Global existence and blow-up for multi-species chemotaxis model in a two-dimensional bounded domain

Abstract This paper deals with the global solvability and finite-time blow-up for both parabolic-elliptic and parabolic-parabolic multi-species chemotaxis models in a two-dimensional bounded domain. In the absence of conflicts, the solution to the initial boundary value problem exists globally for any large initial data if the self-repelling effects are strong enough. However, a critical mass phenomenon nearly exists when all pairs attract mutually. More precisely, the globally bounded solution appears in the sub-critical mass condition, while the finite-time blow-up occurs in the super-critical mass condition. The proof of global well-posedness to the solution relies on the logarithmic Hardy-Littlewood-Sobolev inequality for system and the Moser-Trudinger inequality for system on the bounded domain. 2010 Mathematics Subject Classification. 35A01, 35B44, 35K20, 35K55, 92C17.