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

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
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