期刊
ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE
卷 35, 期 7, 页码 1847-1884出版社
ELSEVIER SCIENCE BV
DOI: 10.1016/j.anihpc.2018.03.004
关键词
Non-linear inverse problem; Size-structured partial differential equation; Fragmentation equation; Mellin transform; Functional equation
资金
- DGES [MTM2014-52347-C2-1-R]
- Basque Government [IT641-13]
- ERC Starting Grant SKIPPERAD [306321]
We consider the fragmentation equation partial derivative f/partial derivative t(t, x) = -B(x)f(t, x) + integral(y=infinity )(y=x)k(y, x)B(y)f(t, y)dy, and address the question of estimating the fragmentation parameters - i.e. the division rate B(x) and the fragmentation kernel k(y, x) - from measurements of the size distribution f (t, .) at various times. This is a natural question for any application where the sizes of the particles are measured experimentally whereas the fragmentation rates are unknown, see for instance Xue and Radford (2013) [26] for amyloid fibril breakage. Under the assumption of a polynomial division rate B(x) = alpha x(gamma) and a self-similar fragmentation kernel k(y, x) = 1/yk(0)(x/y), we use the asymptotic behavior proved in Escobedo et al. (2004) [11] to obtain uniqueness of the triplet (alpha, gamma, k(0)) and a representation formula for k(0). To invert this formula, one of the delicate points is to prove that the Mellin transform of the asymptotic profile never vanishes, what we do through the use of the Cauchy integral. (C) 2018 Elsevier Masson SAS. All rights reserved.
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