On global in time self-similar solutions of Smoluchowski equation with multiplicative kernel

We study the similarity solutions (SS) of Smoluchowski coagulation equation with multiplicative kernel $K(x,y)=(xy)^{s}$ for $s<\frac {1}{2}$. When $s<0$ , the SS consists of three regions with distinct asymptotic behaviours. The appropriate matching yields a global description of the solution consisting of a Gamma distribution tail, an intermediate region described by a lognormal distribution and a region of very fast decay of the solutions to zero near the origin. When $s\in \left ( 0,\frac {1}{2}\right ) $, the SS is unbounded at the origin. It also presents three regions: a Gamma distribution tail, an intermediate region of power-like (or Pareto distribution) decay and the region close to the origin where a singularity occurs. Finally, full numerical simulations of Smoluchowski equation serve to verify our theoretical results and show the convergence of solutions to the selfsimilar regime.