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Existence of bistable waves for a nonlocal and nonmonotone reaction-diffusion equation

Part of: Functional-differential and differential-difference equations Parabolic equations and systems

Published online by Cambridge University Press:  23 January 2019

Sergei Trofimchuk  and
Vitaly Volpert
Sergei Trofimchuk
Affiliation:
Instituto de Matemática y Fisica, Universidad de Talca, Talca, Chile
Vitaly Volpert
Affiliation:
Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, 69622 Villeurbanne, France ([email protected]) and INRIA, Université de Lyon, Université Lyon 1, Institut Camille Jordan, 43 Bd. du 11 Novembre 1918, 69200 Villeurbanne Cedex, France and People's Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation and Marchuk Institute of Numerical Mathematics of the RAS, ul. Gubkina 8, 119333 Moscow, Russian Federation
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Abstract

Reaction-diffusion equation with a bistable nonlocal nonlinearity is considered in the case where the reaction term is not quasi-monotone. For this equation, the existence of travelling waves is proved by the Leray-Schauder method based on the topological degree for elliptic operators in unbounded domains and a priori estimates of solutions in properly chosen weighted spaces.

Keywords

Reaction-diffusion equationnonlocalbistable nonlinearityexistence of wavesnonlinear Fredholm operatorLeray-Schauder method

MSC classification

Primary: 34K60: Qualitative investigation and simulation of models 35K57: Reaction-diffusion equations
Secondary: 92D25: Population dynamics (general)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 2 , April 2020 , pp. 721 - 739
Copyright
Copyright © Royal Society of Edinburgh 2019

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