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

TL;DR

This paper establishes the existence of a family of self-similar solutions with time-dependent tails for the coagulation equation with homogeneous kernels of degree one, identifying critical parameters and decay behaviors.

Contribution

It proves the existence of self-similar solutions with time-dependent tails for a class of coagulation kernels, including the identification of a critical parameter range.

Findings

Existence of a one-parameter family of solutions with specific decay rates.

Identification of a critical parameter er for solution existence.

Non-existence of measure solutions for large tail parameters.

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 kernels $K(x,y)$ which are homogeneous of degree one and satisfy $K(x,1)\to k_0>0$ as $x\to 0$. In particular, we establish the existence of a critical $\rho_*>0$ with the property that for all $\rho\in(0,\rho_*)$ there is a positive and differentiable self-similar solution with finite mass $M$ and decay $A(t)x^{-(2+\rho)}$ as $x\to\infty$, with $A(t)=e^{M(1+\rho)t}$. Furthermore, we show that (weak) self-similar solutions in the class of positive measures cannot exist for large values of the parameter $\rho$.