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 ∈(0,1/2). It is known that such self-similar solutions g ( x ) satisfy that x −1+2 λ 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 λ x −1+2 λ g ( x ) in the limit λ →0. It turns out that as x →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 →∞. Our analysis is based on different approximations of a nonlocal operator, that reduces the original equation in certain regimes to a system of ODE.