The non-monotonicity of the KPP speed with respect to diffusion in the presence of a shear flow

TL;DR

This paper demonstrates through counterexamples that adding shear flow to a reaction-diffusion system can cause the KPP propagation speed to behave non-monotonically with respect to diffusion, even in homogeneous settings.

Contribution

It provides the first known counterexamples showing non-monotonicity of KPP speed with respect to diffusion in the presence of shear flow.

Findings

Shear flow can induce non-monotonicity in KPP speed.

Non-monotonicity occurs even with homogeneous reaction and diffusion.

Results extend to domains with periodic perforations.

Abstract

In this paper, we prove via counterexamples that adding an advection term of the form Shear flow (whose streamlines are parallel to the direction of propagation) to a reaction-diffusion equation will be an enough heterogeneity to spoil the increasing behavior of the KPP speed of propagation with respect to diffusion. The non-monotonicity of the speed with respect to diffusion will occur even when the reaction term and the diffusion matrices are considered homogeneous (do not depend on space variables). For the sake of completeness, we announce our results in a setting which allows domains with periodic perforations that may or may not be equal to the whole space $\mathbb{R}^N.$