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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 $K(ξ,η)= (ξη)^λ$ with $λ\in (0,1/2)$. It is known that such self-similar solutions $g(x)$ satisfy that $x^{-1+2λ} g(x)$ 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 $h(x)=h_λ x^{-1+2λ} g(x)$ in the limit $λ\to 0$. It turns out that $h \sim 1+ C x^{λ/2} \cos(\sqrtλ \log x)$ as $x \to 0$. As $x$ becomes larger $h$ develops peaks of height $1/λ$ 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.