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.
Highlights
Our results show in particular the strong dependence of this asymptotic behavior with respect to the initial data
Fragmentation and growth-fragmentation equations is a family of problems with varied and wide applications: phase transition, aerosols, polymerization processes, bacterial growth, systems with a chemostat etc. [6, 8, 10, 16, 17]
This paper is devoted to the description of the long time asymptotic behavior of two critical cases of these equations that have been left open in the previous literature
Summary
Fragmentation and growth-fragmentation equations is a family of problems with varied and wide applications: phase transition, aerosols, polymerization processes, bacterial growth, systems with a chemostat etc. [6, 8, 10, 16, 17]. Fragmentation and growth-fragmentation equations is a family of problems with varied and wide applications: phase transition, aerosols, polymerization processes, bacterial growth, systems with a chemostat etc. That explains the continuing interest they meet. This paper is devoted to the description of the long time asymptotic behavior of two critical cases of these equations that have been left open in the previous literature. Primary: 35B40, 35Q92; Secondary: 45K05, 92D25, 92C37, 82D60. Structured populations; growth-fragmentation equations; cell division; self-similarity; long-time asymptotics; rate of convergence
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