.
\end{eqnarray}
\quad\hfill$\Box$ \smallskip

Therefore for the interface temperature $U$ we obtained the bound:
\begin{eqnarray*}
U(t)&=&2\{\int_{0}^{t}G(s(t),s(\tau ),t-\tau )[-V(\tau )+U(\tau )V(\tau )]d\tau
\\
&&-\int_{0}^{t}\frac{\partial G}{\partial \xi }(s(t),s(\tau ),t-\tau )U(\tau
)d\tau +\int_{-\infty }^{0}G(s(t),\xi ,t)u^{0}(\xi )d\xi \} \\
&\leq& \frac{\lbrack g^{-1}(-V_{0}/2)+2]V_{0}}{v_{0}}+2\|u^{0}\|\equiv
R_{fb}+2\|u^{0}\|.
\end{eqnarray*}

We have shown that the solution on the free boundary is bounded. In
combination with the boundedness of the initial data it yields boundedness
of the solution everywhere:

\begin{theorem}
\label{apriori}Let the kinetic function $g$ satisfy the kinetic condition in
\ref{kinetics}. If $u(x,t),V(t)$ is a solution of the free boundary problem
( \ref{prob1})-(\ref{prob3}) then the functions $u,V$ are bounded,
\begin{equation}
0\leq u(x,t)\leq R_{fb}+2\|u^{0}\|,  \label{rfb}
\end{equation}
where $R_{fb}$ is an ``absolute'' constant determined by the kinetics.
\end{theorem}

The proof is extremely simple. We ignore the boundary condition on $u_{x
\text{ }}$in (\ref{prob2}) and note that a solution $u(x,t)$ of the free
boundary problem solves the initial value problem for the heat equation with
the given Dirichlet boundary conditions $U(t)=g^{-1}((\dot{s}(t))$ at the
free boundary. Since both initial data and the boundary conditions are
bounded, $u(x,t)$ is also bounded by the maximum principle.

As a corollary we note here \textit{the global existence result} that
follows from the local existence and from the a priori bound (cf. Sec.~5 of
\cite{frsima}).

\begin{remark} \rm
Bounds $V_{0}$ and $v_{0}$ play very different roles in the previous
results. It can be shown that a version of the a priori estimate (\ref{rfb})
holds even if the condition $|g|\leq V_{0}$ is relaxed to
$g(u)/u^{1+\varepsilon }\to 0$ as $u\to \infty $ (see \cite
{all-amer} for details).
\end{remark}

\subsection{A priori estimate for the derivative}

\begin{theorem}
\label{thmux}Consider the ball $\left\| u^{0}\right\| \leq R$. There exists
$\sigma >0$ depending on $R$ such that for any fixed $t$, $0<t\leq \sigma $,
the derivative of the solutions of the free boundary problem with the
initial data from the ball is uniformly bounded. More specifically
\begin{equation*}
\left| u_{x}(x,t)\right| \leq \frac{C}{1+|s(t)-x|}
\end{equation*}
where $C$ is determined by $R$ and $t$.
\end{theorem}

\paragraph{Proof}
Consider the solution $u(x,t)$ given by the single-layer integral
representation (\ref{ro}). By Theorem\ref{local} the representation is valid
on some life span $\sigma $ that is completely determined by $R$. We
differentiate $u$ with respect to $x$
\begin{equation}
u_{x}(x,t)=\frac{\partial }{\partial x}\int_{0}^{t}G(x,s(\tau ),t-\tau
)\varphi (\tau )d\tau +\int_{-\infty }^{0}\frac{\partial }{\partial x}
G(x,\xi ,t)u^{0}(\xi )d\xi  \notag
\end{equation}

It was shown in Sec.\thinspace 3 for $|s(t)-x|>1$, see (\ref{expdec}), that
\begin{equation}
\frac{\partial }{\partial x}\int_{0}^{t}G(x,s(\tau ),t-\tau )\varphi (\tau
)d\tau \leq \frac{C\|\varphi \|_{\sigma }}{|s(t)-x|}  \label{dest1}
\end{equation}
and for $|s(t)-x|\leq 1$, [see (\ref{putnumb})]
\begin{equation}
\frac{\partial }{\partial x}\int_{0}^{t}G(x,s(\tau ),t-\tau )\varphi (\tau
)d\tau \leq C\left\| \varphi \right\| _{\sigma }/\sqrt{t}  \label{dest2}
\end{equation}
Now we need the estimate for the integral of the initial data
\begin{align}
\Big| \int_{-\infty }^{0}\frac{\partial }{\partial x}G(x,\xi ,t)u^{0}(\xi
)d\xi \Big|
&=\int_{-\infty }^{0}\frac{1}{\sqrt{4\pi t}}\frac{|\xi -x|}{2t}
e^{-\frac{(x-\xi )^{2}}{4t}}\left| u^{0}(\xi )\right| d\xi  \label{dest3} \\
&=\frac{1}{\sqrt{t}}\int_{-\infty }^{0}\frac{2}{\sqrt{8\pi t}}e^{-\frac{
(x-\xi )^{2}}{8t}}\frac{|\xi -x|}{\sqrt{8t}}e^{-\frac{(x-\xi )^{2}}{8t}
}\left| u^{0}(\xi )\right| d\xi \nonumber\\
&\leq \frac{2}{e\sqrt{t}}\left\| u^{0}\right\|
\notag
\end{align}
It was shown in the proof of the local existence of solutions that, given a
bound on the initial data $u^{0}$, the density $\varphi $ belongs to the
invariant ball of radius $M$ and therefore is uniformly bounded for all
initial conditions within the bound. Thus, (\ref{dest1})-(\ref{dest3}) yield
the result of the theorem.

\begin{corollary}
\label{corux}For all $t\geq t_{0}$, where $t_{0}\leq \sigma $ the derivative
is uniformly bounded:$\ $\ $\left| u_{x}(x,t)\right| \leq C$.
\end{corollary}

For the proof we note that $u_{x}$ solves the heat equation in the domain
$\{(x,t):t>t_{0},\;x<s(t)\}$ with the initial data $u_{x}(x,t_{0})$ and
boundary conditions $V(t)$ which are both bounded. This yields a bound in
the interior of the domain by the maximum principle.

\section{Absorbing set and attractor\label{absorb-sec}}

In this section we use the estimates obtained above to establish the
existence of a bounded absorbing set and of the attractor which is compact
in the space of continuous functions. In order to establish compactness of
the attractor we need to reinstitute the heat losses. It can be easily
verified that all the analytical properties of the solutions and estimates
established above can be only improved by introducing the heat losses. On
the other hand the problem with the heat losses exhibits uniform exponential
decay in time of the contribution of initial data that is necessary for the
proof of compactness of the attractor.

With the heat loss, the integral representation of the solution in terms of
mixed potential (\ref{mp}) takes the form
\begin{multline*}
u(x,t)\int_{0}^{t}e^{-\gamma (t-\tau )}G(x,s(\tau ),t-\tau )\left[
-V(\tau )+U(\tau )V(\tau )\right] d\tau \\
-\int_{0}^{t}e^{-\gamma (t-\tau )}\frac{\partial G}{\partial \xi }
(x,s(\tau ),t-\tau )U(\tau )d\tau +e^{-\gamma t}\int_{-\infty }^{0}G(x,\xi
,t)u^{0}(\xi )d\xi
\end{multline*}
The representation describes globally in time the evolution of the initial
temperature distribution $u^{0}$: $u(t)=T(t)u^{0}$. We think of the
evolution as taking place for the functions on the fixed interval $(-\infty
,0)$. To achieve this we introduce the moving coordinate system attached to
the free boundary $x'=x-s(t)$.

We split the semigroup operator $T$ into two parts: the contribution of the
free boundary
\begin{eqnarray}
T_{1}(t)u^{0}(x') &=&\int_{0}^{t}e^{-\gamma (t-\tau )}G(x',s(\tau )-s(t),t-\tau )\left[ -V(\tau )+U(\tau )V(\tau )\right] d\tau
\notag \\
&&-\int_{0}^{t}e^{-\gamma (t-\tau )}\frac{\partial G}{\partial \xi }
(x',s(\tau )-s(t),t-\tau )U(\tau )d\tau  \label{t1}
\end{eqnarray}
and that of the initial data
\begin{equation}
T_{2}(t)u^{0}=e^{-\gamma t}\int_{-\infty }^{0}G(x',\xi
-s(t),t)u^{0}(\xi )d\xi  \label{t2}
\end{equation}

As a basic metric space we choose a ball in the space $C(-\infty ,0]$:
\begin{equation*}
X=\{u\in C(-\infty ,0];\quad \|u\|=\sup |u(x')|\leq N\}
\end{equation*}
where the radius $N$ is large enough (it suffices to take $N>R_{fb}+2R_{abs}$
where $R_{abs}$ is the radius of the absorbing ball which is estimated in
the following proposition). Note that by Theorem \ref{apriori}, the
evolution of any ball $B_{R}$ of radius $R\leq (N-R_{fb})/2$ stays in $X$
for all time.

The following result establishes existence of an absorbing set for the
evolution.

\begin{proposition}
(i) The semigroup $T_{2}$ is uniformly contracting:
\begin{equation*}
r_{X}(t)=\sup_{u^{0}\in X}\|T_{2}(t)u^{0}\|\to 0\quad \mathrm{as}
\quad t\to \infty .
\end{equation*}
(ii) There exists a constant, $R_{abs}$, totally determined by the kinetics
such that any ball $B_{a}=\{u\in X: \|u\|\leq a\}$, where
$a=R_{abs}+\varepsilon <N$, is an absorbing set for any ball $B_{R}$ (where
$R\leq (N-R_{fb})/2$) with respect to the evolution by $T_{1}$ (and $T$).
\end{proposition}


\paragraph{Proof}
As is easily seen from (\ref{t2}) the contribution of the initial data
decays uniformly $\|T_{2}(t)u^{0}\|\leq e^{-\gamma t}\|u^{0}\|$. The
contribution from the free boundary is represented through the mixed
potential integral with the densities $U$ and $-V+UV$. Both densities are
bounded$:|V(\tau )|\leq V_{0}$ and $U$ was estimated above in (\ref{rfb}).
In the presence of the heat losses the estimate is modified
\begin{equation*}
 |U(\tau )|\leq R_{fb}+2e^{-\gamma t}\|u^{0}\|
\end{equation*}
(the value of $R_{fb}$ is even slightly less than in (\ref{rfb}): $v_{0}$
should be replaced by $v_{0}+\sqrt{\gamma }$). Next we estimate
contributions of both potentials in the expression for $T_{1}$. The
estimates are very similar to the estimates for the derivative of the single
layer potential in Lemma \ref{lemma1}:
\begin{eqnarray}
\lefteqn{\int_{0}^{t}e^{-\gamma (t-\tau )}\frac{e^{-(x-s(\tau ))^{2}/4(t-\tau )}}{
\sqrt{4\pi (t-\tau )}}\left[ -V(\tau )+U(\tau )V(\tau )\right] d\tau } \notag
\\
&\leq& \int_{0}^{t}V_{0}(R_{fb}+2e^{-\gamma \tau }N+1)e^{-(x-s(\tau
))^{2}/8(t-\tau )}  \notag \\
&&\times \exp \{-\frac{(x')^{2}+2x'(s(t)-s(\tau
))+(s(t)-s(\tau ))^{2}}{8(t-\tau )}\}\frac{e^{-\gamma (t-\tau )}d\tau }{
\sqrt{4\pi (t-\tau )}}  \notag \\
&\leq& e^{-v_{0}|x'|/4}\int_{0}^{t}V_{0}(R_{fb}+2e^{-\gamma \tau
}N+1)e^{-(s(t)-s(\tau ))^{2}/8(t-\tau )}\frac{e^{-\gamma (t-\tau )}d\tau }{
\sqrt{4\pi (t-\tau )}}  \notag \\
&\leq& e^{-v_{0}|x'|/4}\int_{0}^{t}V_{0}(R_{fb}+2e^{-\gamma \tau
}N+1)e^{-v_{0}^{2}(t-\tau )/8}\frac{e^{-\gamma (t-\tau )}d\tau }{\sqrt{4\pi
(t-\tau )}}  \notag \\
&\leq& \frac{\sqrt{2}V_{0}(R_{fb}+2e^{-\gamma t}N+1)}{v_{0}}
e^{-v_{0}|x'|/4}.  \label{t11}
\end{eqnarray}

The estimation for the double layer potential term from (\ref{t1}) is almost
identical to the corresponding estimate in Lemma \ref{dpotential/dt}. For
$|x'|>1$, it produces the bound:
\begin{equation}
\Big| \int_{0}^{t}e^{-\gamma (t-\tau )}\frac{\partial G}{\partial \xi }
(x',s(\tau )-s(t),t-\tau )U(\tau )d\tau \Big|  
\leq \frac{c_{1}(R_{fb}+2e^{-\gamma t}N)}{v_{0}(1+|x'|)}
e^{-v_{0}|x'|/4},  \label{t12}
\end{equation}
while for $|x'|<1$ it is bounded by $c_{2}(R_{fb}+2e^{-\gamma
t}N)/v_{0}$. Both $c_{1}$ and $c_{2}$ are explicit, order one constants.

If now we take $R_{abs}$ equal to the sum of the constants in the above
estimates (\ref{t11})-(\ref{t12}) then
\begin{equation}
|T_{1}(t)u^{0}|\leq R_{abs}e^{-v_{0}|x'|/4}+Ce^{-\gamma
t}e^{-v_{0}|x'|/4}N  \label{estt1}
\end{equation}
if $\|u^{0}\|\leq N$. By choosing $t_{1}$ such that $Ce^{-\gamma
t_{1}}N<a-R_{abs}$ we ensure that the orbit of any bounded subset of $X$
enters $B_{a}$ and remains there after that time $t_{1}$, which means that
$B_{a}$ is absorbing for $X$. \hfill$\Box$ \smallskip

Next we prove that the boundary contribution to the evolution, i.e. the
operators $T_{1}(t)$ are \textit{uniformly compact}. Namely, the following
proposition holds:

\begin{proposition}
There exists $t_{0}>0$ such that $\cup _{t\geq t_{0}}T_{1}(t)X$ is
relatively compact in $X$.
\end{proposition}


\paragraph{Proof}
The proof of the proposition contains the following two basic ingredients:
We establish certain estimates on the functions $T_{1}(t)u$, and their first
spatial derivatives, uniformly in $u\in X$, that are valid for any $t\geq
t_{0}>0$, next we demonstrate that the set determined by the estimates is
relatively compact.

First we recall that by Corollary \ref{corux} for sufficiently small
$t_{0}>0 $ and any $u^{0}\in X$, \begin{equation*}
|(T(t)u^{0})_{x}|\leq C\text{ for }t\geq t_{0},\ x\in (-\infty ,0]
\end{equation*}
On the other hand the contribution from the free boundary
\begin{eqnarray*}
|(T_{1}(t)u^{0})_{x}|&=&|(T(t)u^{0})_{x}-(T_{2}(t)u^{0})_{x}| \\
&\leq &|(T(t)u^{0})_{x}|+|(T_{2}(t)u^{0})_{x}|\\
&\leq& C+C\|u^{0}\|/\sqrt{t}\leq C
\end{eqnarray*}
since the contribution of the initial conditions is also uniformly bounded\break
$|(T_{2}(t)u^{0})_{x}|\leq C\|u^{0}\|/\sqrt{t}$, see (\ref{dest3}). Therefore
the family $\cup _{t\geq t_{0}}T_{1}(t)X$ is equicontinuous.

For \ the version of Arzela-Ascoli theorem appropriate for $(-\infty ,0]$ we
need uniform boundedness and uniform decay of the family of functions as
$|x'|\to \infty $. These properties are provided by the
estimate (\ref{estt1}) that gives a uniform exponential decay. Then it is
easy to construct a finite $\varepsilon $-net by choosing a finite interval
beyond which the functions of the family are smaller than $\varepsilon $ and
extending the elements of the $\varepsilon $-net from this interval by zero.
\hfill$\Box$\smallskip

The properties of the evolution operator $T(t)$ described in the above
propositions allow us to apply the abstract general result (see, for
example, \cite{temam}\ Chap. 1) that in our situation can be stated as
follows:

\begin{theorem}
The continuous semigroup $T(t)$, $T(t)=T_{1}(t)+T_{2}(t)$ with $T_{1}(t)$
uniformly compact and $T_{2}(t)$ uniformly contracting has the following
properties: the $\omega $-limit set $A$ of the absorbing set $B_{a}$ is a
compact attractor for the metric space $X$; $A$ is the maximal attractor in
$X$ and it is connected.
\end{theorem}

\section{Concluding remarks}

Compactness of the attractor and ultimately its finite Hausdorff dimension
(see \cite{fr-hausdorff}) for the free boundary problem modeling
nonequilibrium solidification and SHS is a rather remarkable fact,
especially in view of the surprising wealth of possible dynamical scenarios.
The situation should be compared, perhaps, to the similar facts known for
the Kuramoto-Sivashinsky equation or Navier-Stokes equations. In both cases
the compactness is shown for finite intervals whose length enters also into
the estimate on the Hausdorff dimension. In our case, however, the domain of
the field variable is an infinite interval.

The compactness result was proved here in the presence of heat losses for
any nonzero heat loss. Although we chose to operate in spaces of continuous
uniformly bounded functions on the infinite interval, we believe that
compactness can be established in spaces with weaker topology, specifically
in the space of continuous functions bounded on each finite interval. In
this case we would not need the heat loss term, but we would have less
control over the behavior of solutions at infinity.

Results of this paper are proved for the kinetic function satisfying the
bounds in (\ref{kinetics}). These bounds are quite physical and cover a wide
range of important applications. Nonetheless, our numerical experimentation
with different types of kinetic functions, including unbounded ones
demonstrate that the asymptotic dynamics are insensitive to the behavior of
the kinetic function for large temperatures. On the other hand, our results
from \cite{all-amer} provide global existence for a wider class of kinetic
functions, than in (\ref{kinetics}), namely for sublinear kinetics.
Therefore it seems plausible that a compact attractor should exists for this
case as well.

Finally, we should remark that the one-phase problem is to a degree a
particular case of a more general two-phase problem (\ref{kinetics}). There
are technical difficulties in implementation of the construction of this
paper for the two-phase problem, as the field extends behind the interface
where it is not necessarily decaying. At the same time numerical experiments
show a great similarity in dynamical behavior of both problems. It would be,
therefore, interesting to extend results of the present paper to the
two-phase problem.

\paragraph{Acknowledgments}
The authors would like to acknowledge support in part by NSF through grants
DMS-9623006 and DMS-9704325. Part of this work was performed while V.
Roytburd was visiting Institute for Mathematics and its Applications,
University of Minnesota. Hospitality of the Institute and of its director,
Willard Miller, is gratefully acknowledged. Some results of the paper were
announced in \cite{amlet}.

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\noindent\textsc{Michael L. Frankel }\\
Department of Mathematical Sciences, \\
Indiana University--Purdue University \\
Indianapolis, Indianapolis, IN 45205 USA \smallskip

\noindent\textsc{Victor Roytburd }\\
Department of Mathematical Sciences, \\
Rensselaer Polytechnic Institute,\\
Troy, NY 12180-3590 USA \\
e-mail: roytbv@rpi.edu

\end{document}