On self-similarity and stationary problem for fragmentation and coagulation models

Abstract

We prove the existence of a stationary solution of any given mass to the coagulation–fragmentation equation without assuming a detailed balance condition, but assuming instead that aggregation dominates fragmentation for small particles while fragmentation predominates for large particles. We also show the existence of a self-similar solution of any given mass to the coagulation equation and to the fragmentation equation for kernels satisfying a scaling property. These results are obtained, following the theory of Poincaré–Bendixson on dynamical systems, by applying the Tykonov fixed point theorem on the semigroup generated by the equation or by the associated equation written in “self-similar variables”. Moreover, we show that the solutions to the fragmentation equation with initial data of a given mass behaves, as t→+∞, as the unique self similar solution of the same mass.