2013 / December Volume 8 No.4
Spreading Profile and Nonlinear Stefan Problems
| Published Date |
2013 / December
|
|---|---|
| Title | Spreading Profile and Nonlinear Stefan Problems |
| Author | |
| Keyword | |
| Download | |
| Pagination | 413-430 |
| 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.
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| AMS Subject Classification |
35K20, 35R35, 35J60
|
| Received |
2013-07-05
|
| Accepted |
2013-07-15
|