Nonunique solutions of kinetic equations

Two very hard particle models are solved and the nonuniqueness of the initial value problem for these (model) kinetic equations is explicitly demonstrated, when distribution functions decaying sufficiently slowly are permitted. The intimate connection between nonuniqueness and violation of conservation laws is made evident. The associated eigenvalue problems are solved. Finally, the general implications of these results for kinetic equations with transition rates that are increasing functions of the state variable, are stated in the form of a number of conjectures. They affect the solution of the Boltzmann equation for realistic intermolecular interactions when the collision rategI(g, ) is an increasing function of the relative velocityg.