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Degenerate Linear Evolution Equations with the Riemann–Liouville Fractional Derivative

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Abstract

We study the unique solvability of the Cauchy and Schowalter–Sidorov type problems in a Banach space for an evolution equation with a degenerate operator at the fractional derivative under the assumption that the operator acting on the unknown function in the equation is p-bounded with respect to the operator at the fractional derivative. The conditions are found ensuring existence of a unique solution representable by means of the Mittag-Leffler type functions. Some abstract results are illustrated by an example of a finite-dimensional degenerate system of equations of a fractional order and employed in the study of unique solvability of an initial-boundary value problem for the linearized Scott-Blair system of dynamics of a medium.

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Authors and Affiliations

  1. Chelyabinsk State University South Ural State University, Chelyabinsk, Russia

    V. E. Fedorov & M. V. Plekhanova

  2. Chelyabinsk State University, Chelyabinsk, Russia

    R. R. Nazhimov

Authors
  1. V. E. Fedorov
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  2. M. V. Plekhanova
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  3. R. R. Nazhimov
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Correspondence to V. E. Fedorov.

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Original Russian Text Copyright © 2018 Fedorov V.E., Plekhanova M.V., and Nazhimov R.R.

Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 59, No. 1, pp. 171–184, January–February, 2018; DOI: 10.17377/smzh.2018.59.115

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Fedorov, V.E., Plekhanova, M.V. & Nazhimov, R.R. Degenerate Linear Evolution Equations with the Riemann–Liouville Fractional Derivative. Sib Math J 59, 136–146 (2018). https://doi.org/10.1134/S0037446618010159

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  • DOI: https://doi.org/10.1134/S0037446618010159

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