Diameters of the level sets for reaction-diffusion equations in nonperiodic slowly varying media *

TL;DR

This paper investigates reaction-diffusion equations with slowly varying, non-periodic heterogeneity, revealing that large oscillations cause the interface width to diverge linearly over time, preventing the existence of generalized transition fronts.

Contribution

It demonstrates how large heterogeneity oscillations affect interface width growth and the non-existence of generalized transition fronts in such media.

Findings

Interface width diverges linearly with time under large oscillations.

Sublinear growth of interface width occurs along certain sequences.

Generalized transition fronts do not exist in highly oscillatory media.

Abstract

We consider in this article reaction-diffusion equations of the Fisher-KPP type with a nonlinearity depending on the space variable x, oscillating slowly and non-periodically. We are interested in the width of the interface between the unstable steady state 0 and the stable steady state 1 of the solutions of the Cauchy problem. We prove that, if the heterogeneity has large enough oscillations, then the width of this interface, that is, the diameter of some level sets, diverges linearly as t $\rightarrow$ +$\infty$ along some sequences of times, while it is sublinear along other sequences. As a corollary, we show that under these conditions generalized transition fronts do not exist for this equation.