Abstract
We prove the correct solvability of the initial problems for integrodifferential equations with unbounded operator coefficients in Hilbert spaces. Such equations occur in many problems of the theory of viscoelasticity with memory and the heat transfer theory.
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References
R. O. Hryniv and A. A. Shkalikov, “On an Operator Pencil Arising in the Problem of Beam Oscillation with Internal Damping,” Matem. Zametki, 56(2), 114–131 (1994).
A. I. Miloslavskii, Spectral Properties of an Operator Bundle Arising in Viscoelasticity, Available from Ukrainian NIINTI, No. 1229-UK87 (Kharkov, 1987).
A. I. Miloslavskii, “On the Stability of Some Classes of Evolution Equations,” Sib. Matem. Zhurn. 26, 118–132 (1985).
A. I. Miloslavskii, “Instability Spectrum of an Operator Bundle,” Matem. Zametki 49(4), 88–94 (1991).
V. V. Vlasov, N. A. Rautian, and A. S. Shamaev, “Solvability and Spectral Analysis of Integrodifferential Equations Arising in the Theory of Heat Transfer and Acoustics,” Dokl. Phys. 434(1), 12–15 (2010).
V. V. Vlasov, N. A. Rautian, and A. S. Shamaev, “Spectral Analysis and Correct Solvability of Abstract Integrodifferential Equations Arising in the Theory of Heat Transfer and Acoustics,” Sovremen. Matem. Fundamental’nye Napravleniya 39, 36–65 (2011).
J.-P. Lions and E. Magenes, Problèmes Aux Limites Non Homogènes Et Applications (Dunod, Paris, 1970, Mir,Moscow, 1971).
V. V. Vlasov, “Solubility and Properties of Solution of Functional-Differential Equations in Hilbert Space,” Matem. Sborn. 186(8), 67–92 (1995).
V. V. Vlasov and D. A. Medvedev, “Functional-Differential Equations in Sobolev Spaces and Related Problems in Spectral Theory,” Sovremen.Matem. Fundamental’nye Napravleniya 30, 3–173 (2008).
J. Prüss, Evolutionary Integral Equations and Applications, Monographs in Mathematics 87 (Birkhäuser Verlag, Basel, 1993).
L. Pandolfi, “The Controllability of the Gurtin-Pipkin Equations: a CosineOperator Approach,” Appl.Math. Optim. 52, 143–165 (2005).
S. Ivanov and L. Pandolfi, “Heat Equations withMemory: Lack of Controllability to the Rest,” J.Math. Anal. and Appl. 355, 1–11 (2009).
V. V. Vlasov and J. Wu, “Spectral Analysis and Solvability of Abstract Hyperbolic Equations with Aftereffect,” Differents. Uravneniya 45(4), 524–533 (2009).
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Original Russian Text © V.V. Vlasov and N.A. Rautian, 2012, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2012, No. 6, pp. 56–60.
Submitted by V.P.Maksimov
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Vlasov, V.V., Rautian, N.A. Integrodifferential equations in viscoelasticity theory. Russ Math. 56, 48–51 (2012). https://doi.org/10.3103/S1066369X12060060
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DOI: https://doi.org/10.3103/S1066369X12060060