A nonlocal model describing tumor angiogenesis

Abstract

In this paper, we derive and study a new mathematical model that describes the onset of angiogenesis. 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 ∂tp −1/2(−Δ)(α−1)/2H∂tp = −1/2(−Δ)α/2p + p∂xp − ∂xp, where H denotes the Hilbert transform. In addition to the derivation of the new model, the main novelty of the present paper is that we also prove a number of well-posedness results. Finally, some preliminary numerical results are shown. These numerical results suggest that the dynamics of the equation is rich enough to have solutions that blow up in finite time.