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

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...

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Bibliographic Details
Published inEuropean journal of applied mathematics Vol. 33; no. 6; pp. 1083 - 1116
Main Authors LAURENÇOT, PH, WALKER, CH
Format Journal Article
LanguageEnglish
Published Cambridge Cambridge University Press 01.12.2022
Cambridge University Press (CUP)
Subjects
Analysis of PDEs
Applied mathematics
Cauchy problems
Fragmentation
Infinity
Investigations
Linear operators
Mathematics
Operators (mathematics)
Perturbation
Semigroups
semigroup
well-posedness
convergence
perturbation
fragmentation
size diffusion
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ISSN0956-7925
1469-4425
DOI10.1017/S0956792521000346

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Summary: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.
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ISSN:0956-7925
1469-4425
DOI:10.1017/S0956792521000346