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 gamma is an element of (-infinity, 1) and satisfy K(x,1) similar to x(-a) as x -> 0, for a = 1 - gamma. In particular, for small values of a parameter rho > 0 we establish the existence of a positive self-similar solution with finite mass and asymptotics A(t)x(-(2+rho)) as x -> infinity, with A(t) similar to rho t(1-gamma/rho).