# American Institute of Mathematical Sciences

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June  2016, 9(2): 251-297. doi: 10.3934/krm.2016.9.251

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

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

Received  January 2015 Revised  August 2015 Published  March 2016

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.
Citation: Marie Doumic, Miguel Escobedo. Time asymptotics for a critical case in fragmentation and growth-fragmentation equations. Kinetic & Related Models, 2016, 9 (2) : 251-297. doi: 10.3934/krm.2016.9.251
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