Self-similar solutions of decaying Keller-Segel systems for several populations

It is known that solutions of the parabolic elliptic Keller-Segel equations in the two dimensional plane decay, as time goes to infinity, provided the initial data admits sub-critical mass and finite second moments, while such solution concentrate, as $t\rightarrow\infty$, in the critical mass. 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 for the concentration in the critical case. This paper is motivated by the Keller-Segel system of several interacting populations, under the existence of an additional drift for each component which decays in time at the rate $O(1/\sqrt{t})$. We show that self-similar solutions always exists in the sub-critical case, while the existence of such self-similar solution in the critical case depends on the gap between the decaying drifts for each of the components. For this, we study the conditions for existence/non existence of solutions for the corresponding Liouville's systems, which, in turn, is related to the existence/non existence of minimizers to a corresponding Free Energy functional.