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Inverse Problems for a Class of Degenerate Evolution Equations with Riemann - Liouville Derivative

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Abstract

Unique solvability and well-posedness issues are studied for linear inverse problems with a constant unknown parameter to fractional order differential equations with Riemann - Liouvlle derivative in Banach spaces. Firstly, well-posedness criteria for the inverse problem with the Cauchy type initial conditions to the differential equation in a Banach space that solved with respect to the fractional derivative is obtained. This result is applied to search of sufficient conditions for the unique solution existence of the inverse problem for equation with linear degenerate operator at the Riemann - Liouville fractional derivative. It is shown that the presence of the matching conditions for the data of the problem excludes the possibility of the well-posedness consideration for the degenerate inverse problem with the Cauchy type condition. But for the inverse problem with the Showalter - Sidorov type conditions it is found the criteria of the well-posedness. Abstract results are used to the search of conditions of the unique solvability for an inverse problem to a class of partial differential equations of time-fractional order with polynomials of elliptic differential operators with respect to the spatial variables.

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Authors and Affiliations

  1. Mathematical Analysis Dept., Chelyabinsk State University, Kashirin Brothers St., 129, Chelyabinsk, 454001, Russia

    Vladimir E. Fedorov

  2. Laboratory of Functional Materials, South Ural State University, Lenin Av., 76, Chelyabinsk, 454080, Russia

    Roman R. Nazhimov

Authors
  1. Vladimir E. Fedorov
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  2. Roman R. Nazhimov
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Correspondence to Vladimir E. Fedorov.

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Fedorov, V.E., Nazhimov, R.R. Inverse Problems for a Class of Degenerate Evolution Equations with Riemann - Liouville Derivative. FCAA 22, 271–286 (2019). https://doi.org/10.1515/fca-2019-0018

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