在論文中，探討的是一個兩點自由邊界問題的自我相似解之存在性和唯一性。其中，將a(s)分成兩部分來探討：
(A)若a(s)在無窮大處的一次微分大於0，則初始問題的解會完全地存在。
(B)若a(s)在無窮大處的一次微分等於0，且存在一個有限的常數s0，使得a(s)在大於或等於s0時，它的二次微分都小於或等於0，則如果p足夠大時，初始問題的解是不會越過y軸，但是解一旦越過了y軸，它就會完全地存在。

\section{Abstract}
\hspace*{17pt}In this paper, we study the following two-point free boundary problem
\begin{equation}\label{eq:(01)}
\left\{
\begin{array}{ll}
u_t=(a(u_x))_x, \; -\xi_1(t) < x < \xi_2(t),\ t>0, & \\
u_x(-\xi_1(t),t)=\tan\alpha_1, \;t>0,& \\
u(-\xi_1(t),t)=\xi_1(t)\tan \beta_1,\; t>0, & \\
u_x(\xi_2(t),t)=\tan\alpha_2, \; t>0, & \\
u(\xi_2(t),t)=\xi_2(t)\tan \beta_2,\; t>0, & \\
u(x, 0)=u_0(x),\ -\xi_{01}\le x\le \xi_{02},&\\
\xi_1(0)=\xi_{01}, \; \xi_2(0)=\xi_{02}, &
\end{array}
\right.
\end{equation}
where $a\in C^2(\bf R)$ such that $a(0)=0$ and
$a'>0$ in $\bf{R}$,
$\beta_1, \beta_2 \in [0,\pi/2)$,
$\alpha_1\in (-\beta_1,\pi/2)$,
$\alpha_2\in(-\pi/2,\beta_2)$,
$\xi_{01}$ and $\xi_{02}$ are positive constants,
$u_0\in C^{1+\alpha}[-\xi_{01},\xi_{02}]$
for some $\alpha\in(0,1)$,
$u_0$ satisfies the compatibility conditions,
and $u_0 > 0$ in $(-\xi_{01},\xi_{02})$.
In this problem, $u,\xi_1,\xi_2$ are unknown functions to be determined.
We are concerned with the existence of forward self-similar positive solutions of (\ref{eq:(01)}) in the form
\begin{eqnarray}
u(x,t)=\sqrt{2(t+1)}\varphi(\frac{x}{\sqrt{2(t+1)}})
\end{eqnarray}
for some smooth function $\varphi$.
Then $u$ satisfies (\ref{eq:(01)}) if and only if $\varphi$ satisfies the following nonlinear free boundary problem (P):
\begin{eqnarray}\label{eq:(02)}
&&(a(\varphi'(y))'
+y\varphi'(y) -\varphi(y) =0, -p <y < q, \\
&& \varphi(-p)=p\tan\beta_1, \varphi'(-p) =\tan\alpha_1,\\
&& \varphi(q)=q\tan\beta_2, \varphi'(q)=\tan\alpha_2
\end{eqnarray}
for some positive constants $p$ and $q$.
The purpose of this paper is to study the existence and uniqueness of solution for the nonlinear free boundary problem (P).
In this paper, we divide $a(s)$ into two case:
$(\mbox{A})\;a^{'}(\infty)>0\label{eq:(04)}$ ;
$(\mbox{B})\;a^{'}(\infty)=0\label{eq:(05)}$
and there is a constant $s_{0}<\infty$
such that $a^{''}(s)\leq 0$ for all $s\geq s_{0}$.
In (A), the solution exists globally for all $p>0$.
The special case "$a(s)=s$" of (A) has been proved in (1).
For the case (B) it is more complicated.
If $p$ is sufficiently large,
then the solution of (IVP) cannot be extended over $y=0$.
Moreover,
the solution of (IVP) is global as long as it can be
extended over $y=0$.
By the comparison principle for nonlinear equations, we can
obtain the existence and uniqueness of the self-similar
solution.
The special case "$a(s)=tan^{-1}s$" of (B) has been proved in (3).

(1) J.-S. Guo and Y. Kohsaka,
Self-similar solutions of two-point free boundary Problem
for the heat equation, Nonlinear Diffusive Systems and
Related Topics, RIMS Kokyuroku 1258, Research Institute
for Mathematical Sciences, Kyoto University, April, 2002,
pp.94-107.
(2) H.-H. Chern, J.-S. Guo and C.-P. Lo,
The self-similar expanding curve for the curvature flow
equation, Proc. Amer. Math. Soc.(to appear).
(3) M. H. Protter and H. F. Weinberger, Maximum Principles
in Differential Equations, Springer-Verlag, 1984.