2020/11/14 by Hong, Guangyi, Peng, Hongyun, Wang, Zhi-An +1 · 1 citation
#35B40 #35L04 #35L60 #35Q92 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2011.07258
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 \citeambrosi2005review, gamba2003percolation to describe the coherent vascular network formation observed \it in vitro experiment. Considering the system in the half line ℝ+=(0,∞) with Dirichlet boundary conditions, we first prove the existence \textcolorblackand 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 \it a priori assumption and a weighted Hardy-type inequality.