Lattice Dynamical Systems Associated with a Fractional Laplacian. (18th August 2019)

• Record Type: Journal Article
• Authors: Keyantuo, Valentin; Lizama, Carlos; Warma, Mahamadi
• Is Part Of: Numerical functional analysis and optimization. Volume 40:Number 11(2019)
• Page Start: 1315
• Page End: 1343
• Publication Date: 2019-08-18
• Subjects: Lattice dynamical systems -- discrete fractional Laplacian -- optimal well-posedness -- fractional discrete Fisher and Newell-Whitehead-Segel equations; 49K40 -- 34K31 -- 47D07 -- 26A33; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2019.1602542
• Languages: English
• ISSNs: 0163-0563

Abstract

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.