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 as x→ 0. In particular, we establish the existence of a critical ρ ∗ > 0 with the property that for all ρ∈ (0 , ρ ∗ ) there is a positive and differentiable self-similar solution with finite mass M and decay A(t) x -(2+ρ) as x→ ∞, with A(t) = e M(1+ρ)t . Furthermore, we show that (weak) self-similar solutions in the class of positive measures cannot exist for large values of the parameter ρ.