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Sharp discontinuous traveling waves in a hyperbolic Keller–Segel equation

Xiaoming Fu

Institut de Mathématiques de Bordeaux, Université de Bordeaux, CNRS, IMB, UMR 5251, Talence 33400, France

E-mail Address: [email protected]

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Quentin Griette

Institut de Mathématiques de Bordeaux, Université de Bordeaux, CNRS, IMB, UMR 5251, Talence 33400, France

E-mail Address: [email protected]

Corresponding author

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, and

Pierre Magal

Institut de Mathématiques de Bordeaux, Université de Bordeaux, CNRS, IMB, UMR 5251, Talence 33400, France

E-mail Address: [email protected]

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https://doi.org/10.1142/S0218202521500214Cited by:4 (Source: Crossref)

Abstract

In this work, we describe a hyperbolic model with cell–cell repulsion with a dynamics in the population of cells. More precisely, we consider a population of cells producing a field (which we call “pressure”) which induces a motion of the cells following the opposite of the gradient. The field indicates the local density of population and we assume that cells try to avoid crowded areas and prefer locally empty spaces which are far away from the carrying capacity. We analyze the well-posedness property of the associated Cauchy problem on the real line. We start from bounded initial conditions and we consider some invariant properties of the initial conditions such as the continuity, smoothness and monotony. We also describe in detail the behavior of the level sets near the propagating boundary of the solution and we find that an asymptotic jump is formed on the solution for a natural class of initial conditions. Finally, we prove the existence of sharp traveling waves for this model, which are particular solutions traveling at a constant speed, and argue that sharp traveling waves are necessarily discontinuous. This analysis is confirmed by numerical simulations of the PDE problem.

Communicated by O. Diekmann

Keywords:

AMSC: 92C17, 35L60, 35D30

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