Front blocking and propagation in cylinders with varying cross section

TL;DR

This paper studies how geometry influences wave propagation in reaction-diffusion equations within cylinders of varying cross sections, identifying conditions for complete invasion or blocking phenomena.

Contribution

It establishes conditions under which wave propagation is complete or blocked in domains with changing cross sections, including star-shaped geometries.

Findings

Complete propagation occurs in domains with decreasing cross section.

Wave blocking happens at abrupt geometric changes.

Star-shaped domains ensure complete invasion.

Abstract

In this paper we consider a bistable reaction-diffusion equation in unbounded domains and we investigate the existence of propagation phenomena, possibly partial, in some direction or, on the contrary, of blocking phenomena. We start by proving the well-posedness of the problem. Then we prove that when the domain has a decreasing cross section with respect to the direction of propagation there is complete propagation. Further, we prove that the wave can be blocked as it comes to an abrupt geometry change. Finally we discuss various general geometrical properties that ensure either partial or complete invasion by 1. In particular, we show that in a domain that is "star-shaped" with respect to an axis, there is complete invasion by 1.