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Maximum Principles for Discrete and Semidiscrete Reaction-Diffusion Equation

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  • Petr Stehlík
  • Jonáš Volek

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.

Suggested Citation

  • Petr Stehlík & Jonáš Volek, 2015. "Maximum Principles for Discrete and Semidiscrete Reaction-Diffusion Equation," Discrete Dynamics in Nature and Society, Hindawi, vol. 2015, pages 1-13, September.
    • Handle: RePEc:hin:jnddns:791304
    DOI: 10.1155/2015/791304
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    Cited by:

    1. Hošek, Radim & Volek, Jonáš, 2019. "Discrete advection–diffusion equations on graphs: Maximum principle and finite volumes," Applied Mathematics and Computation, Elsevier, vol. 361(C), pages 630-644.

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