  We consider an approximation of the linearised equation of the homogeneous
Boltzmann equation that describes the distribution of quasiparticles in a
dilute gas of bosons at low temperature. The corresponding collision frequency
is neither bounded from below nor from above. We prove the existence and
uniqueness of solutions satisfying the conservation of energy. We show that
these solutions converge to the corresponding stationary state, at an algebraic
rate as time tends to infinity.
