On the mean speed of bistable transition fronts in unbounded domains

TL;DR

This paper investigates the propagation speeds of bistable transition fronts in unbounded domains, establishing conditions under which these fronts propagate with a unique mean speed, related to planar speeds, in complex geometries.

Contribution

It demonstrates that in exterior and cylindrical-branch domains, all transition fronts propagate with a unique planar speed, providing geometric and scaling conditions for mean speed existence.

Findings

All transition fronts in exterior domains propagate with the same global mean speed.

In domains with multiple cylindrical branches, some solutions propagate with a unique planar speed.

Geometrical and scaling conditions guarantee the existence of a global mean speed for transition fronts.

Abstract

This paper is concerned with the existence and further properties of propagation speeds of transition fronts for bistable reaction-diffusion equations in exterior domains and in some domains with multiple cylindrical branches. In exterior domains we show that all transition fronts with complete propagation propagate with the same global mean speed, which turns out to be equal to the uniquely defined planar speed. In domains with multiple cylindrical branches, we show that the solutions emanating from some branches and propagating completely are transition fronts propagating with the unique planar speed. We also give some geometrical and scaling conditions on the domain, either exterior or with multiple cylindrical branches, which guarantee that any transition front has a global mean speed.