Absence of gelation and self-similar behavior for a coagulation-fragmentation equation

TL;DR

This paper demonstrates that in a coagulation-fragmentation system with specific kernels, fragmentation prevents gelation, ensures mass conservation, and leads to self-similar long-term behavior, with finite second moments at all positive times.

Contribution

It proves the absence of gelation and constructs mass-conserving solutions for a coagulation-fragmentation equation with singular fragmentation.

Findings

Fragmentation prevents gelation in the studied model.

Mass-conserving self-similar solutions exist and describe long-term behavior.

The second moment remains finite for all positive times.

Abstract

The dynamics of a coagulation-fragmentation equation with multiplicative coagulation kernel and critical singular fragmentation is studied. In contrast to the coagulation equation, it is proved that fragmentation prevents the occurrence of the gelation phenomenon and a mass-conserving solution is constructed. The large time behavior of this solution is shown to be described by a self-similar solution. In addition, the second moment is finite for positive times whatever its initial value. The proof relies on the Laplace transform which maps the original equation to a first-order nonlinear hyperbolic equation with a singular source term. A precise study of this equation is then performed with the method of characteristics.