Transition fronts of combustion reaction-diffusion equations in domains with multiple cylindrical branches

TL;DR

This paper studies the propagation of combustion reaction-diffusion fronts in domains with multiple cylindrical branches, proving the existence, convergence, and uniqueness of transition fronts and their speeds.

Contribution

It introduces new results on the existence, convergence, and uniqueness of transition fronts in complex cylindrical domains, extending understanding of combustion wave propagation.

Findings

Existence of entire solutions from planar traveling fronts.

Transition fronts converge to planar fronts with finite shifts.

All transition fronts have a unique global mean speed equal to the planar wave speed.

Abstract

This paper is concerned with the propagation phenomenon of the combustion reaction-diffusion equations in domains with multiple cylindrical branches. We first show that there is an entire solution emanating from planar traveling fronts in some branches. Then we prove that the entire solution is a transition front and converges to some planar traveling fronts (with some finite shifts) in the rest branches as time goes to $+\infty$ if the propagation is complete.In addition, by providing the complete propagation of every front-like solution coming from one branch, it is proved that any transition front connecting $0$ and $1$ in domains with multiple cylindrical branches propagates completely and has a unique global mean speed which turns out to be equal to the planar wave speed. Finally, we give some sufficient conditions to ensure that the assumptions on complete propagation are not empty.