Stationary solutions with vacuum for a hyperbolic–parabolic chemotaxis model in dimension two

In this research, we study the existence of stationary solutions with vacuum to a hyperbolic–parabolic chemotaxis model with nonlinear pressure in dimension two that describes vasculogenesis. We seek radially symmetric solutions in the whole space, in which the system will be reduced to a system of ODE’s on $[0,\infty )$. The fundamental solutions to the ODE system are the Bessel functions of different types. We find two nontrivial solutions. One is formed by half bump (positive density region) starting at $r=0$ and a region of vacuum on the right. Another one is a full nonsymmetric bump away from $r=0$. These solutions bear certain resemblance to in vitro vascular network and the numerically produced structure by Gamba et al. (2003). We also show the nonexistence of full bump starting at $r=0$ and nonexistence of full symmetric bump away from $r=0$.