Time Asymptotics for a Critical Case in Fragmentation and Growth-Fragmentation Equations
Résumé
Fragmentation and growth-fragmentation equations is a family of problems with varied and wide applications. This paper is devoted to the time asymptotics of critical cases of these equations, namely when the division rate is constant and the growth rate is linear or is zero. The study of these cases may be reduced to the study of the following fragmentation equation: ∂ ∂t u(t, x) + u(t, x) = ∞ x k 0 (x y)u(t, y)dy. It is well-known that the solution of this equation converges to a Dirac mass at zero. By means of the Mellin transform of the equation, we determine the zone on the space (t, x) where the mass concentrates and establish estimates on the rates of convergence, which happen to be either exponential or polynomial according to the position on curves (t, x(t)).
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Origine : Fichiers produits par l'(les) auteur(s)