This article is cited in 20 papers

On a periodic inner layer in the reaction–diffusion problem with a modular cubic source

N. N. Nefedov, E. I. Nikulin, A. O. Orlov

Faculty of Physics, Lomonosov Moscow State University, Moscow, 119991 Russia

Abstract: The article studies a singularly perturbed periodic problem for the parabolic reaction–diffusion equation in the case of a discontinuous source: a nonlinearity describing the reaction (interaction). The case of the existence of an inner transition layer under conditions of an unbalanced and a balanced reaction is considered. An asymptotic approximation is constructed, and the asymptotic Lyapunov stability of periodic solutions in each of the cases is investigated. To prove the existence of a solution and its asymptotic stability, the asymptotic method of differential inequalities is used. The theoretical result is illustrated by an example and numerical calculations.

Key words: singularly perturbed parabolic problems, periodic problems, reaction–diffusion equations, two-dimensional contrast structures, balanced nonlinearity, inner layers, fronts, asymptotic methods, differential inequalities, asymptotic Lyapunov stability, discontinuous reaction.

UDC: 517.9

Received: 11.11.2019
Revised: 10.01.2020
Accepted: 09.04.2020

DOI: 10.31857/S0044466920090136

 English version: Computational Mathematics and Mathematical Physics, 2020, 60:9, 1461–1479

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