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The fragmentation equation with size diffusion: Well posedness and long-term behaviour

Part of: Groups and semigroups of linear operators, their generalizations and applications Special classes of linear operators Partial differential equations Miscellaneous applications of operator theory

Published online by Cambridge University Press:  16 December 2021

PH. LAURENÇOT Open the ORCID record for PH. LAURENÇOT [Opens in a new window]  and
CH. WALKER
PH. LAURENÇOT
Affiliation:
Institut de Mathématiques de Toulouse, UMR 5219, Université de Toulouse, CNRS, F–31062 Toulouse Cedex 9, France email: laurenco@math.univ-toulouse.fr
CH. WALKER
Affiliation:
Leibniz Universität Hannover, Institut für Angewandte Mathematik, Welfengarten 1, D-30167 Hannover, Germany email: walker@ifam.uni-hannover.de
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Abstract

The dynamics of the fragmentation equation with size diffusion is investigated when the size ranges in $(0,\infty)$ . The associated linear operator involves three terms and can be seen as a nonlocal perturbation of a Schrödinger operator. A Miyadera perturbation argument is used to prove that it is the generator of a positive, analytic semigroup on a weighted $L_1$ -space. Moreover, if the overall fragmentation rate does not vanish at infinity, then there is a unique stationary solution with given mass. Assuming further that the overall fragmentation rate diverges to infinity for large sizes implies the immediate compactness of the semigroup and that it eventually stabilizes at an exponential rate to a one-dimensional projection carrying the information of the mass of the initial value.

Keywords

Fragmentationsize diffusionwell posednessconvergencesemigroupperturbation

MSC classification

Primary: 45K05: Integro-partial differential equations 47D06: One-parameter semigroups and linear evolution equations
Secondary: 47B65: Positive operators and order-bounded operators 47N50: Applications in the physical sciences 35B40: Asymptotic behavior of solutions
Type
Papers
Information
European Journal of Applied Mathematics , Volume 33 , Issue 6 , December 2022 , pp. 1083 - 1116
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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