Stable transition layers in a balanced bistable equation. (English) Zbl 0981.34011

Nakashima, Kimie

Differ. Integral Equ. 13, No. 7-9, 1025-1038 (2000).

Summary: This paper is concerned with the existence of steady-state solutions to the problem \[ \begin{alignedat}{2} &u_t= \varepsilon^2 u_{xx}- (u- a(x))(u- b(x))(u- c(x))\quad && \text{in }(0,1)\times (0,\infty),\\ & u_x(0, t)= u_x(1,t)= 0\quad &&\text{in }(0,\infty).\end{alignedat} \] 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.

MSC:

34B15	Nonlinear boundary value problems for ordinary differential equations
34E15	Singular perturbations for ordinary differential equations
35B25	Singular perturbations in context of PDEs
35B40	Asymptotic behavior of solutions to PDEs
35J60	Nonlinear elliptic equations
35K57	Reaction-diffusion equations

Keywords:

steady-state solutions; upper and lower solutions; transition layers