Communications in Mathematical Sciences

Volume 21 (2023)

Number 8

Stability of planar rarefaction wave for viscous vasculogenesis model

Pages: 2261 – 2299



Qingqing Liu (School of Mathematics, South China University of Technology, Guangzhou, China)

Yuxiu Tian (School of Mathematics, South China University of Technology, Guangzhou, China)


In this paper, we are concerned with a two-dimensional quasi-linear hyperbolicparabolic-elliptic system modelling vasculogenesis. We first derive a two-dimensional inviscid system as the asymptotic equations in large time by ignoring all the viscous terms. Then we show that this inviscid system admits a planar rarefaction wave when the pressure function satisfies some suitable structure conditions. By using elaborate energy estimates, we further prove that the solution of the concerned system will asymptotically converge to this planar rarefaction wave under the same assumptions on pressure function.


vasculogenesis model, planar rarefaction wave, asymptotic stability

2010 Mathematics Subject Classification

35B35, 35B40, 35L65, 35Q92

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Q.Q. Liu was supported by the National Natural Science Foundation of China (No. 12071153); by the Guangdong Basic and Applied Basic Research Foundation (No. 2021A1515012360, and No. 2020B1515310015); and by the Guangzhou Municipal Science and Technology Project (No. 202102021137).

Received 18 October 2022

Accepted 2 April 2023

Published 15 November 2023