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We consider solutions to the initial value problem for the spatially homogeneous Boltzmann equation for pseudo-Maxwell molecules and show uniform in time propagation of upper Maxwellians bounds if the initial distribution function is bounded by a given Maxwellian. First we prove the corresponding integral estimate and then transform it to the desired local estimate. We remark that propagation of such upper Maxwellian bounds were obtained by Gamba, Panferov and Villani for the case of hard spheres and hard potentials with angular cut-off. That manuscript introduced the main ideas and tools needed to prove such local estimates on the basis of similar integral estimates. The case of pseudo-Maxwell molecules needs, however, a special consideration performed in the present paper.

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