Time Asymptotics for a Critical Case in Fragmentation and Growth-Fragmentation Equations

Marie Doumic (1, 2) , Miguel Escobedo (3)

1 MAMBA - Modelling and Analysis for Medical and Biological Applications
2 LJLL - Laboratoire Jacques-Louis Lions
3 Departamento de Matemáticas [Bilbao]

Fragmentation and growth-fragmentation equations is a family of problems with varied and wide applications. This paper is devoted to description of the long time 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.

Mots clés: Structured populations; growth-fragmentation equations; cell division; self-similarity; long-time behaviour; fragmentation models; Mellin transform; Asymptotic Analysis

Domaines: Mathématiques [math]; Equations aux dérivées partielles [math.AP]; Analyse numérique [cs.NA]

Marie Doumic, Miguel Escobedo. Time Asymptotics for a Critical Case in Fragmentation and Growth-Fragmentation Equations. 2015. ⟨hal-01080361v2⟩