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

We consider the nonlinear Stefan problem $$\{\begin{array}{ll}-d\mathrm{\Delta }u=au-b{u}^2& forx\in \mathrm{\Omega }(t),t>0,\\ u=0and{u}_t=\mu \mathrm{\mid }{\mathrm{\nabla }}_{x}u{\mathrm{\mid }}^2& forx\in \mathrm{\partial }\mathrm{\Omega }(t),t>0,\\ u(0,x)=u_0(x)& forx\in {\mathrm{\Omega }}_0,\end{array}$$ where $\mathrm{\Omega }(0)={\mathrm{\Omega }}_0$ is an unbounded smooth domain in ${\mathbb{R}}^N$, $u_0>0$ in ${\mathrm{\Omega }}_0$ and $u_0$ vanishes on $\mathrm{\partial }{\mathrm{\Omega }}_0$. When ${\mathrm{\Omega }}_0$ 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 ${\mathrm{\Omega }}_0$. We also give a unified approach for a weak solution theory to this kind of free boundary problems with bounded or unbounded ${\mathrm{\Omega }}_0$.