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Discrete Dynamics in Nature and Society
Volume 2015, Article ID 791304, 13 pages
http://dx.doi.org/10.1155/2015/791304
Research Article

Maximum Principles for Discrete and Semidiscrete Reaction-Diffusion Equation

Department of Mathematics and NTIS, Faculty of Applied Sciences, University of West Bohemia, Univerzitni 8, 30614 Pilsen, Czech Republic

Received 24 March 2015; Accepted 29 July 2015

Academic Editor: Cengiz Çinar

Copyright © 2015 Petr Stehlík and Jonáš Volek. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study reaction-diffusion equations with a general reaction function on one-dimensional lattices with continuous or discrete time , . 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.