Front propagation for reaction-diffusion equations in composite structures

TL;DR

This paper studies the asymptotic behavior of interface motion in reaction-diffusion equations within composite media, revealing conditions under which the interface follows a Finsler metric or exhibits non-local jumps, especially for KPP-type nonlinearities.

Contribution

It provides new limit theorems describing interface motion in composite structures, including non-local jumps and Finsler metric-based propagation, for reaction-diffusion equations with KPP nonlinearities.

Findings

Interface motion can be described by Finsler or Riemannian metrics under certain conditions.

The interface may exhibit jump discontinuities in its motion.

Results are based on large deviation limit theorems.

Abstract

We consider asymptotic problems concerning the motion of interface separating the regions of large and small values of the solution of a reaction-diffusion equation in the media consisting of domains with different characteristics (composites). Under certain conditions, the motion can be described by the Huygens principle in the appropriate Finsler (e.g., Riemannian) metric. In general, the motion of the interface has, in a sense, non-local nature. In particular, the interface may move by jumps. We are mostly concerned with the nonlinear term that is of KPP type. The results are based on limit theorems for large deviations.