Self-similar solutions with fat tails for a coagulation equation with diagonal kernel

We consider self-similar solutions of Smoluchowski’s coagulation equation with a diagonal kernel of homogeneity $\gamma <1$. We show that there exists a family of second-kind self-similar solutions with power-law behavior $x^{-(1+\rho )}$ as $x\to \mathrm{\infty }$ with $\rho \in (\gamma ,1)$. To our knowledge this is the first example of a non-solvable kernel for which the existence of such a family has been established.