Uniform in time lower bound for solutions to a quantum Boltzmann   equation of bosons
  We consider the quantum Boltzmann equation, which describes the growth of the condensate, or in other words, models the interaction between excited atoms and a condensate. In this work, the full form of Bogoliubov dispersion law is considered, which leads to a detailed study of surface integrals inside the collision operator on energy manifolds. We prove that positive radial solutions of the quantum Boltzmann equation are bounded from below by a Gaussian, uniformly in time. 
Toan T. Nguyen, Minh-Binh Tran
math.AP math-ph math.MP