Novel Dynamics in Models of Angiogenesis with p-Laplacian diffusion

Abstract

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&#34; ($1<p<2$) and ``slow&#34; ($p>2$), 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.