Reaction-diffusion equations in periodic media: convergence to pulsating fronts

TL;DR

This paper studies reaction-diffusion equations in periodic media, showing convergence to pulsating fronts and analyzing the asymptotic invasion shapes under certain stability assumptions.

Contribution

It establishes convergence of solutions to pulsating fronts in periodic media and derives a generalized Freidlin-Gärtner formula for invasion shapes.

Findings

Solutions exhibit pulsating front profiles at large times.

The asymptotic invasion shapes are related to the solutions' upper level sets.

The results apply to combustion and bistable reaction-diffusion equations.

Abstract

This paper is concerned with reaction-diffusion-advection equations in spatially periodic media. Under an assumption of weak stability of the constant states 0 and 1, and of existence of pulsating traveling fronts connecting them, we show that fronts' profiles appear, along sequences of times and points, in the large-time dynamics of the solutions of the Cauchy problem, whether their initial supports are bounded or unbounded. The types of equations that fit into our assumptions are the combustion and the bistable ones. We also show a generalized Freidlin-G{\"a}rtner formula and other geometrical properties of the asymptotic invasion shapes, or spreading sets, of invading solutions, and we relate these sets to the upper level sets of the solutions.