具p-拉普拉斯扩散的血管生成模型中的新动力学
Novel Dynamics in Models of Angiogenesis with p-Laplacian diffusion
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中文总结 AI 辅助
本文研究具p-拉普拉斯扩散的血管生成两物种模型，证明p>3/2且初始数据足够小时系统弱适定，发现p-拉普拉斯带来的新动力学，并探讨其在心脏健康数字孪生框架中的应用。
中文摘要 AI 辅助
缺血性心脏病是全球死亡率最高的疾病，血管重建术（恢复阻塞部位血流的过程）展现出良好前景。为此，针对血管生成（现有血管形成新血管的过程）的数学模型已得到深入研究。本文研究经典的血管生成两物种模型，包含细胞和VEGF（血管内皮生长因子）种群，假设细胞按p-拉普拉斯扩散运动，涵盖“快扩散”（1<p<2）、“慢扩散”（p>2）及正常扩散（p=2）情形。首先证明当p>3/2且初始数据足够小时，该系统在弱意义下适定；其次表明p-拉普拉斯会产生多种此前未报道的新动力学，包括通过双峰及多尖峰解增强细胞增殖、正则性提升、避免有限时间爆破、通过有限时间灭绝导致细胞耗竭，以及图灵模式。本文还讨论了这些结果在心脏健康领域通过数字孪生框架的应用。
英文摘要
Ischemic heart diseases represent the leading cause of mortality worldwide. Revascularization, the process to restore blood flow in blockages, shows promise. To this end, mathematical models for angiogenesis, the process by which new blood vessels form from existing ones, have been extremely well investigated. In the current work, we consider a classical two species model for angiogenesis, consisting of cell and VEGF populations. However, we assume the cells move according to p-Laplacian diffusion, which could be both ``fast" ($1<p<2$) and ``slow" ($p>2$), in addition to normal diffusion ($p=2$). We first show that the system is well posed in a weak sense when $p>\frac{3}{2}$, for sufficiently small initial data. Next, we show that the p-Laplacian can lead to several novel dynamics not reported earlier; these include increased cellular proliferation via bi-modal and multi spike solutions, gain of regularity, prevention of finite time blow-up, cell depletion via finite time extinction, and Turing patterns. We discuss applications of these results for cardiac health via a digital twins framework.
