Exponential tail behaviour of self-similar solutions to Smoluchowski's coagulation equation

TL;DR

This paper proves that self-similar solutions to Smoluchowski's coagulation equation exhibit exponential decay at infinity, under mild conditions on the kernel, and analyzes their asymptotic behavior.

Contribution

It establishes exponential tail decay and asymptotic properties of self-similar solutions for a class of kernels with homogeneity zero, including singular cases.

Findings

Solutions decay exponentially at infinity.

Existence of the limit of (1/x) log(1/f(x)) as x approaches infinity.

Properties of the prefactor when the kernel is bounded below.

Abstract

We consider self-similar solutions with finite mass to Smoluchowski's coagulation equation for rate kernels that have homogeneity zero but are possibly singular such as Smoluchowski's original kernel. We prove pointwise exponential decay of these solutions under rather mild assumptions on the kernel. If the support of the kernel is sufficiently large around the diagonal we also proof that $\lim_{x \to \infty} \frac{1}{x} \log \Big(\frac{1}{f(x)}\Big)$ exists. In addition we prove properties of the prefactor if the kernel is uniformly bounded below.