Local properties of self-similar solutions to Smoluchowski’s coagulation equation with sum kernels

Abstract

The regularity of the scaling profiles ψ to Smoluchowski’s coagulation equation is studied when the coagulation kernel K is given by K(x, y) = xλ + yλ with λ∈ (0, 1). More precisely, ψ is C1-smooth on (0,∞) and decays exponentially fast for large x. Furthermore, the singular behaviour of ψ(x) as x → 0 is identified, thus giving a rigorous proof of physical conjectures.