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 I center dot) (lambda) with lambda a(0,1/2). 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 -> 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) in the limit lambda -> 0. It turns out that h similar to 1 + Cx(lambda/2) cos(root lambda log x) as x -> 0. As x becomes larger h develops peaks of height 1/lambda that are separated by large regions where h is small. Finally, h converges to zero exponentially fast as x -> a. Our analysis is based on different approximations of a nonlocal operator, that reduces the original equation in certain regimes to a system of ODE.