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

Matsuzawa, Hiroshi

Electronic journal of differential equations, Tome 2006 (2006)

Résumé

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 $|a(x)|$ 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 $h$.

Détail

Classification : 35B40, 35J25, 35J55, 35J50, 35K57
Keywords: transition layer, Allen-Cahn equation, bistable equation, unbalanced