Long–time asymptotics of kinetic models of granular flows

We analyze the long-time asymptotics of certain one-dimensional kinetic models of granular flows, which have been recently introduced in [22] in connection with the quasi elastic limit of a model Boltzmann equation with dissipative collisions and variable coefficient of restitution. These nonlinear equations, classified as nonlinear friction equations, split naturally into two classes, depending whether their similarity solutions (homogeneous cooling state) extinguish or not in finite time. For both classes, we show uniqueness of the solution by proving decay to zero in the Vasershtein metric of any two solutions with the same mass and mean velocity. Furthermore, if the similarity solution extinguishes in finite time, we prove that any other solution with initially bounded support extinguishes in finite time, by computing explicitly upper bounds for the life-time of the solution in terms of the length of the support.