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We prove a local-in-time existence and uniqueness theorem for a smoothclassical solution to the spatially homogeneous Boltzmann equation with cutoffsoft potentials. Our proof is based on a series of bilinear estimates for theintegrability and Sobolev regularity of the associated collision operator.While the global-in-time existence is left inconclusive, we give a lower boundof the maximal time of existence and a necessary condition for finite timeextinction of existence.

Local Existence for the Spatially Homogeneous Boltzmann Equation with Soft Potentials