Nonlinear stability of phase transition steady states to a hyperbolic–parabolic system modeling vascular networks

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 line (Formula presented.) 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.