On a solution with transition layers for a bistable reaction-diffusion equation with spatially heterogeneous environments
Hiroshi Matsuzawa · AIMS Press eBooks · 2009
In this paper we consider a boundary value problem for the following semilinear elliptic equation :$-\varepsilon^{2}\Delta u=h(|x|)^2(u-a(|x|))(1-u^2)$ in $B_1(0)$ with homogeneous Neumann boundary condition. The function $a$ is a $C^1$ function satisfying $|a(r)|0$ is very small, there exist stationary solutions with sharp transition layers. We investigate asymptotic locations of transition layers of a global minimizer corresponding to an 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 $h$.