Spreading, flattening and logarithmic lag for reaction-diffusion equations in R^N: old and new results

TL;DR

This paper reviews and extends results on the large-time behavior of reaction-diffusion solutions in R^N, focusing on spreading speeds, flattening, and logarithmic lag, including new estimates for unbounded initial supports.

Contribution

It introduces new logarithmic-in-time lag estimates for solutions with unbounded initial supports, extending previous results to broader initial conditions.

Findings

Logarithmic lag estimates for unbounded initial supports

Flattening properties of level sets for subgraph initial conditions

Extension of known results to initial supports with logarithmic growth

Abstract

This paper is concerned with the large-time dynamics of bounded solutions of reaction-diffusion equations with bounded or unbounded initial support in R N. We start with a survey of some old and recent results on the spreading speeds of the solutions and their asymptotic local one-dimensional symmetry. We then derive some flattening properties of the level sets of the solutions if initially supported on subgraphs. We also investigate the special case of asymptotically conical-shaped initial conditions. Lastly, we reclaim some known results about the logarithmic lag between the position of the solutions and that of planar or spherical fronts expanding with minimal speed, for almost-planar or compactly supported initial conditions. We then prove some new logarithmic-in-time estimates of the lag of the position of the solutions with respect to that of a planar front, for initial conditions which are supported on subgraphs with logarithmic growth at infinity. These estimates entail in particular that the same lag as for compactly supported initial data holds true for a class of unbounded initial supports. The paper also contains some related conjectures and open problems.