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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
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\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]
\newtheorem{remark}[theorem]{Remark}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{proposition}[theorem]{Proposition}
\renewcommand{\theequation}{\thesection.\arabic{equation}}
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\@addtoreset{equation}{section}
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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 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 $01$ 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 $0<\delta 0$, 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 $00$
\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 $x~~0$, $u(x,t)$ 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 $0~~__0\}$, 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 $S(t)__~~0$, 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 $t$, $01$, 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
$\{(x,t):t>t_{0},\;x~~~~R_{fb}+2R_{abs}$
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
$a=R_{abs}+\varepsilon 1$, 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 $Ce^{-\gamma
t_{1}}N0$ 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}
~~