Existence of self-similar profile for a kinetic annihilation model
Abstract
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.