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 R+=(0,infinity) 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.