Critical mass phenomenon for a parabolic-elliptic multispecies chemotaxis system in a two-dimensional disk

A parabolic-elliptic Keller-Segel model for multipopulations in a two-dimensional unit ball will be considered in this paper. For the initial boundary value problem with mutually attractive populations, the global existence of bounded solutions has been proved through the logarithmic Hardy-Littlewood-Sobolev inequality for system when the initial masses satisfy the subcritical condition. Moreover, based on the moments technique, there exists a blow-up solution when the initial masses satisfy the supercritical case. In a word, a critical mass phenomenon has been established in this multispecies chemotaxis system.