# Stability of semi-wavefronts for delayed reaction-diffusion equations

## TL;DR

This paper investigates the stability of semi-wavefront solutions in delayed reaction-diffusion equations, establishing conditions for global stability of wavefronts and the stability of their leading edges.

## Contribution

It provides new stability results for semi-wavefronts in delayed reaction-diffusion equations under specific conditions on the reaction term.

## Key findings

- Global stability of fast wavefronts proved
- Stability of the leading edge of semi-wavefronts established
- Conditions on the reaction term g for stability are identified

## Abstract

This paper deals with the asymptotic behavior of solutions to the delayed monostable equation: $(*)$ $u_{t}(t,x) = u_{xx}(t,x) - u(t,x) + g(u(t-h,x)),$ $x \in \mathbb{R},\ t >0,$ where $h>0$ and the reaction term $g: \mathbb{R}_+ \to \mathbb{R}_+$ has exactly two fixed points (zero and $\kappa >0$). Under certain condition on the derivative of $g$ at $\kappa$, the global stability of fast wavefronts is proved. Also, the stability of the $leading \ edge$ of semi-wavefronts for $(*)$ with $g$ satisfying $g(u)\leq g'(0)u, u\in\R_+,$ is established