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

Philippe Laurencot (1) , Henry van Roessel (2)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 Dpt of Mathematical and Statistical Sciences [Alberta]

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.

Mots clés: coagulation; singular fragmentation; mass conservation; self-similarity; Laplace transform; characteristics

Domaines: Equations aux dérivées partielles [math.AP]

Philippe Laurencot, Henry van Roessel. Absence of gelation and self-similar behavior for a coagulation-fragmentation equation. SIAM Journal on Mathematical Analysis, 2015, 47, pp.2355--2374. ⟨hal-01018909⟩