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Steady-states with transition layers and spikes for a bistable reaction-diffusion equation. (English) Zbl 1154.35389

Aiki, T. (ed.) et al., Mathematical approach to nonlinear phenomena: modelling, analysis and simulations, Proceedings of the 3rd Polish-Japanese days held in Chiba, Japan, November 29–December 3, 2004. Tokyo: Gakkotosho (ISBN 4-7625-0432-7/hbk). GAKUTO International Series. Mathematical Sciences and Applications 23, 267-279 (2005).
Summary: This paper is concerned with steady-states of a spatially inhomogeneous bistable reaction-diffusion equation in one space dimension. This equation involves small parameter \(\varepsilon>0\) as a diffusion coefficient and its nonlinearity has the form \(u(1- u) (u -a(x))\), where a is a \(C^2\)-function lying between 0 and 1. It is well known that this problem admits solutions with transition layers and spikes when \(\varepsilon\) is sufficiently small. Furthermore, under some appropriate conditions, multiple transition layers and spikes can appear as a cluster in a neighborhood of a certain point in \((0, 1)\). We will give some information on locations and profiles of transition layers, spikes, multi-layers and multi-spikes. Furthermore, stability properties of these solutions will be also studied in terms of Morse index.
For the entire collection see [Zbl 1091.00006].

MSC:

35K57 Reaction-diffusion equations
35B25 Singular perturbations in context of PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34E15 Singular perturbations for ordinary differential equations
35A15 Variational methods applied to PDEs
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