Asymptotics of the heat kernels on 2D lattices

Article type: Research Article

Authors: Gurevich, Pavel^{a; b; *}

Affiliations: [a] Free University of Berlin, Germany. E-mail: [email protected] | [b] Peoples’ Friendship University of Russia, Russia

Correspondence: [*] Corresponding author. E-mail: [email protected].

Abstract: We obtain asymptotic expansions of the spatially discrete 2D heat kernels, or Green’s functions on lattices, with respect to powers of time variable up to an arbitrary order and estimate the remainders uniformly on the whole lattice. Unlike in the 1D case, the asymptotics contains a time independent term. The derivation of its spatial asymptotics is the technical core of the paper. Besides numerical applications, the obtained results play a crucial role in the analysis of spatio-temporal patterns for reaction-diffusion equations on lattices, in particular rattling patterns for hysteretic diffusion systems.

Keywords: Discrete heat kernel, Green’s function, lattice dynamics, asymptotics

DOI: 10.3233/ASY-181498

Journal: Asymptotic Analysis, vol. 112, no. 1-2, pp. 107-124, 2019