Time asymptotics for a critical case in fragmentation and growth-fragmentation equations

Marie  Doumic; Miguel  Escobedo

doi:10.3934/krm.2016.9.251

Article Contents

2016, Volume 9, Issue 2: 251-297. Doi: 10.3934/krm.2016.9.251

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Time asymptotics for a critical case in fragmentation and growth-fragmentation equations

Marie Doumic^1, and
Miguel Escobedo^2,

1.
Sorbonne Universités, Inria, UPMC Univ Paris 06, Lab. J.-L. Lions, UMR CNRS 7598, 75005 Paris
2.
Departamento de Matemáticas, Universidad del País Vasco, (UPV/EHU), Apartado 644, E-48080 Bilbao

Received: December 31, 2014

Revised: July 31, 2015

Published: May 2016

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Abstract

Fragmentation and growth-fragmentation equations is a family of problems with varied and wide applications. This paper is devoted to the description of the long-time asymptotics of two critical cases of these equations, when the division rate is constant and the growth rate is linear or zero. The study of these cases may be reduced to the study of the following fragmentation equation: $$\frac{\partial }{\partial t} u(t,x) + u(t,x) = \int\limits_x^\infty k_0 (\frac{x}{y}) u(t,y) dy.$$ Using the Mellin transform of the equation, we determine the long-time behavior of the solutions. Our results show in particular the strong dependence of this asymptotic behavior with respect to the initial data.

Keywords:

Mathematics Subject Classification: Primary: 35B40, 35Q92; Secondary: 45K05, 92D25, 92C37, 82D60.

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