# The speed of propagation for KPP reaction-diffusion equations within large drift

## TL;DR

This paper investigates the asymptotic behavior of the minimal propagation speed of pulsating traveling fronts in KPP reaction-diffusion equations with large drift, within periodic heterogeneous media, providing explicit formulas and conditions for positivity.

## Contribution

It extends the understanding of minimal speed asymptotics in reaction-diffusion equations with large drift, using variational formulas and detailed analysis in two dimensions.

## Key findings

- Derived asymptotic formulas for minimal speed in large drift regimes.
- Identified conditions for the positivity of the asymptotic speed in 2D.
- Connected the minimal speed limits to variational quantities over first integrals.

## Abstract

This paper is devoted to the study of the asymptotic behaviors of the minimal speed of propagation of pulsating traveling fronts solving the Fisher-KPP reaction-advection-diffusion equation within either a large drift, a mixture of large drift and small reaction, or a mixture of large drift and large diffusion. We consider a periodic heterogenous framework and we use the formula of Berestycki, Hamel and Nadirashvili \cite{bhn2} for the minimal speed of propagation to prove the asymptotics in any space dimension $N.$ We express the limits as the maxima of certain variational quantities over the family of "first integrals" of the advection field. Then, we perform a detailed study in the case N=2 which leads to a necessary and sufficient condition for the positivity of the asymptotic limit of the minimal speed within a large drift.