Speed Selection for Reaction Diffusion Equations in Heterogeneous Environments

TL;DR

This paper investigates how to determine the minimal propagation speed in reaction-advection-diffusion equations within heterogeneous environments, especially when nonlinearities are complex and do not follow the KPP type, providing new criteria for speed selection.

Contribution

It introduces novel selection criteria for the minimal speed in heterogeneous reaction-diffusion equations with general nonlinearities, extending beyond classical KPP assumptions.

Findings

Derived criteria for minimal speed selection in heterogeneous media.

Established conditions under which linearization predicts the minimal speed.

Provided practical methods using upper/lower solutions for complex nonlinearities.

Abstract

Reaction-advection-diffusion equations, in periodic settings and with general type nonlinearities, admit a threshold known as the minimal speed of propagation. The minimal speed does not have an accessible formula when the nonlinearity is not of KPP type, for instance. The question becomes whether the minimal speed can be obtained through a linearization procedure or not. In this paper, we derive selection criteria for the minimal speed: a key feature of the nonlinear selection is unveiled. Moreover, we use upper/lower solution techniques in order to derive practical criteria determining the minimal speed in the presence of advection and a general type nonlinearilty.