Abstract
Under consideration is some problem for inhomogeneous differential evolution equation in Banach space with an operator that generates a C 0-continuous semigroup and a nonlocal integral condition in the sense of Stieltjes. In case the operator has continuous inhomogeneity in the graph norm. We give the necessary and sufficient conditions for existence of a generalized solution for the problem of whether the nonlocal data belong to the generator domain. Estimates on solution stability are given, and some conditions are obtained for existence of the classical solution of the nonlocal problem. All results are extended to a Sobolev-type linear equation, the equation in Banach space with a degenerate operator at the derivative. The time nonlocal problem for the partial differential equation, modeling a filtrating liquid free surface, illustrates the general statements.
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Original Russian Text Copyright © 2014 Fedorov V.E., Ivanova N.D., and Fedorova Yu.Yu.
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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 55, No. 4, pp. 882–897, July–August, 2014.
The first author was partially supported by the Laboratory of Quantum Topology of Chelyabinsk State University (Grant 14.Z50.31.0020 of the Government of the Russian Federation).
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Fedorov, V.E., Ivanova, N.D. & Fedorova, Y.Y. On a time nonlocal problem for inhomogeneous evolution equations. Sib Math J 55, 721–733 (2014). https://doi.org/10.1134/S0037446614040144
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DOI: https://doi.org/10.1134/S0037446614040144