The Stefan problem for the Fisher-KPP equation with unbounded initial range

We consider the nonlinear Stefan problem $$ left { begin{array} {ll} -d Delta u=a u-b u^2 ;; & mbox{for } x in Omega (t), ; t>0, u=0 mbox{ and } u_t=mu| abla_x u |^2 ;;&mbox{for } x in partialOmega (t), ; t>0, u(0,x)=u_0 (x) ;; & mbox{for } x in Omega_0, end{array}right. $$ 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 $partialOmega_0$. When $Omega_0$ is bounded, the long-time behavior of this problem has been rather well-understood by cite{DG1,DG2,DLZ, DMW}. 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 $Omega_0$.