Premium
Nonlinear stability of phase transition steady states to a hyperbolic–parabolic system modeling vascular networks
Author(s) -
Hong Guangyi,
Peng Hongyun,
Wang ZhiAn,
Zhu Changjiang
Publication year - 2021
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms.12415
Subject(s) - uniqueness , mathematics , mathematical analysis , dirichlet boundary condition , phase transition , steady state (chemistry) , nonlinear system , constant (computer programming) , perturbation (astronomy) , boundary value problem , physics , condensed matter physics , quantum mechanics , chemistry , computer science , programming language
This paper is concerned with the existence and stability of phase transition steady states to a quasi‐linear hyperbolic–parabolic system of chemotactic aggregation, which was proposed in [Ambrosi, Bussolino and Preziosi, J. Theoret. Med . 6 (2005) 1–19; Gamba et al ., Phys. Rev. Lett . 90 (2003) 118101.] to describe the coherent vascular network formation observed in vitro experiment. Considering the system in the half lineR + = ( 0 , ∞ )with Dirichlet boundary conditions, we first prove the existence and uniqueness of non‐constant phase transition steady states under some structure conditions on the pressure function. Then we prove that this unique phase transition steady state is nonlinearly asymptotically stable against a small perturbation. We prove our results by the method of energy estimates, the technique of a priori assumption and a weighted Hardy‐type inequality.