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Linear equations of the Sobolev type with integral delay operator

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Abstract

We establish sufficient conditions of the local and global solvability of initial value problems for a class of linear operator-differential equations of the first order in a Banach space. Equations are assumed to have a degenerate operator at the derivative and an integral delay operator. We apply methods of the theory of degenerate semigroups of operators and the contraction mapping theorem. As examples illustrating the general results we consider the evolution equation for a free surface of a filtered liquid with a delay and a linearized quasistationary system of equations for a phase field with a delay.

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Authors and Affiliations

  1. Chelyabinsk State University, ul. Br. Kashirinykh 129, Chelyabinsk, 454001, Russia

    V. E. Fedorov

  2. Ural Branch of the Russian Academy of Justice, pr. Pobedy 160, Chelyabinsk, 454084, Russia

    E. A. Omel’chenko

Authors
  1. V. E. Fedorov
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  2. E. A. Omel’chenko
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Correspondence to V. E. Fedorov.

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Original Russian Text © V.E. Fedorov and E.A. Omel’chenko, 2014, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2014, No. 1, pp. 71–81.

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Fedorov, V.E., Omel’chenko, E.A. Linear equations of the Sobolev type with integral delay operator. Russ Math. 58, 60–69 (2014). https://doi.org/10.3103/S1066369X14010071

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