Self-similar gelling solutions for the coagulation equation with diagonal kernel

TL;DR

This paper investigates self-similar solutions to Smoluchowski's coagulation equation with a diagonal kernel, demonstrating existence for large homogeneity values and analyzing solution behavior as parameters vary.

Contribution

It rigorously proves the existence of physically relevant self-similar solutions for large homogeneity in the diagonal kernel case and explores their behavior across parameter ranges.

Findings

Existence of self-similar solutions for large γ

Characterization of solution behavior as parameter b varies

Analysis of gelation phenomena in the model

Abstract

We consider Smoluchowski's coagulation equation in the case of the diagonal kernel with homogeneity $\gamma>1$. In this case the phenomenon of gelation occurs and solutions lose mass at some finite time. The problem of the existence of self-similar solutions involves a free parameter $b$, and one expects that a physically relevant solution (i.e. nonnegative and with sufficiently fast decay at infinity) exists for a single value of $b$, depending on the homogeneity $\gamma$. We prove this picture rigorously for large values of $\gamma$. In the general case, we discuss in detail the behaviour of solutions to the self-similar equation as the parameter $b$ changes.