Solutions of the Boltzmann equation with infinite energy: Existence, stability and long time behavior

This paper is devoted to studying a class of solutions to the nonlinear Boltzmann equation having infinite kinetic energy, these solutions have an upper Maxwellian bound with infinite kinetic energy. Firstly, the existence and stability of this kind of solutions are established near vacuum. Secondly, it is proved that this kind of solutions are stable for any initial data, as a consequence, the Boltzmann equation has at most one solution with infinite kinetic energy. Finally, the long time behavior of the solutions is also established.