We consider energy dissipative Boltzmann transport Equation (BTE) type. Examples are inelastic BTE modeling granular gases or Elastic BTE for
gas mixtures. We study self-similar solutions (Homogeneous cooling states) and classify the high energy tails decay depending on the collisional cross section. In the case of wide class of Maxwell models we show all self-similar
solutions have finite energy and power like tails, and so they can not
have finite moments for all orders. We study self-similar asymptotics and find some explicit solutions to some special choice of parameters. These explicit solutions are fundamental markers for the design of numerical approximations to non-equilibrium collisional systems.
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