Asymptotic stability of diffusion waves of a quasi-linear hyperbolic-parabolic model for vasculogenesis

TL;DR

This paper analyzes the long-term behavior of solutions to a hyperbolic-parabolic model for vasculogenesis, demonstrating that solutions tend to diffusion waves over time with specific decay rates.

Contribution

It introduces a novel combination of Fourier energy methods and spectral analysis to establish the asymptotic stability of diffusion waves in a quasi-linear hyperbolic-parabolic system.

Findings

Solutions tend to diffusion waves asymptotically.

Algebraic decay rates are established.

Conditions on the pressure function influence stability.

Abstract

In this paper, we derive the large-time profile of solutions to the Cauchy problem of a hyperbolic-parabolic system modeling the vasculogenesis in $\R^3$. When the initial data are prescribed in the vicinity of a constant ground state, by constructing a time-frequency Lyapunov functional and employing the Fourier energy method and spectral analysis, we show that solution of the Cauchy problem tend time-asymptotically to linear diffusion waves around the constant ground state with algebraic decaying rates under certain conditions on the density-dependent pressure function.