Sharp critical mass criteria for weak solutions to a degenerate cross-attraction system

The qualitative study of solutions to the coupled parabolic-elliptic chemotaxis system with nonlinear diffusion for two species will be considered in the whole Euclidean space $\mathbb{R}^d$ ($d\geq 3$). It was proven in \cite{CK2021-ANA} that there exist two critical curves that separate the global existence and blow-up of weak solutions to the above problem. We improve this result by providing sharp criteria for the dicothomy: global existence of weak solution versus blow-up below and at these curves. Besides, there exist sharp critical masses of initial data at the intersection of the two critical lines, which extend the well-known critical mass phenomenon in one-species Keller-Segel system in \cite{BCL09-CVPDE} to two-species case.