# Global continuation of monotone waves for a unimodal bistable reaction-diffusion equation with delay

## TL;DR

This paper proves the existence of monotone wavefronts in a class of delayed bistable reaction-diffusion equations without assuming monotonicity of the reaction term, using a functional-analytic approach.

## Contribution

It introduces a novel method based on Hale-Lin functional analysis to establish wavefront existence without monotonicity assumptions on the delay term.

## Key findings

- Existence of a maximal family of monotone wavefronts for certain reaction functions.
- Wavefront monotonicity depends on delay size and wave speed sign.
- Different behaviors observed for moderate and large delays.

## Abstract

We study the existence of monotone wavefronts for a general family of bistable reaction-diffusion equations with delayed reaction term $g$. Differently from previous works, we do not assume the monotonicity of $g(u,v)$ with respect to the delayed variable $v$ that does not allow to apply the comparison techniques. Thus our proof is based on a variant of the Hale-Lin functional-analytic approach to heteroclinic solutions of functional differential equations where Lyapunov-Schmidt reduction is done in appropriate weighted spaces of $C^2$-smooth functions. This method requires a detailed analysis of associated linear differential Fredholm operators and their formal adjoints. For two different types of $v-$unimodal functions $g(u,v)$, we prove the existence of a maximal continuous family of bistable monotone wavefronts.. Depending on the type of unimodality (equivalently, on the sign of the wave speed), two different scenarios can be observed for the bistable waves: 1) independently on the size of delay, each bistable wavefront is monotone; 2) wavefronts are monotone for moderate values of delays and can oscillate for large delays.