论文标题
双曲线凯勒（segel）方程中的尖锐不连续的行进波
Sharp discontinuous traveling waves in a hyperbolic Keller--Segel equation
论文作者
论文摘要
在这项工作中，我们描述了一个具有细胞细胞排斥的双曲线模型，细胞群中具有动力学。更确切地说，我们考虑了产生一个场的细胞群（我们称为“压力”），该细胞诱发了梯度相反的细胞运动。该领域表示人口的局部密度，我们假设细胞试图避免拥挤的区域，并且更喜欢远离承载能力的本地空间。我们分析了实际线路上相关问题的适合性属性。我们从有限的初始条件开始，我们考虑了初始条件的一些不变特性，例如连续性，平滑度和单调性。我们还详细描述了解决方案的传播边界附近的水平集的行为，并发现在自然初始条件类别的溶液上形成了渐近跳跃。最后，我们证明了该模型的急剧流动波，这是以恒定速度行进的特定解决方案，并认为锋利的行进波必定是不连续的。通过对PDE问题的数值模拟证实了该分析。
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.
