Stationary solutions of Keller-Segel type crowd motion and herding models: multiplicity and dynamical stability

TL;DR

This paper analyzes two Keller-Segel type models for crowd motion, classifies their stationary solutions, and investigates their multiplicity and stability properties through theoretical and numerical methods.

Contribution

It provides a comprehensive classification of radial stationary solutions, establishes multiplicity results, and links variational and dynamical stability for crowd motion models.

Findings

Multiple stable stationary solutions exist for certain parameters.

A Lyapunov functional links variational and dynamical stability.

Numerical simulations support theoretical stability results.

Abstract

In this paper we study two models for crowd motion and herding. Each of the models is of Keller-Segel type and involves two parabolic equations, one for the evolution of the density and one for the evolution of a mean field potential. We classify all radial stationary solutions, prove multiplicity results and establish some qualitative properties of these solutions, which are characterized as critical points of an energy functional. A notion of variational stability is associated to such solutions. The dynamical stability in a neighborhood of a stationary solution is also studied in terms of the spectral properties of the linearized evolution operator. For one of the two models, we exhibit a Lyapunov functional which allows to make the link between the two notions of stability. Even in that case, for certain values of the mass parameter and all other parameters taken in an appropriate range, we find that two dynamically stable stationary solutions exist. We further discuss qualitative properties of the solutions using theoretical methods and numerical computations.