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The Stefan problem for the Fisher–KPP equation with unbounded initial range
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\(\ mathbb {R} ^ N \) \(U_0> 0 \)\(\欧米茄_0 \)\( u_0 \)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\(\ Omega _0 \)

更新日期:2021-04-05
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