# Existence and non-existence of transition fronts in mixed ignition-monostable media

## TL;DR

This paper investigates the conditions under which transition fronts exist or do not exist in one-dimensional reaction-diffusion equations with mixed ignition-monostable reactions, highlighting the impact of reaction inhomogeneity.

## Contribution

It provides an almost sharp criterion for the existence and non-existence of transition fronts in perturbed ignition-monostable media.

## Key findings

- Strong inhomogeneity prevents front formation
- Weak inhomogeneity allows transition fronts
- Extends previous results with new sharp conditions

## Abstract

We study transition fronts for one-dimensional reaction-diffusion equations with compactly perturbed ignition-monostable reactions. We establish an almost sharp condition on reactions which characterizes the existence and non-existence of fronts. In particular, we prove that a strong inhomogeneity in the reaction prevents formation of transition fronts, while a weak inhomogeneity gives rise to a front. Our work extends results and methods introduced by J. Nolen, J.M. Roquejoffre, L. Ryzhik, and A. Zlato\v{s}.