A nonlocal model describing tumor angiogenesis

TL;DR

This paper introduces a new nonlocal Burgers-type model for tumor angiogenesis, analyzes its mathematical properties, and provides preliminary numerical simulations indicating complex dynamics including potential finite-time blow-up.

Contribution

The paper derives a novel nonlocal model for tumor angiogenesis and establishes well-posedness results along with initial numerical explorations.

Findings

Model exhibits rich dynamics with potential finite-time blow-up.

Well-posedness results are established for the model.

Numerical simulations suggest complex solution behaviors.

Abstract

In this paper we study the onset of angiogenesis and derive a new model to describe it. This new model takes the form of a nonlocal Burgers equation with both diffusive and dispersive terms. For a particular value of the parameters, the equation reduces to $$ \partial_t p-\frac{1}{2}(-\Delta)^{(\alpha-1)/2}H \partial_t p=-\frac{1}{2}(-\Delta)^{\alpha/2} p+ p\partial_x p-\partial_x p, $$ where $H$ denotes the Hilber transform. In addition to the derivation of the new model, we also prove a number of well-posedness results. Finally, some preliminary numerics are shown. These numerics suggest that the dynamics of the equation is rich enough to have solutions that blow up in finite time.