On Liouville’s systems corresponding to self similar solutions of the Keller-Segel systems of several populations

Abstract: The Keller-Segel system in two dimensions represents the evolution of living cells under self- attraction and diffusive forces. In its simplest form, it is a conservative drift-diffusion equation for the cell density coupled to an elliptic equation for the chemo-attractant concentration. It is known that in two space dimension there is a critical mass βc such that for initial mass β ≤ βc there is global in time existence of solutions while for β > βc finite time blow-up occurs. In the sub-critical regime (β < βc), the solutions decay as time t goes to infinity, while such solution concentrate, as t goes to infinity for the critical initial mass (β = βc). In the sub-critical case, this decay can be resolved by a steady, self-similar solution, while no such self-similar solution is known to exist in the critical case.