Electron. J. Diff. Eqns., Vol. 2006(2006), No. 05, pp. 1-12.

Asymptotic profile of a radially symmetric solution with transition layers for an unbalanced bistable equation

Hiroshi Matsuzawa

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 $|a(x)|$ lees than 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.

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Hiroshi Matsuzawa
Numazu National College of Technology
Ooka 3600, Numazu-city, Shizuoka 410-8501, Japan
email: hmatsu@numazu-ct.ac.jp

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