Lattice Dynamical Systems Associated with a Fractional Laplacian. (18th August 2019) - Keyantuo, Valentin; Lizama, Carlos; Warma, Mahamadi
Abstract: We derive optimal well-posedness results and explicit representations of solutions in terms of special functions for the linearized version of the equation ( * ) D t β u ( n, t ) = − ( − Δ d ) α u ( n, t ) + f ( n − ct, u ( n, t ) ), n ∈ Z, t > 0, 0 < α, β < 1, u ( n, 0 ) = φ ( n ), n ∈ Z, for some constant c ≥ 0, where D t β denotes the Caputo fractional derivative in time of order β and ( − Δ d ) α denotes the discrete fractional Laplacian of order α ∈ ( 0, 1 ] . We also prove a comparison principle. A special case of this equation is the discrete Fisher- KPP equation with and without delay. We show that if 0 ≤ φ ( n ) ≤ γ for every n ∈ Z, and the function f ( x, · ) is concave on [ 0, γ ], f ( x, s ) is nonnegative for every x ∈ R, s ∈ [ 0, γ ], and satisfies f ( x, 0 ) = 0 and f ( x, γ ) ≤ 0, ∀ x ∈ R, for some γ > 0, then the system ( * ) has a nonnegative unique solution u satisfying 0 ≤ u ( n, t ) ≤ γ for every n ∈ Z and t ≥ 0 . 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.,