# Stability and uniqueness of self-similar profiles in $L^1$ spaces for perturbations of the constant kernel in Smoluchowski's coagulation equation

## TL;DR

This paper demonstrates the stability and uniqueness of self-similar profiles in Smoluchowski's coagulation equation when the kernel is a small perturbation of the constant kernel, improving existing results with relaxed conditions.

## Contribution

It establishes stability and improves the uniqueness results for self-similar profiles under less restrictive perturbation conditions in the coagulation kernel.

## Key findings

- Self-similar profiles are stable under perturbations in weighted L^1 spaces.
- The paper relaxes conditions on kernel perturbations compared to previous results.
- The proof of uniqueness is shortened and the conditions are significantly relaxed.

## Abstract

In this work, we consider self-similar profiles for Smoluchowski's coagulation equation for kernels which are possibly unbounded perturbations of the constant one. For this model, we show that the self-similar solutions for the perturbed kernel are close in weighted $L^1$ spaces to the profile of the unperturbed equation, i.e. the profiles are stable with respect to the perturbation. Additionally, we revisit the problem of uniqueness for these coagulation kernels. In fact, we will improve a corresponding result (see arXiv:1510.03361 and ref. [22]) by relaxing the conditions on the perturbation significantly while at the same time the corresponding proof can also be notably shortened.