  An Euler-type hyperbolic-parabolic system of chemotactic aggregation
describing the vascular network formation is investigated in the critical
regularity setting. For small initial data around a constant equilibrium state,
the well-posedness of the global classical solution to the Cauchy problem with
general pressure laws is established in homogeneous hybrid Besov spaces. Then,
the optimal time-decay rates of the global solution are analyzed under an
additional regularity assumption on the initial data. Furthermore, the
relaxation limit (large friction limit) of the hyperbolic-parabolic system is
justified rigorously. It is shown that as the friction coefficient tends to
zero, the global solution of the hyperbolic-parabolic chemotaxis system
converges to the global solution of the Keller-Segel equations with an explicit
convergence rate. To capture the dissipative properties of the nonlinear
system, our approach relies on the introduction of new effective unknowns in
low frequencies and the construction of a Lyapunov functional in the spirit of
Beauchard and Zuazua's in [5] to treat the high frequencies.
