# American Institute of Mathematical Sciences

doi: 10.3934/krm.2021032
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## The fragmentation equation with size diffusion: Small and large size behavior of stationary solutions

 1 Institut de Mathématiques de Toulouse, UMR 5219, Université de Toulouse, CNRS, F–31062 Toulouse Cedex 9, France 2 Leibniz Universität Hannover, Institut für Angewandte Mathematik, Welfengarten 1, D–30167 Hannover, Germany

Received  May 2021 Revised  September 2021 Early access November 2021

Fund Project: The first author is partially supported by Deutscher Akademischer Austauschdienst funding programme Research Stays for University Academics and Scientists, 2021 (57552334)

The small and large size behavior of stationary solutions to the fragmentation equation with size diffusion is investigated. It is shown that these solutions behave like stretched exponentials for large sizes, the exponent in the exponential being solely given by the behavior of the overall fragmentation rate at infinity. In contrast, the small size behavior is partially governed by the daughter fragmentation distribution and is at most linear, with possibly non-algebraic behavior. Explicit solutions are also provided for particular fragmentation coefficients.

Citation: Philippe Laurençot, Christoph Walker. The fragmentation equation with size diffusion: Small and large size behavior of stationary solutions. Kinetic & Related Models, doi: 10.3934/krm.2021032
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