Nontrivial dynamics beyond the logarithmic shift in two-dimensional Fisher-KPP equations

TL;DR

This paper investigates the long-term behavior of solutions to the 2D Fisher-KPP equation with Heaviside-like initial conditions, revealing conditions for convergence to traveling waves or oscillatory behaviors beyond the classical logarithmic shift.

Contribution

It demonstrates that solutions can either converge to a traveling wave or oscillate, depending on the initial condition’s behavior at infinity, extending understanding of 2D Fisher-KPP dynamics.

Findings

Solutions may converge to a translated wave.

Solutions may oscillate between two translates.

Behavior depends on initial condition at infinity.

Abstract

We study the asymptotic behaviour, as time goes to infinity, of the Fisher-KPP equation $\partial_t u=\Delta u +u-u^2$ in spatial dimension $2$, when the initial condition looks like a Heaviside function. Thus the solution is, asymptotically in time, trapped between two planar critical waves whose positions are corrected by the Bramson logarithmic shift. The issue is whether, in this reference frame, the solutions will converge to a travelling wave, or will exhibit more complex behaviours. We prove here that both convergence and nonconvergence may happen: the solution may converge towards one translate of the planar wave, or oscillate between two of its translates. This relies on the behaviour of the initial condition at infinity in the transverse direction.