Abstract
We study reaction-diffusion equations with a general reaction functionfon one-dimensional lattices with continuous or discrete timeux′ (or Δtux)=k(ux-1-2ux+ux+1)+f(ux),x∈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.
Highlights
Reaction-diffusion equation ∂tu = k∂xxu + f(u) serves as a nonlinear model to describe a class of phenomena in which two factors are combined
We study reaction-diffusion equations with a general reaction function f on one-dimensional lattices with continuous or discrete time ux = k(ux−1 − 2ux + ux+1) + f(ux), x ∈ Z
We prove weak and strong maximum and minimum principles for corresponding initial-boundary value problems
Summary
In certain situations (e.g., spatially structured environment) it is natural to study reaction-diffusion equations with discretized space variable and continuous time (we refer to it as a semidiscrete problem and use ux(t) = u(x, t)): ux (t) = k (ux−1 (t) − 2ux (t) + ux+1 (t)) + f (ux (t)) , (1). As in the case of (non)existence of traveling wave solutions for Nagumo equations, it has been shown that discrete and semidiscrete structures influence the validity of maximum principles in a significant way.
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