# Travelling waves for a non-monotone bistable equation with delay: existence and oscillations

## TL;DR

This paper studies travelling wave solutions in a delayed bistable reaction-diffusion equation relevant to population dynamics, revealing existence, oscillations, and convergence behaviors without relying on comparison principles.

## Contribution

It introduces a novel approach combining a priori estimates and topological degree theory to establish existence and detailed asymptotic behaviors of travelling waves in non-monotone delayed equations.

## Key findings

- Existence of travelling waves connecting 0 to a state above the unstable equilibrium.
- Conditions under which waves converge to 1 at infinity.
- Demonstration of oscillatory behavior around 1 for large delays.

## Abstract

We consider a bistable ($0\textless{}\theta\textless{}1$ being the three constant steady states) delayed reaction diffusion equation, which serves as a model in population dynamics. The problem does not admit any comparison principle. This prevents the use of classical technics and, as a consequence, it is far from obvious to understand the behaviour of a possible travelling wave in $+\infty$. Combining refined {\it a priori} estimates and a Leray Schauder topological degree argument, we construct a travelling wave connecting 0 in $-\infty$ to \lq\lq something" which is strictly above the unstable equilibrium $\theta$ in $+\infty$. Furthemore, we present situations (additional bound on the nonlinearity or small delay) where the wave converges to 1 in $+\infty$, whereas the wave is shown to oscillate around 1 in $+\infty$ when, typically, the delay is large.