Lattice Dynamical Systems Associated With A Fractional Laplacian

We derive optimal well-posedness results and explicit representations of solutions in terms of special functions for the linearized version of the equation (Formula presented.) for some constant (Formula presented.) where (Formula presented.) denotes the Caputo fractional derivative in time of order β and (Formula presented.) denotes the discrete fractional Laplacian of order (Formula presented.) We also prove a comparison principle. A special case of this equation is the discrete Fisher- (Formula presented.) equation with and without delay. We show that if (Formula presented.) for every (Formula presented.) and the function (Formula presented.) is concave on (Formula presented.) f(x, s) is nonnegative for every (Formula presented.) and satisfies (Formula presented.) for some (Formula presented.) then the system (Formula presented.) has a nonnegative unique solution u satisfying (Formula presented.) for every (Formula presented.) and (Formula presented.) 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.