Sharp discontinuous traveling waves in a hyperbolic Keller–Segel equation

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