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

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 $γ\in(-\infty,1)$ and satisfy $K(x,1)\sim x^{-a}$ as $x\to 0$, for $a=1-γ$. In particular, for small values of a parameter $ρ>0$ we establish the existence of a positive self-similar solution with finite mass and asymptotics $A(t)x^{-(2+ρ)}$ as $x\to\infty$, with $A(t)\simρt^\fracρ{1-γ}$.