On a solution with transition layers for a bistable reaction-diffusion equation with spatially heterogeneous environments

Pages: 516 - 525, Issue Special, September 2009

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Hiroshi Matsuzawa - Numazu National College of Technology, Ooka 3600, Numazu-city, Shizuoka 410-8501, Japan (email)

Abstract: 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 $|a(r)|< 1$ for $r\in [0,1]$ and $a^'(0)=0$. The function $h$ is a positive $C^1$ function satisfying $h^'(0)=0$. The nonlinear function in the equation is a typical example of the so-called {\it bistable} nonlinearity. Functions $a$ and $h$ in the nonlinearity represent spatial inhomogeneity. In particular, we consider the case where $a(r)=0$ on some interval $I\subset (0,1)$. When $\varepsilon>0$ 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 $h$.

Keywords:  transition layer, Allen-Cahn equation, bistable equation, unbalanced
Mathematics Subject Classification:  Primary:35K57, 35J50,; Secondary:35J25

Received: July 2008;      Revised: June 2009;      Published: September 2009.