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.

中文翻译：

---

双曲 Keller-Segel 方程中的尖锐不连续行波

在这项工作中，我们描述了一个双曲线模型，该模型具有细胞 - 细胞排斥和细胞群的动力学。更准确地说，我们考虑一群细胞产生一个场（我们称之为“压力”），该场诱导细胞按照与梯度相反的方向运动。该字段表示人口的局部密度，我们假设细胞试图避开拥挤的区域，并更喜欢远离承载能力的局部空白空间。我们分析了相关柯西问题在实线上的适定性。我们从有界初始条件开始，我们考虑初始条件的一些不变性质，例如连续性、平滑性和单调性。我们还详细描述了解的传播边界附近水平集的行为，我们发现在自然类初始条件的解上形成了渐近跳跃。最后，我们证明了该模型存在尖锐行波，它们是以恒定速度传播的特定解，并认为尖锐行波必然是不连续的。PDE 问题的数值模拟证实了这一分析。