Convergence to a propagating front in a degenerate Fisher-KPP equation with advection

TL;DR

This paper analyzes a Fisher-KPP equation with density-dependent diffusion and advection, focusing on the emergence and propagation of transition layers, and proves convergence to a free-boundary problem with precise layer thickness estimates.

Contribution

It introduces a detailed analysis of layer emergence and propagation in a degenerate Fisher-KPP equation with advection, establishing convergence to a free-boundary limit and providing sharp layer thickness estimates.

Findings

Transition layers emerge due to reaction and drift balance.

Propagation of layers is characterized and analyzed.

Convergence to a free-boundary problem is rigorously proved.

Abstract

We consider a Fisher-KPP equation with density-dependent diffusion and advection, arising from a chemotaxis-growth model. We study its behavior as a small parameter, related to the thickness of a diffuse interface, tends to zero. We analyze, for small times, the emergence of transition layers induced by a balance between reaction and drift effects. Then we investigate the propagation of the layers. Convergence to a free-boundary limit problem is proved and a sharp estimate of the thickness of the layers is provided.