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

Abstract

We consider the nonlinear Stefan problem $$\begin{aligned} \left\{ \begin{array} {ll} u_t-d \Delta u=a u-b u^2 &{} \hbox {for}\quad x \in \Omega (t), \; t>0,\\ u=0\,\, \hbox {and}\,\,u_t=\mu |\nabla _x u |^2 &{}\hbox {for}\quad x \in \partial \Omega (t), \; t>0, \\ u(0,x)=u_0 (x) &{} \hbox {for}\quad x \in \Omega _0, \end{array} \right. \end{aligned}$$where \(\Omega (0)=\Omega _0\) is an unbounded Lipschitz 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 by Du et al. (J Differ Equ 250:4336–4366, 2011; J Differ Equ 253:996–1035, 2012; J Ellip Par Eqn 2:297–321, 2016; Arch Ration Mech Anal 212:957–1010, 2014). 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 \(\Omega _0\).