论文标题
Fisher-KPP方程的Stefan问题，具有无限的初始范围
The Stefan problem for the Fisher-KPP equation with unbounded initial range
论文作者
论文摘要
我们认为非线性Stefan问题$$ \ left \ {\ begin {array} {ll} -dΔU= a u-b u^2 \; \; ＆\ mbox {for} x \ inω（t），\; t> 0， \\ u = 0 \ mbox {and} u_t =μ| \ nabla_x u |^2 \; \;＆\ mbox {for} x \ in \partialΩ（t），\;; t> 0，\\ u（0，x）= u_0（x）\; \; ＆\ mbox {for} x \ inω_0，\ end {array} \ right。 $$ 其中$ω（0）=ω_0$是$ \ Mathbb r^n $，$ u_0> 0 $ in $ω_0$和$ u_0 $ in $ \partialΩ_0$ nistanes in $ \ mathbb r^n $中的无界平滑域。当$ω_0$有限时，\ cite {dg1，dg2，dlz，dmw}已经非常理解了此问题的长期行为。在这里，我们揭示了某些无界$ω_0$的有趣不同行为。我们还为弱解决方案理论提供了一种统一的方法，以使用有限或无限的$ω_0$来解决这种自由边界问题。
We consider the nonlinear Stefan problem $$ \left \{ \begin{array} {ll} -d Δu=a u-b u^2 \;\; & \mbox{for } x \in Ω(t), \; t>0, \\ u=0 \mbox{ and } u_t=μ|\nabla_x u |^2 \;\;&\mbox{for } x \in \partialΩ(t), \; t>0, \\ u(0,x)=u_0 (x) \;\; & \mbox{for } x \in Ω_0, \end{array}\right. $$ where $Ω(0)=Ω_0$ is an unbounded smooth domain in $\mathbb R^N$, $u_0>0$ in $Ω_0$ and $u_0$ vanishes on $\partialΩ_0$. When $Ω_0$ 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 $Ω_0$. We also give a unified approach for a weak solution theory to this kind of free boundary problems with bounded or unbounded $Ω_0$.
