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

Abstract

<p>We consider the nonlinear Stefan problem</p> <table><tr><td>{</td><td><i>u_t-dΔu = au − bu²</i></td> <td> for <i>x Ω</i>(<i>t</i>),<i> t</i> > 0,</td></tr><tr><td> </td><td><i>u</i> = 0 and <i>u_t = μ|∇_xu</i>|²</td><td>for <i>x ∂Ω</i>(<i>t</i>), <i>t</i> > 0,</td></tr><tr><td> </td><td><i>u</i>(0,<i>x</i>) = <i>u</i>₀(<i>x</i>)</td> <td> for <i>x</i> Ω₀,</td></tr></table> <p>where Ω(0)=Ω0 is an unbounded Lipschitz domain in ℝ^<i>N</i>, <i>u</i>₀ > 0 in Ω₀ and <i>u</i>₀ vanishes on ∂Ω₀. When Ω₀ 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 Ω₀. We also give a unified approach for a weak solution theory to this kind of free boundary problems with bounded or unbounded Ω₀.