A nonlocal model describing tumor angiogenesis

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 $\partial_t p-\frac{1}{2}(-Δ)^{(α-1)/2}H \partial_t p=-\frac{1}{2}(-Δ)^{α/2} p+ p\partial_x p-\partial_x p,$ 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.

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