Transition fronts in unbounded domains with multiple branches

TL;DR

This paper investigates the existence, uniqueness, and behavior of transition fronts in reaction-diffusion-advection equations within complex, multi-branch unbounded domains, extending the concept of planar fronts to more general geometries.

Contribution

It introduces the notion of almost-planar fronts in multi-branch domains and proves the existence and convergence of transition fronts under certain conditions, generalizing previous results.

Findings

Existence of transition fronts in complex domains.

Convergence of solutions to almost-planar fronts in some branches.

Uniqueness of transition fronts under complete propagation conditions.

Abstract

This paper is concerned with the existence and uniqueness of transition fronts of a general reaction-diffusion-advection equation in domains with multiple branches. In this paper, every branch in the domain is not necessary to be straight and we use the notions of almost-planar fronts to generalize the standard planar fronts. Under some assumptions of existence and uniqueness of almost-planar fronts with positive propagating speeds in extended branches, we prove the existence of entire solutions emanating from some almost-planar fronts in some branches. Then, we get that these entire solutions converge to almost-planar fronts in some of the rest branches as time increases if no blocking occurs in these branches. Finally, provided by the complete propagation of every front-like solution emanating from one almost-planar front in every branch, we prove that there is only one type of transition fronts, that is, the entire solutions emanating from some almost-planar fronts in some branches and converging to almost-planar fronts in the rest branches.