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

### Marco Bonacini

University of Bonn, Institute for Applied Mathematics, Endenicher Allee 60, 53115 Bonn, Germany### Barbara Niethammer

University of Bonn, Institute for Applied Mathematics, Endenicher Allee 60, 53115 Bonn, Germany### Juan J.L. Velázquez

University of Bonn, Institute for Applied Mathematics, Endenicher Allee 60, 53115 Bonn, Germany

## 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.

## Cite this article

Marco Bonacini, Barbara Niethammer, Juan J.L. Velázquez, Self-similar gelling solutions for the coagulation equation with diagonal kernel. Ann. Inst. H. Poincaré Anal. Non Linéaire 36 (2019), no. 3, pp. 705–744

DOI 10.1016/J.ANIHPC.2018.09.001