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Study of degenerate evolution equations with memory by operator semigroup methods

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Abstract

We reduce the problem with some history prescribed for an integrodifferential equation in a Banach space including memory effect to the Cauchy problem for some evolution system with a constant operator in a larger space that possesses a resolvent (C 0)-semigroup. This enables us to state conditions for the existence of a unique classical solution to the original problem. We use the results to study the unique solvability of problems with history prescribed for degenerate linear evolution equations with memory in Banach spaces. We show that the initial-boundary value problem for the linearized integrodifferential Oskolkov system describing the dynamics of Kelvin–Voigt fluids in linear approximation belongs to this class of problems.

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Authors and Affiliations

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

    V. E. Fedorov

  2. Chelyabinsk State University, Chelyabinsk, Russia

    L. V. Borel

Authors
  1. V. E. Fedorov
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  2. L. V. Borel
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Correspondence to V. E. Fedorov.

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Original Russian Text Copyright © 2016 Fedorov V.E. and Borel L.V.

The authors were supported by the Laboratory of Quantum Topology of Chelyabinsk State University (Grant 14.Z50.31.0020 of the Government of the Russian Federation).

Chelyabinsk. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 57, No. 4, pp. 899–912, July–August, 2016; DOI: 10.17377/smzh.2016.57.412. Original article submitted July 15, 2015.

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Fedorov, V.E., Borel, L.V. Study of degenerate evolution equations with memory by operator semigroup methods. Sib Math J 57, 704–714 (2016). https://doi.org/10.1134/S0037446616040121

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  • DOI: https://doi.org/10.1134/S0037446616040121

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