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 $α\in (0,1)$ 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.

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