Finite time blow up in Kaniadakis-Quarati model of Bose-Einstein particles

We study a Fokker-Planck equation with linear diusion and super-linear drift introduced by Kaniadakis and Quarati to describe the evolution of a gas of Bose-Einstein particles. For kinetic equation of this type it is well-known that, in the physical space R3, the structure of the equilibrium Bose-Einstein distribution depends upon a parameter m∗, the critical mass. We are able to describe the time-evolution of the solution in two dierent situations, which correspond to m ≪ m∗ and m ≫ m∗ respectively. In the former case, it is shown that the solution remains regular, while in the latter we prove that the solution starts to blow up at some nite time, for which we give an upper bound in terms of the initial mass. The results are in favor of the validation of the model, which, in the supercritical regime, could produce in nite time a transition from a normal uid to one with a condensate component. The research that led to the present paper was partially supported by a grant of the group GNFM of INdA