Calculus of Variations and Partial Differential Equations ( IF 2.1 ) Pub Date : 2021-04-05 , DOI: 10.1007/s00526-020-01877-4 Weiwei Ding , Yihong Du , Zongming Guo
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 \(\Omega _0\).
中文翻译：
初始范围无界的Fisher–KPP方程的Stefan问题
我们考虑非线性Stefan问题
$$ \ begin {aligned} \ left \ {\ begin {array} {ll} u_t-d \ Delta u = a ub u ^ 2＆{} \ hbox {for} \ quad x \ in \ Omega（t）， \; t> 0，\\ u = 0 \，\，\ hbox {and} \，\，u_t = \ mu | \ nabla _x u | ^ 2＆{} \ hbox {for} \ quad x \ in \ partial \ Ω（t），\；t> 0，\\ u（0，x）= u_0（x）＆{} \ hbox {for} \ quad x \ in \ Omega _0，\ end {array} \ right。\ end {aligned} $$其中\（\欧米茄（0）= \欧米茄_0 \）是无界李普希茨域\（\ mathbb {R} ^ N \） ，\（U_0> 0 \）在\（\欧米茄_0 \）和\（ u_0 \）消失在\（\ partial \ Omega _0 \）上。当\（\ Omega _0 \）有界时，Du等人已经很好地理解了此问题的长期行为。（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）。在这里，我们揭示了某些无界\（\ Omega _0 \）的一些有趣的不同行为。对于带有边界或无边界的这种自由边界问题，我们还为弱解理论提供了统一的方法\（\ Omega _0 \）。
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