Self-similar solutions with fat tails for a coagulation equation with diagonal kernel

TL;DR

This paper establishes the existence of a family of self-similar solutions with fat tails for a specific class of coagulation equations with a diagonal kernel, expanding understanding of solutions for non-solvable kernels.

Contribution

It demonstrates the existence of second-kind self-similar solutions with power-law tails for a coagulation equation with a non-solvable diagonal kernel, a novel result in the field.

Findings

Existence of self-similar solutions with fat tails for the given kernel.

First example of such solutions for a non-solvable kernel.

Solutions exhibit power-law decay with specific exponent range.

Abstract

We consider self-similar solutions of Smoluchowski's coagulation equation with a diagonal kernel of homogeneity $\gamma < 1$. We show that there exists a family of second-kind self-similar solutions with power-law behavior $x^{-(1+\rho)}$ as $x \to \infty$ with $\rho \in (\gamma,1)$. To our knowledge this is the first example of a non-solvable kernel for which the existence of such a family has been established.