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

We consider the nonlinear Stefan problem $$ \left \{ \begin{array} {ll} -d Δu=a u-b u^2 \;\; & \mbox{for } x \in Ω(t), \; t>0, \\ u=0 \mbox{ and } u_t=μ|\nabla_x u |^2 \;\;&\mbox{for } x \in \partialΩ(t), \; t>0, \\ u(0,x)=u_0 (x) \;\; & \mbox{for } x \in Ω_0, \end{array}\right. $$ where $Ω(0)=Ω_0$ is an unbounded smooth domain in $\mathbb R^N$, $u_0>0$ in $Ω_0$ and $u_0$ vanishes on $\partialΩ_0$. When $Ω_0$ 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 $Ω_0$. We also give a unified approach for a weak solution theory to this kind of free boundary problems with bounded or unbounded $Ω_0$.