Electronic Journal of Differential Equations,
Vol. 2006(2006), No. 05, pp. 1-12.
Title: Asymptotic profile of a radially symmetric solution
with transition layers for an unbalanced bistable equation
Author: Hiroshi Matsuzawa (Numazu National College of Tech., Japan)
Abstract:
In this article, we consider the semilinear elliptic problem
$$
-\varepsilon^{2}\Delta u=h(|x|)^2(u-a(|x|))(1-u^2)
$$
in $B_1(0)$ with the Neumann boundary condition.
The function $a$ is a $C^1$ function satisfying $|a(x)|< 1$
for $x\in [0,1]$ and $a'(0)=0$. In particular we consider
the case $a(r)=0$ on some interval $I\subset [0,1]$.
The function $h$ is a positive $C^1$ function satisfying $h'(0)=0$.
We investigate an asymptotic profile of the global minimizer
corresponding to the energy functional as $\varepsilon\to 0$.
We use the variational procedure used in [4] with a few
modifications prompted by the presence of the function $h$.
Submitted August 31, 2005. Published January 11, 2006.
Math Subject Classifications: 35B40, 35J25, 35J55, 35J50, 35K57.
Key Words: Transition layer; Allen-Cahn equation;
bistable equation; unbalanced