Self-similar solutions to coagulation equations with time-dependent tails: The case of homogeneity smaller than one

We prove the existence of a one-parameter family of self-similar solutions with time-dependent tails for Smoluchowski’s coagulation equation, for a class of rate kernels K(x,y) which are homogeneous of degree γ∈(−∞,1) and satisfy K(x,1)∼x−a as x→0, for a = 1−γ. In particular, for small values of a parameter ρ>0 we establish the existence of a positive self-similar solution with finite mass and asymptotics A(t)x−(2+ρ) as x→∞, with A(t) ~ ρtρ/1-γ.