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

Summary: 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