Sharp thresholds between finite spread and uniform convergence for a reaction-diffusion equation with oscillating initial data

TL;DR

This paper studies the long-term behavior of reaction-diffusion equations with ignition nonlinearities, revealing sharp thresholds between uniform convergence and finite-speed spreading based on initial data oscillations.

Contribution

It establishes precise conditions under which solutions either converge uniformly or spread, including the transition thresholds related to initial data oscillations and periodicity.

Findings

Solutions either converge uniformly or spread with finite speed.

Transition thresholds depend sharply on initial data oscillation period.

Convergence to planar fronts occurs with asymptotically periodic initial data.

Abstract

We investigate the large-time dynamics of solutions of multi-dimensional reaction-diffusion equations with ignition type nonlinearities. We consider solutions which are in some sense locally persistent at large time and initial data which asymptotically oscillate around the ignition threshold. We show that, as time goes to infinity, any solution either converges uniformly in space to a constant state, or spreads with a finite speed uniformly in all directions. Furthermore, the transition between these two behaviors is sharp with respect to the period vector of the asymptotic profile of the initial data. We also show the convergence to planar fronts when the initial data are asymptotically periodic in one direction.