Long time strong convergence to bose-einstein distribution for low temperature

We study the long time behavior of measure-valued isotropic solutions F-t of the Boltzmann equation for Bose-Einstein particles for low temperature. The global in time existence of such solutions F-t that converge at least semi-strongly to equilibrium (the Bose-Einstein distribution) has been proven in previous work and it has been known that the long time strong convergence to equilibrium is equivalent to the long time convergence to the Bose-Einstein condensation. Here we show that if such a solution F-t as a family of Borel measures satisfies a uniform double-size condition (which is also necessary for the strong convergence), then F-t converges strongly to equilibrium as t tends to infinity. We also propose a new condition on the initial datum F-0 such that a corresponding solution F-t converges strongly to equilibrium.