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

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