Bistable pulsating fronts in slowly oscillating environments *

TL;DR

This paper investigates the behavior of bistable reaction-diffusion fronts in environments with large spatial periods, establishing the existence and explicit formula for their limiting speeds and profiles as the period tends to infinity.

Contribution

It provides the first explicit characterization of the limit of bistable front speeds in large-period media, addressing a complex open problem in the field.

Findings

Existence of the limit of front speeds as period approaches infinity

Explicit formula for the limiting front speed

Convergence of front profiles to homogeneous equation solutions

Abstract

We consider reaction-diffusion fronts in spatially periodic bistable media with large periods. Whereas the homogenization regime associated with small periods had been well studied for bistable or Fisher-KPP reactions and, in the latter case, a formula for the limit minimal speeds of fronts in media with large periods had also been obtained thanks to the linear formulation of these minimal speeds and their monotonicity with respect to the period, the main remaining open question is concerned with fronts in bistable environments with large periods. In bistable media the unique front speeds are not linearly determined and are not monotone with respect to the spatial period in general, making the analysis of the limit of large periods more intricate. We show in this paper the existence of and an explicit formula for the limit of bistable front speeds as the spatial period goes to infinity. We also prove that the front profiles converge to a family of front profiles associated with spatially homogeneous equations. The main results are based on uniform estimates on the spatial width of the fronts, which themselves use zero number properties and intersection arguments.