# Uniqueness of fat-tailed self-similar profiles to Smoluchowski's coagulation equation for a perturbation of the constant kernel

## TL;DR

This paper proves the uniqueness of fat-tailed self-similar profiles for a perturbed Smoluchowski coagulation equation with a kernel close to constant, under certain regularity and small perturbation conditions.

## Contribution

It establishes the uniqueness of self-similar solutions with algebraic decay for a class of perturbed coagulation kernels, extending previous results to more singular and general kernels.

## Key findings

- Uniqueness holds for small perturbations of the constant kernel.
- Self-similar profiles exhibit algebraic decay at infinity.
- Results apply under specific regularity assumptions on the perturbation W.

## Abstract

This article is concerned with the question of uniqueness of self-similar profiles for Smoluchowski's coagulation equation which exhibit algebraic decay (fat tails) at infinity. More precisely, we consider a rate kernel $K$ which can be written as $K=2+\varepsilon W$. The perturbation is assumed to have homogeneity zero and might also be singular both at zero and at infinity. Under further regularity assumptions on $W$, we will show that for sufficiently small $\varepsilon$ there exists, up to normalisation of the tail behaviour at infinity, at most one self-similar profile.