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A class of inverse problems for evolution equations with the Riemann‐Liouville derivative in the sectorial case
Author(s) -
Fedorov Vladimir E.,
Nagumanova Anna V.,
Avilovich Anna S.
Publication year - 2021
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.6794
Subject(s) - mathematics , mathematical analysis , bounded function , inverse problem , operator (biology) , norm (philosophy) , inverse , partial differential equation , biochemistry , chemistry , geometry , repressor , political science , transcription factor , law , gene
We investigate the unique solvability of a class of linear inverse problems with a time‐independent unknown coefficient in an evolution equation in Banach space, which is resolved with respect to the fractional Riemann‐Liouville derivative. We assume that the operator in the right‐hand side of the equation generates a family of resolving operators for the corresponding homogeneous equation, which is exponentially bounded and analytic in a sector containing the positive semiaxis. It is shown that the inverse problem is well‐posed with respect to the graph norm of the generating operator only. A well‐posedness criterion is found. The obtained abstract results are applied to the unique solvability study of an inverse problem for a class of time‐fractional partial differential equations. An example, in particular, shows that in the case of an unbounded generating operator, the inverse problem can be ill‐posed with respect to the norm of the whole space.