Sharp condition for blow-up and global existence in a two species chemotactic Keller–Segel system in 2

For the parabolic–elliptic Keller–Segel system in 2 it has been proved that if the initial mass is less than 8π/χ, a global solution exists, and in case the initial mass is larger than 8π/χ, 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 two species case that allows us to determine if the system is a blow-up or a global in time solution. No radial symmetry is assumed.

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