Sharp conditions for blowup of solutions of a chemotactical model for two species in R2R2

We consider a model system of Keller–Segel type for the evolution of two species in the whole space R2R2 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∞m∞ and n∞n∞ be the total masses of the species. Then there exists a critical curve γγ in the m∞−n∞m∞−n∞ plane such that the solution blows up if and only if (m∞,n∞)(m∞,n∞) is above γγ. 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.