# Propagation in a fractional reaction-diffusion equation in a periodically hostile environment

## TL;DR

This paper analyzes a fractional reaction-diffusion equation in a periodic environment, establishing the existence of a stable state and its exponential invasion speed into an unstable state.

## Contribution

It provides the first asymptotic analysis of a fractional Fisher-KPP type equation in periodic media with Dirichlet conditions, including existence, uniqueness, and invasion speed.

## Key findings

- Existence and uniqueness of a non-trivial stationary state
- Stable state invades at exponential speed
- Analysis in periodic non-connected media

## Abstract

We provide an asymptotic analysis of a fractional Fisher-KPP type equation in periodic non-connected 1-dimensional media with Dirichlet conditions outside the domain. After demonstrating the existence and uniqueness of a non-trivial bounded stationary state $n\_+$ , we prove that the stable state $n\_+$ invades the unstable state 0 with a speed which is exponential in time.