Finite Time Blow Up in Kaniadakis-Quarati Model of Bose-Einstein Particles

We study a Fokker-Planck equation with linear diffusion and super-linear drift introduced by Kaniadakis and Quarati [12, 13] 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 R-3, 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 different 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 finite time t(c), 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 finite time a transition from a normal fluid to one with a condensate component.