Asymptotics of Self-similar Solutions to Coagulation Equations with Product Kernel - McLeod, J; Niethammer, B; Velazquez, J (2011)
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+2λg(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 λx -1+2λg(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. © 2011 Springer Science+Business Media, LLC.