Fast propagation for KPP equations with slowly decaying initial conditions

TL;DR

This paper investigates the large-time behavior of solutions to one-dimensional Fisher-KPP equations with slowly decaying initial conditions, revealing infinitely fast spreading and asymptotic flatness of solutions.

Contribution

It provides the first systematic analysis of KPP equations with slowly decaying initial conditions, contrasting with the classical exponential decay case.

Findings

Level sets move infinitely fast as time increases.

Spatial profiles become asymptotically flat.

Level set locations depend on initial decay rate.

Abstract

This paper is devoted to the analysis of the large-time behavior of solutions of one-dimensional Fisher-KPP reaction-diffusion equations. The initial conditions are assumed to be globally front-like and to decay at infinity towards the unstable steady state more slowly than any exponentially decaying function. We prove that all level sets of the solutions move infinitely fast as time goes to infinity. The locations of the level sets are expressed in terms of the decay of the initial condition. Furthermore, the spatial profiles of the solutions become asymptotically uniformly flat at large time. This paper contains the first systematic study of the large-time behavior of solutions of KPP equations with slowly decaying initial conditions. Our results are in sharp contrast with the well-studied case of exponentially bounded initial conditions.