Freidlin-G\"artner formula and asymptotic profile in reaction-diffusion equations

TL;DR

This paper investigates the large-time behavior of reaction-diffusion equations in periodic media, focusing on the asymptotic invasion shape described by the Freidlin-G"artner formula and related profile convergence results.

Contribution

It provides a proof of the Freidlin-G"artner formula applicable to general reaction terms and presents new results on profile convergence for bistable equations.

Findings

The asymptotic invasion shape is characterized by the Freidlin-G"artner formula.

A regular version of the Freidlin-G"artner formula is established.

Solutions converge in profile towards pulsating traveling fronts.

Abstract

We address the question of the large-time behavior of solutions to reaction-diffusion equations in periodic media. We start with the description of the asymptotic shape of the invasion set, which is characterized by the Freidlin-G\"artner formula. We outline a proof of the formula that holds true for general types of reaction terms. We then present some recent results, obtained in collaboration with H. Guo and F. Hamel, for (weakly) bistable equations. They include a regular version of the Freidlin-G\"artner formula and the convergence in profile towards pulsating traveling fronts for solutions with either bounded or unbounded initial support.