Spreading Profile and Nonlinear Stefan Problems
by Yihong Du  

Vol. 8 No. 4 (2013) P.413~P.430

ABSTRACT

We report some recent progress on the study of the following nonlinear Stefan problem \[ \left\{ \begin{aligned} &u_t-\Delta u=f(u) & &\mbox{for}~~x\in\Omega(t),\;t>0,\& u=0~\mbox{and}~u_t=\mu|\nabla_x u|^2 && \mbox{for}~~x\in\Gamma(t),\; t>0,\ &u(0,x)=u_0(x) && \mbox{for}~~x\in\Omega_0, \end{aligned} \right. \] where $\Omega(t)\subset\mathbb{R}^N$ ($N\geq 1$) is bounded by the free boundary $\Gamma(t)$, with $\Omega(0)=\Omega_0$, $\mu$ is a given positive constant. The initial function $u_0$ is positive in $\Omega_0$ and vanishes on $\partial\Omega_0$. The class of nonlinear functions $f(u)$ includes the standard monostable, bistable and combustion type nonlinearities. When $\mu\to\infty$, it can be shown that this free boundary problem converges to the corresponding Cauchy problem \[ \left\{ \begin{aligned} & u_t-\Delta u=f(u) & &\mbox{for}~~x\in\mathbb{R}^N,\; t>0,\ &u(0,x)=u_0(x) && \mbox{for}~~x\in\mathbb{R}^N. \end{aligned} \right. \] We will discuss the similarity and differences of the dynamical behavior of these two problems by closely examining their spreading profiles, which suggest that the Stefan condition is a stabilizing factor in the spreading process.

KEYWORDS
Free boundary, Stefan problem, Spreading-vanishing dichotomy, Spreading profile

MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 35K20, 35R35, 35J60

MILESTONES

Received: 2013-07-05
Revised :
Accepted: 2013-07-15

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