On instability and stability of a quasi-linear hyperbolic-parabolic model for vasculogenesis

TL;DR

This paper investigates the conditions under which a hyperbolic-parabolic model for vasculogenesis is stable or unstable, providing criteria for both linear and nonlinear stability and demonstrating global existence and convergence rates.

Contribution

It establishes new stability and instability criteria for the vasculogenesis model based on pressure conditions, extending previous results with improved conditions for stability.

Findings

Steady-state is linearly unstable when rac{ P'(ar ho)}{eta} < 1.

Steady-state is nonlinearly unstable under the same condition.

Global existence and optimal convergence rates are proven when rac{ P'(ar ho)}{eta} > 1.

Abstract

In this paper, we are concerned with the instability and stability of a quasi-linear hyperbolic-parabolic system modeling vascular networks. Under the assumption that the pressure satisfies $\frac{\nu P'(\bar\rho)}{\gamma \bar\rho} < \beta$, we first show that the steady-state is linear unstable (i.e., the linear solution grows in time in $L^2$) by constructing an unstable solution. Then based on the lower grow estimates on the solution to the linear system, we prove that the steady-state is nonlinear unstable in the sense of Hadamard. On the contrary, if the pressure satisfies $\frac{\nu P'(\bar\rho)}{\gamma \bar\rho} > \beta$, we establish the global existence for small perturbations and the optimal convergent rates for all-order derivatives of the solution by slightly getting rid of the condition proposed in [Liu-Peng-Wang, SIAM J. MATH. ANAL 54:1313--1346, 2022].