  The global dynamics and regularity of parabolic-hyperbolic systems is an
interesting topic in PDEs due to the coupling of competing dissipation and
hyperbolic effects. This paper is concerned with the Cauchy problem of a
parabolic-hyperbolic system derived from a chemotaxis model describing the
dynamics of the initiation of tumor angiogenesis. It is shown that, as time
tends to infinity, the Cauchy problem with large-amplitude discontinuous data
admit global weak solutions which converge to a constant state (resp. a viscous
shock wave) if the asymptotic states of initial values at far field are equal
(resp. unequal). Our results improve the previous results where initial value
was required to be continuous and have small amplitude. Numerical simulations
are performed to verify our analytical results, illustrate the possible
regularity of solutions and speculate the minimal regularity of initial data
required to obtain the smooth (classical) solutions of the concerned
parabolic-hyperbolic system.
