# Propagation dynamics of Fisher-KPP equation with time delay and free boundaries

## TL;DR

This paper studies how time delay and free boundaries influence the propagation of solutions in Fisher-KPP equations, revealing a dichotomy between spreading and vanishing, and showing delays slow down spreading speed.

## Contribution

It establishes a well-posedness condition for delayed reaction-diffusion equations with free boundaries and characterizes the spreading speed and profile affected by delays.

## Key findings

- A vanishing-spreading dichotomy is proven.
- Spreading speed is slowed by time delay.
- Spreading profile is determined by a nonlocal semi-wave problem.

## Abstract

Incorporating free boundary into time-delayed reaction-diffusion equations yields a compatible condition that guarantees the well-posedness of the initial value problem. With the KPP type nonlinearity we then establish a vanishing-spreading dichotomy result. Further, when the spreading happens, we show that the spreading speed and spreading profile are nonlinearly determined by a delay-induced nonlocal semi-wave problem. It turns out that time delay slows down the spreading speed.