# Self-similar solutions to coagulation equations with time-dependent tails: the case of homogeneity smaller than one

## TL;DR

This paper establishes the existence of a family of self-similar solutions with time-dependent tails for Smoluchowski's coagulation equation, focusing on kernels with homogeneity less than one and specific tail behaviors.

## Contribution

It proves the existence of self-similar solutions with time-dependent tails for a class of coagulation kernels with homogeneity less than one, including finite mass solutions with specific asymptotics.

## Key findings

- Existence of self-similar solutions with time-dependent tails.
- Asymptotic behavior of solutions as x→∞.
- Finite mass solutions with specific tail decay.

## Abstract

We prove the existence of a one-parameter family of self-similar solutions with time-dependent tails for Smoluchowski's coagulation equation, for a class of rate kernels $K(x,y)$ which are homogeneous of degree $\gamma\in(-\infty,1)$ and satisfy $K(x,1)\sim x^{-a}$ as $x\to 0$, for $a=1-\gamma$. In particular, for small values of a parameter $\rho>0$ we establish the existence of a positive self-similar solution with finite mass and asymptotics $A(t)x^{-(2+\rho)}$ as $x\to\infty$, with $A(t)\sim\rho t^\frac{\rho}{1-\gamma}$.