Uniqueness of the self-similar profile for a kinetic annihilation model

TL;DR

This paper proves the existence and uniqueness of a self-similar profile for a probabilistic ballistic annihilation model described by a modified Boltzmann equation, expanding understanding of the system's long-term behavior.

Contribution

It establishes the uniqueness of the self-similar profile for the model for certain annihilation probabilities, complementing previous existence results.

Findings

Existence of self-similar profile for small enough annihilation probability.

Uniqueness of the self-similar profile for a range of probabilities.

Explicit threshold for the annihilation probability ensuring uniqueness.

Abstract

We show the existence of a self-similar solution for a We prove the uniqueness of the self-similar profile 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 $\alpha \in (0,1)$ or they undergo an elastic collision with probability $1-\alpha$. The existence of a self-similar profile for $\alpha$ smaller than an explicit threshold value $\underline{\alpha}_1$ has been obtained in our previous contribution (J. Differential Equations, 254, 3023--3080, 2013). . We complement here our analysis of such a model by showing that, for some $\alpha^{\sharp} $ explicit, the self-similar profile is unique for $\alpha \in (0,\alpha^{\sharp})$.