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

We prove the existence of a stationary solution of any given mass to the coagulationfragmentation 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.