Sharp Condition for blow-up and global existence in a two species chemotactic Keller-Segel system in $\mathbb{R}^2$

Abstract

For the parabolic-elliptic Keller-Segel system in R^2 it has been proved that if the initial mass is less than 8\pi/\chi\ global solution exist and in the case that the initial mass is larger than 8\pi/\chi\ blow-up happens. The case of several chemotactic species introduces an additional question: What is the analog for the critical mass obtained for the single species system? We find a threshold curve in the case of two especies case that allows us to determine if the system has blow-up or has a global in time solution.