Details
| Original language | English |
|---|---|
| Pages (from-to) | 1083-1116 |
| Number of pages | 34 |
| Journal | European Journal of Applied Mathematics |
| Volume | 33 |
| Issue number | 6 |
| Early online date | 16 Dec 2021 |
| Publication status | Published - Dec 2022 |
Abstract
The dynamics of the fragmentation equation with size diffusion is investigated when the size ranges in <![CDATA[ $(0,\infty)$ ]]>. 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 <![CDATA[ $L_1$ ]]> -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
- convergence, Fragmentation, perturbation, semigroup, size diffusion, well posedness
ASJC Scopus subject areas
- Mathematics(all)
- Applied Mathematics
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In: European Journal of Applied Mathematics, Vol. 33, No. 6, 12.2022, p. 1083-1116.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - The fragmentation equation with size diffusion
T2 - Well posedness and long-term behaviour
AU - Laurençot, Philippe
AU - Walker, Christoph
N1 - Publisher Copyright: © The Author(s), 2021. Published by Cambridge University Press.
PY - 2022/12
Y1 - 2022/12
N2 - The dynamics of the fragmentation equation with size diffusion is investigated when the size ranges in <![CDATA[ $(0,\infty)$ ]]>. 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 <![CDATA[ $L_1$ ]]> -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.
AB - The dynamics of the fragmentation equation with size diffusion is investigated when the size ranges in <![CDATA[ $(0,\infty)$ ]]>. 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 <![CDATA[ $L_1$ ]]> -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.
KW - convergence
KW - Fragmentation
KW - perturbation
KW - semigroup
KW - size diffusion
KW - well posedness
UR - http://www.scopus.com/inward/record.url?scp=85121384181&partnerID=8YFLogxK
U2 - 10.48550/arXiv.2104.14798
DO - 10.48550/arXiv.2104.14798
M3 - Article
VL - 33
SP - 1083
EP - 1116
JO - European Journal of Applied Mathematics
JF - European Journal of Applied Mathematics
SN - 0956-7925
IS - 6
ER -