Maximum principles for discrete and semidiscrete reaction-diffusion equation
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.
scientific article; zbMATH DE number 7025493
scientific article