Monostable pulled fronts and logarithmic drifts

TL;DR

This paper explores the occurrence of logarithmic drifts in the position of level sets of solutions to monostable reaction-diffusion equations, revealing new behaviors when the front speed is linearly determined.

Contribution

It extends the understanding of logarithmic drift phenomena to cases where the minimal front speed is linearly determined, beyond classical KPP and nonlinear cases.

Findings

Logarithmic drift always occurs when the speed is linearly determined.

The factor in front of the logarithmic term differs from the classical KPP case.

The drift phenomenon disappears in nonlinearly determined minimal speed cases.

Abstract

In this work we investigate the issue of logarithmic drifts in the position of the level sets of solutions of monostable reaction-diusion equations, with respect to the traveling front with minimal speed. On the one hand, it is a celebrated result of Bramson that such a logarithmic drift occurs when the reaction is of the KPP (or sublinear) type. On the other hand, it is also known that this drift phenomenon disappears when the minimal front speed is nonlinearly determined. However, some monostable reaction-diusion equations fall in neither of those cases and our aim is to fill that gap. We prove that a logarithmic drift always occurs when the speed is linearly determined, but surprisingly we find that the factor in front of the logarithmic term may be different from the KPP case.