Stable transition layers in a balanced bistable equation

Abstract

This paper is concerned with the existence of steady-state solutions for $$ \left\{ \begin{array}{ll} u_t = \epsilon^2 u_{xx} - (u-a(x))(u-b(x))(u-c(x))\quad & \mbox{in}~(0,1)\times(0,\infty),\\ u_x(0,t) = u_x(1,t) = 0\quad & \mbox{in}~(0,\infty). \end{array} \right. $$ Here $a, b$ and $c$ are $C^2$-functions satisfying $b = (a+c)/2$ and $c > a$. By using upper and lower solutions it is proved that there exist stable steady states with transition layers near any points where $c(x)-a(x)$ has its local minimum.