  We study a quantum Boltzmann-Condensation system that describes the evolution
of the interaction between a well formed Bose-Einstein condensate and the
quasi-particles cloud. The kinetic model is valid for a dilute regime at which
the temperature of the gas is very low compared to the Bose-Einstein
condensation critical temperature. In particular, our system couples the
density of the condensate from a Gross-Pitaevskii type equation to the kinetic
equation through the dispersion relation in the kinetic model and the
corresponding transition probability rate from pre to post collision momentum
states. We rigorously show the following three properties (1) the
well-posedness of the Cauchy problem for the system in the case of a radially
symmetric initial configuration, (2) find qualitative properties of the
solution such as instantaneous creation of exponential tails and, (3) prove the
uniform condensate stability related to the initial mass ratio between
condensed particles and quasi-particles. The stability result from (3) leads to
global in time existence of the initial value problem for the quantum
Boltzmann-Condensation system.
