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

One dimensional dissipative Boltzmann equation: measure solutions, cooling rate and self-similar profile