Existence of self-similar profile for a kinetic annihilation model

We show the existence of a self-similar solution for a modified Boltzmann equation describing probabilistic ballistic annihilation. Such a model describes a system of hard spheres such that, whenever two particles meet, they either annihilate with probability α∈(0,1) or they undergo an elastic collision with probability 1−α. For such a model, the number of particles, the linear momentum and the kinetic energy are not conserved. We show that, for α smaller than some explicit threshold value α⁎, a self-similar solution exists.