| Abstract: |
| We consider the nonlinear Stefan problem
\[
\left \{ \begin{array} {ll}
u_t-d \Delta u=a u-bu^2 \;\; & \mbox{for $x \in \Omega (t), \; t>0$},\
u=0 \hbox{ and } u_t=\mu|\nabla_x u |^2 \;\;&\mbox{for $x \in \partial\Omega (t),\; t>0$}, \
u(0,x)=u_0 (x) \;\; & \mbox{for $x \in \Omega_0$},
\end{array} \right.
\]
where $\Omega(0)=\Omega_0$ is an unbounded smooth domain in $\mathbb{R}^N$, $u_0>0$ in $\Omega_0$ and $u_0$ vanishes on $\partial\Omega_0$. When $\Omega_0$ is bounded, the long-time behavior of this problem has been rather well-understood. Here we reveal some interesting different behavior for certain unbounded $\Omega_0$. We also give a unified
approach for a weak solution theory to this kind of free boundary problems with bounded or unbounded $\Omega_0$. |
|