Probabilistic aspects of critical growth-fragmentation equations

@article{Bertoin2016ProbabilisticAO,
  title={Probabilistic aspects of critical growth-fragmentation equations},
  author={Jean Bertoin and Alexander R. Watson},
  journal={Advances in Applied Probability},
  year={2016},
  volume={48},
  pages={37 - 61}
}
Abstract The self-similar growth-fragmentation equation describes the evolution of a medium in which particles grow and divide as time proceeds, with the growth and splitting of each particle depending only upon its size. The critical case of the equation, in which the growth and division rates balance one another, was considered in Doumic and Escobedo (2015) for the homogeneous case where the rates do not depend on the particle size. Here, we study the general self-similar case, using a… 
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Time Asymptotics for a Critical Case in Fragmentation and Growth-Fragmentation Equations
TLDR
Using the Mellin transform of the equation, the long time behavior of the solutions of the fragmentation and growth-fragmentation equations is determined and the results show the strong dependence of this asymptotic behavior with respect to the initial data.
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