Self-Similar Solutions to Coagulation Equations with Time-Dependent Tails: The Case of Homogeneity 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 kernels K (x, y) which are homogeneous of degree one and satisfy K (x, 1) -> k(0) > 0 as x -> 0. In particular, we establish the existence of a critical rho(*) > 0 with the property that for all rho is an element of(0, rho(*)) there is a positive and differentiable self-similar solution with finite mass M and decay A(t)x(-(2+rho)) as x -> infinity, with A(t) = e(M(1+rho)t). Furthermore, we show that (weak) self-similar solutions in the class of positive measures cannot exist for large values of the parameter rho.