  We consider the Boltzmann equations for mixtures ofMaxwell gases. It is shown
that in certain limiting case the equations admit self-similar solutions that
can be constructed in explicit form. More precisely, the solutions have simple
explicit integral representations. The most interesting solutions have finite
energy and power like tails. This shows that power like tails can appear not
just for granular particles (Maxwell models are far from reality in this case),
but also in the system of particles interacting in accordance with laws of
classical mechanics. In addition, non-existence of positive self-similar
solutions with finite moments of any order is proven for a wide class of
Maxwell models.
