Reaction-diffusion fronts in funnel-shaped domains

TL;DR

This paper investigates the behavior of reaction-diffusion fronts in funnel-shaped domains, establishing conditions for blocking or spreading, and characterizing the large-time dynamics and transition fronts in such geometries.

Contribution

It provides new Liouville type results, characterizes the global mean speed of spreading solutions, and identifies parameter conditions for propagation or blocking in funnel-shaped domains.

Findings

Spreading solutions converge to well-formed fronts in the conical part.

Conditions on domain size and opening angle determine blocking or spreading.

The set of parameters allowing complete propagation is open.

Abstract

We consider bistable reaction-diffusion equations in funnel-shaped domains of R N made up of straight parts and conical parts with positive opening angles. We study the large time dynamics of entire solutions emanating from a planar front in the straight part of such a domain and moving into the conical part. We show a dichotomy between blocking and spreading, by proving especially some new Liouville type results on stable solutions of semilinear elliptic equations in the whole space R N. We also show that any spreading solution is a transition front having a global mean speed, which is the unique speed of planar fronts, and that it converges at large time in the conical part of the domain to a well-formed front whose position is approximated by expanding spheres. Moreover, we provide sufficient conditions on the size R of the straight part of the domain and on the opening angle $\alpha$ of the conical part, under which the solution emanating from a planar front is blocked or spreads completely in the conical part. We finally show the openness of the set of parameters (R, $\alpha$) for which the propagation is complete.