Convergence to nonlinear diffusion waves for a hyperbolic-parabolic chemotaxis system modelling vasculogenesis

In this paper, we are concerned with a quasi-linear hyperbolic-parabolic system of persistence and endogenous chemotaxis modelling vasculogenesis. Under some suitable structural assumption on the pressure function, we first predict and derive the system admits a nonlinear diffusion wave in R driven by the damping effect. Then we show that the solution of the concerned system will locally and asymptotically converge to this nonlinear diffusion wave if the wave strength is small. By using the time-weighted energy estimates, we further prove that the convergence rate of the nonlinear diffusion wave is algebraic.