Entire Solutions for Bistable Lattice Differential Equations with Obstacles

TL;DR

This paper proves the existence of wave-like solutions in bistable lattice differential equations with obstacles, extending continuous domain results to discrete lattices using sub/super-solution methods.

Contribution

It generalizes wave propagation results around obstacles from continuous to discrete lattice systems for bistable reaction-diffusion equations.

Findings

Existence of wave-like solutions in lattice systems with obstacles

Development of sub and super-solution techniques for discrete equations

Extension of classical spreading results to discrete spatial settings

Abstract

We consider scalar lattice differential equations posed on square lattices in two space dimensions. Under certain natural conditions we show that wave-like solutions exist when obstacles (characterized by "holes") are present in the lattice. Our work generalizes to the discrete spatial setting the results obtained in a paper of Berestycki, Hamel and Matano for the propagation of waves around obstacles in continuous spatial domains. The analysis hinges upon the development of sub and super-solutions for a class of discrete bistable reaction-diffusion problems and on a generalization of a classical result due to Aronson and Weinberger that concerns the spreading of localized disturbances.