Self-similar asymptotic behavior for the solutions of a linear coagulation equation

TL;DR

This paper studies the long-term behavior of solutions to a linear coagulation equation modeling particle interactions, revealing a unique stable self-similar profile for certain parameter ranges.

Contribution

It establishes the well-posedness and asymptotic stability of self-similar solutions for the linear Smoluchowski equation within a specific range of the power-law exponent.

Findings

Existence of a unique self-similar profile for 5/3<σ<2.

Proof of asymptotic stability of the self-similar solution.

Well-posedness of the equation in the specified parameter range.

Abstract

In this paper we consider the long time asymptotics of a linear version of the Smoluchowski equation which describes the evolution of a tagged particle moving at constant speed in a random distribution of fixed particles. The volumes $v$ of the particles are independently distributed according to a probability distribution which decays asymptotically as a power law $v^{-\sigma}$. The validity of the equation has been rigorously proved in \cite{NoV} for values of the exponent $\sigma>3$. The solutions of this equation display a rich structure of different asymptotic behaviours according to the different values of the exponent $\sigma$. Here we show that for $\frac{5}{3}<\sigma<2$ the linear Smoluchowski equation is well posed and that there exists a unique self-similar profile which is asymptotically stable.