Sharp conditions for blowup of solutions of a chemotactical model for two species in

We consider a model system of Keller–Segel type for the evolution of two species in the whole space R 2 which are driven by chemotaxis and diffusion. It is well known that this problem admits global and blowup solutions. We show that there exists a sharp condition which allows to distinguish global and blowup solutions in the radially symmetric case. More precisely, let m ∞ and n ∞ be the total masses of the species. Then there exists a critical curve γ in the m ∞ − n ∞ plane such that the solution blows up if and only if ( m ∞ , n ∞ ) is above γ . This gives an answer to a question raised by Conca et al. (2011) in [8]. We also study the asymptotic behaviour of global solutions in the subcritical case, showing that they are asymptotically self-similar.