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On self-similarity and stationary problem for fragmentation and coagulation models. (English) Zbl 1130.35025
Summary: 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 Tikhonov 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 \(t\to +\infty\) as the unique self similar solution of the same mass.

MSC:
35F30 Boundary value problems for nonlinear first-order PDEs
35B40 Asymptotic behavior of solutions to PDEs
60G18 Self-similar stochastic processes
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82C99 Time-dependent statistical mechanics (dynamic and nonequilibrium)
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