Asymptotics of self-similar solutions to coagulation equations with product kernel

We consider mass-conserving self-similar solutions for Smoluchowski’s coagulation equation with kernel $K(\xi ,\eta )=(\xi \eta {)}^{\lambda }$ with $\lambda \in (0,1\mathrm{/}2)$. It is known that such self-similar solutions $g(x)$ satisfy that $x^{-1+2\lambda }g(x)$ is bounded above and below as $x\to 0$. In this paper we describe in detail via formal asymptotics the qualitative behavior of a suitably rescaled function $h(x)=h_{\lambda} x^{-1+2\lambda} g(x)$inthelimit$\lambda \to 0$\mathrm{.}Itturnsoutthat$h \sim 1+ C x^{\lambda/2} \cos(\sqrt{\lambda} \log x)$as$x \to 0$\mathrm{.}As$x$becomeslarger$h$developspeaksofheight$1/\lambda$ that are separated by large regions where $h$ is small. Finally, $h$ converges to zero exponentially fast as $x \to \infty$. Our analysis is based on different approximations of a nonlocal operator, that reduces the original equation in certain regimes to a system of ODE.