Convective stability of the critical waves of an FKPP-type model for self-organized growth

TL;DR

The paper analyzes the stability of critical waves in a model of self-organized growth involving two types of particles.

Contribution

A novel Feynman–Kac approach is used to prove stability of critical waves in a reaction–diffusion system with unbounded weights.

Findings

Critical traveling waves are locally stable under exponentially growing perturbations.

A modified semi-group approach with unbounded weights was successfully applied.

A Feynman–Kac formula provides a new exponential a priori estimate for the PDE tail.

Abstract

We construct the traveling wave solutions of an FKPP growth process of two densities of particles, and prove that the critical traveling waves are locally stable in a space where the perturbations can grow exponentially at the back of the wave. The considered reaction–diffusion system was introduced by Hannezo et al. (Cell 171(1):242–255, 2017) in the context of branching morphogenesis: active, branching particles accumulate inactive particles, which do not react. Thus, the system features a continuum of steady state solutions, complicating the analysis. We adopt a result by Faye and Holzer (J Differ Equ 269(9):6559–6601, 2020) for proving the stability of the critical traveling waves, and modify the semi-group estimates to spaces with unbounded weights. We use a Feynman–Kac formula to get an exponential a priori estimate for the tail of the PDE, a novel and simple approach.

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