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

TL;DR

This paper studies a hyperbolic-parabolic chemotaxis system modeling vasculogenesis, proving solutions converge to nonlinear diffusion waves with algebraic rates under certain conditions.

Contribution

It establishes the existence and asymptotic convergence of nonlinear diffusion waves for a chemotaxis system, with algebraic convergence rates.

Findings

Solutions converge to nonlinear diffusion waves

Convergence is local and asymptotic

Convergence rate is algebraic

Abstract

In this paper, we are concerned with a quasi-linear hyperbolic-parabolic system of persistence and endogenous chemotaxis modelling vasculogenesis in $\mathbb{R}$. Under some suitable structural assumption on the pressure function, we first predict the system admits a nonlinear diffusion wave in $\mathbb{R}$ based on the empirical results in the literature. Then we show that the solution of the concerned system will locally and asymptotically converges 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.