Self-similar gelling solutions for the coagulation equation with diagonal kernel

Abstract

We consider Smoluchowski's coagulation equation in the case of the diagonal kernel with homogeneity $\gamma >1$. In this case the phenomenon of gelation occurs and solutions lose mass at some finite time. The problem of the existence of self-similar solutions involves a free parameter b, and one expects that a physically relevant solution (i.e. nonnegative and with sufficiently fast decay at infinity) exists for a single value of b, depending on the homogeneity γ. We prove this picture rigorously for large values of γ. In the general case, we discuss in detail the behavior of solutions to the self-similar equation as the parameter b changes.