  This manuscript investigates the following aspects of the one dimensional
dissipative Boltzmann equation associated to variable hard-spheres kernel: (1)
we show the optimal cooling rate of the model by a careful study of the system
satisfied by the solution's moments, (2) give existence and uniqueness of
measure solutions, and (3) prove the existence of a non-trivial self-similar
profile, i.e. homogeneous cooling state, after appropriate scaling of the
equation. The latter issue is based on compactness tools in the set of Borel
measures. More specifically, we apply a dynamical fixed point theorem on a
suitable stable set, for the model dynamics, of Borel measures.
