Entire solution in cylinder-like domains for a bistable reaction-diffusion equation

TL;DR

This paper constructs entire solutions for a bistable reaction-diffusion equation in unbounded, cylinder-like domains, extending previous results by considering domains that approximate cylinders and analyzing related one-dimensional problems.

Contribution

It introduces new entire solutions in domains that tend to cylinders, broadening understanding of propagation phenomena in more general unbounded domains.

Findings

Existence of entire solutions in cylinder-like domains.

Extension of propagation results to more general unbounded domains.

Existence of solutions for a 1D problem with non-homogeneous linear term.

Abstract

We construct nontrivial entire solutions for a bistable reaction-diffusion equation in a class of domains that are unbounded in one direction. The motivation comes from recent results of Berestycki, Bouhours, and Chapuisat concerning propagation and blocking phenomena in infinite domains. A key assumption in their study was the "cylinder-like" assumption: their domains are supposed to be straight cylinders in a half space. The purpose of this paper is to consider domains that tend to a straight cylinder in one direction. We also prove the existence of an entire solution for a one-dimensional problem with a non-homogeneous linear term.