Differential and Integral Equations

Stable transition layers in a balanced bistable equation

Kimie Nakashima

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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.

Article information

Differential Integral Equations, Volume 13, Number 7-9 (2000), 1025-1038.

First available in Project Euclid: 21 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34E15: Singular perturbations, general theory
Secondary: 34B15: Nonlinear boundary value problems 35B25: Singular perturbations 35B40: Asymptotic behavior of solutions 35J60: Nonlinear elliptic equations 35K57: Reaction-diffusion equations


Nakashima, Kimie. Stable transition layers in a balanced bistable equation. Differential Integral Equations 13 (2000), no. 7-9, 1025--1038. https://projecteuclid.org/euclid.die/1356061208

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