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Zbl 1130.35025
Escobedo, M.; Mischler, S.; Rodriguez Ricard, M.
On self-similarity and stationary problem for fragmentation and coagulation models. (English)
[J] Ann. Inst. Henri Poincaré, Anal. Non Linéaire 22, No. 1, 99-125 (2005). ISSN 0294-1449

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 2000:: *35F30 Boundary value problems for first order nonlinear PDE
35B40 Asymptotic behavior of solutions of PDE
60G18 Self-similar processes
60K35 Interacting random processes
82C99 Time-dependent statistical mechanics

Keywords: equilibrium; no detailed balance condition; Poincaré-Bendixson's Theory; Tikhonov fixed point theorem; self-similar solutions; uniqueness; existence; convergence to self-similarity

Cited in: Zbl 1154.82024 Zbl 1081.35122

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