Maximum principles for discrete and semidiscrete reaction-diffusion equation
type: Item
instance of: scholarly article
title: Maximum principles for discrete and semidiscrete reaction-diffusion equation
zbMATH Open document ID: 1418.35238
DOI: 10.1155/2015/791304
author: Petr Stehlík
author: Jonáš Volek
published in: Discrete Dynamics in Nature and Society
publication date: 2019-02-19T00:00:00Z
review text: Summary: We study reaction-diffusion equations with a general reaction function \(f\) on one-dimensional lattices with continuous or discrete time \(u_x' (\text{or } \Delta_t u_x) = k(u_{x - 1} - 2 u_x + u_{x + 1}) + f(u_x)\), \(x \in \mathbb{Z}\). We prove weak and strong maximum and minimum principles for corresponding initial-boundary value problems. Whereas the maximum principles in the semidiscrete case (continuous time) exhibit similar features to those of fully continuous reaction-diffusion model, in the discrete case the weak maximum principle holds for a smaller class of functions and the strong maximum principle is valid in a weaker sense. We describe in detail how the validity of maximum principles depends on the nonlinearity and the time step. We illustrate our results on the Nagumo equation with the bistable nonlinearity.
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 35B50
Mathematics Subject Classification ID: 65M06
Mathematics Subject Classification ID: 65M12
zbMATH DE Number: 7025493
Wikidata QID: Q59105446
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2015/791304
OpenAlex ID: W1939601086
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description: scientific article; zbMATH DE number 7025493