Lattice Dynamical Systems Associated with a Fractional Laplacian

Abstract

We derive optimal well-posedness results and explicit representations of solutions in terms of special functions for the linearized version of the equationfor some constant where denotes the Caputo fractional derivative in time of order β and denotes the discrete fractional Laplacian of order We also prove a comparison principle. A special case of this equation is the discrete Fisher- equation with and without delay. We show that if for every and the function is concave on f(x, s) is nonnegative for every and satisfiesfor some then the system has a nonnegative unique solution u satisfying for every and Our results include cubic nonlinearities and incorporate new results for the discrete Newell-Whitehead-Segel equation. We use Lévy stable processes as well as Mittag-Leffler, Wright and modified Bessel functions to describe the solutions of the linear lattice model, providing a useful framework for further study. For the nonlinear model, we use a generalization of the upper–lower solution method for reaction–diffusion equations in order to prove existence and uniqueness of solutions.