Sharp discontinuous traveling waves in a hyperbolic Keller--Segel equation

TL;DR

This paper investigates a hyperbolic Keller--Segel model describing cell movement driven by cell-cell repulsion, demonstrating the existence of sharp, discontinuous traveling wave solutions and analyzing their properties through mathematical and numerical methods.

Contribution

It introduces and proves the existence of sharp discontinuous traveling wave solutions in a hyperbolic Keller--Segel model, highlighting their formation and properties.

Findings

Existence of sharp traveling waves with discontinuities.

Asymptotic jump formation near propagating boundaries.

Numerical simulations confirming theoretical results.

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 monotonicity. 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.