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Front blocking and propagation in cylinders with varying cross section

H. Berestycki,J. Bouhours,Guillemette Chapuisat

Published 2015 in Calculus of Variations and Partial Differential Equations

ABSTRACT

In this paper we consider a bistable reaction–diffusion equation in unbounded domains and we investigate geometric conditions under which propagation, possibly partial, takes place in some direction or, on the contrary, there is a blocking phenomenon. 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 through an abrupt opening. 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.

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