The dissipative linear Boltzmann equation for hard spheres

We prove the existence and uniqueness of an equilibrium state with unit mass to the dissipative linear Boltzmann equation with hard–spheres collision kernel describing inelastic interactions of a gas particles with a fixed background. The equilibrium state is a universal Maxwellian distribution function with the same velocity as field particles and with a non–zero temperature lower than the background one, which depends on the details of the binary collision. Thanks to the H-theorem we then prove strong convergence of the solution to the Boltzmann equation towards the equilibrium.