Geodesic
Analysis of operator models arising in problems of hereditary mechanics
Contemporary Mathematics. Fundamental Directions, Proceedings of the Sixth International Conference on Differential and Functional-Differential Equations (Moscow, August 14–21, 2011). Part 1, Tome 45 (2012), pp. 43-61.

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The correct solvability of initial-value problems for integrodifferential equations with unbounded operator coefficients in Hilbert spaces is determined. Numerous problems of hereditary mechanics and thermal physics have motivated the study of such equations. Models taking into account a Kelvin–Voight friction are considered. A spectral analysis of the operator functions, which are the symbols of integro-differential equations considered in this paper, is made. 
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V. V. Vlasov; N. A. Rautian; A. S. Shamaev. Analysis of operator models arising in problems of hereditary mechanics. Contemporary Mathematics. Fundamental Directions, Proceedings of the Sixth International Conference on Differential and Functional-Differential Equations (Moscow, August 14–21, 2011). Part 1, Tome 45 (2012), pp. 43-61. http://geodesic.mathdoc.fr/item/CMFD_2012_45_a3/

[1] Azizov T. Ya., Kopachevskii N. D., Orlova L. D., “Operatornyi podkhod k issledovaniyu gidrodinamicheskoi modeli Oldroita”, Mat. zametki, 65:6 (1999), 924–928 | DOI | MR | Zbl

[2] Vlasov V. V., “O razreshimosti i svoistvakh reshenii funktsionalno-differentsialnykh uravnenii v gilbertovom prostranstve”, Mat. sb., 186:8 (1995), 67–92 | MR | Zbl

[3] Vlasov V. V., Vu Dzh., “Spektralnyi analiz i razreshimost abstraktnykh giperbolicheskikh uravnenii s posledeistviem”, Diff. uravn., 45:4 (2009), 524–533 | MR | Zbl

[4] Vlasov V. V., Medvedev D. A., “Funktsionalno-differentsialnye uravneniya v prostranstvakh Soboleva i svyazannye s nimi voprosy spektralnoi teorii”, Sovrem. mat. Fundam. napravl., 30, 2008, 3–173 | MR

[5] Vlasov V. V., Rautian N. A., “Issledovanie integrodifferentsialnykh uravnenii, voznikayuschikh v teorii vyazkouprugosti”, Izv. vuzov. Ser. Mat., 2012, no. 6, 56–60 | Zbl

[6] Vlasov V. V., Rautian N. A., “Korrektnaya razreshimost i spektralnyi analiz abstraktnykh giperbolicheskikh integrodifferentsialnykh uravnenii”, Trudy seminara im. I. G. Petrovskogo, 28, 2011, 75–114

[7] Vlasov V. V., Rautian N. A., Shamaev A. C., “Razreshimost i spektralnyi analiz integrodifferentsialnykh uravnenii, voznikayuschikh v teplofizike i akustike”, Dokl. RAN, 434:1 (2010), 12–15 | MR | Zbl

[8] Vlasov V. V., Rautian N. A., Shamaev A. C., “Spektralnyi analiz i korrektnaya razreshimost abstraktnykh integrodifferentsialnykh uravnenii, voznikayuschikh v teplofizike i akustike”, Sovrem. mat. Fundam. napravl., 39, 2011, 36–65 | MR

[9] Griniv R. O., Shkalikov A. A., “O puchke operatorov, voznikayuschem v zadache o kolebaniyakh sterzhnya s vnutrennim treniem”, Mat. zametki, 56:2 (1994), 114–131 | MR | Zbl

[10] Lions Zh.-L., Madzhenes E., Neodnorodnye granichnye zadachi i ikh prilozheniya, Mir, M., 1971 | Zbl

[11] Lykov A. V., Problema teplo- i massoobmena, Nauka i tekhnika, Minsk, 1976

[12] Miloslavskii A. I., “Ob ustoichivosti nekotorykh klassov evolyutsionnykh uravnenii”, Sib. mat. zh., 26:5 (1985), 118–132 | MR

[13] Miloslavskii A. I., “O spektre neustoichivosti operatornogo puchka”, Mat. zametki, 49:4 (1991), 88–94 | MR | Zbl

[14] Miloslavskii A. I., Spektralnye svoistva operatornogo puchka, voznikayuschego v vyazkouprugosti, Deponirovano v Ukr. NIINTI, No 1229-UK87, Kharkov, 13.07.87

[15] Palin V. V., Radkevich E. V., “Zakony sokhraneniya i ikh giperbolicheskie regulyarizatsii”, Sovrem. problemy mat. i mekh., v. 5, Sekts. 1: Diff. uravn., 2009, 88–115

[16] Sanches-Palensiya E., Neodnorodnye sredy i teoriya kolebanii, Mir, M., 1984 | MR

[17] Shamaev A. S., Shumilova V. V., “Usrednenie uravnenii akustiki dlya vyazkouprugogo materiala s kanalami, zapolnennymi vyazkoi szhimaemoi zhidkostyu”, Izv. RAN. MZhG, 2011, no. 2, 92–103 | MR | Zbl

[18] Dafermos C. M., “Asymptotic Stability in Viscoelasticity”, Arch. Ration. Mech. Anal., 37 (1970), 297–308 | DOI | MR | Zbl

[19] Desch W., “Exponential Stabilization of Volterra Integrodifferential Equations in Hilbert Space”, J. Differential Equations, 70 (1987), 366–389 | DOI | MR | Zbl

[20] Eremenko A., Ivanov S., “Spectra of the Gurtin-Pipkin type equations”, SIAM J. Math. Anal., 43 (2011), 2296–2306 | DOI | MR | Zbl

[21] Gurtin M. E., Pipkin A. C., “Theory of heat conduction with finite wave speed”, Arch. Ration. Mech. Anal., 31 (1968), 113–126 | DOI | MR | Zbl

[22] Ivanov S., Pandolfi L., “Heat equations with memory: lack of controllability to the rest”, J. Math. Anal. Appl., 355 (2009), 1–11 | DOI | MR | Zbl

[23] Muñoz Rivera J. E., Naso M. G., Vegni F. M., “Asymptotic behavior of the energy for a class of weakly dissipative second-order systems with memory”, J. Math. Anal. Appl., 286 (2003), 692–704 | DOI | MR | Zbl

[24] Pandolfi L., “The controllability of the Gurtin-Pipkin equations: a cosine operator approach”, Appl. Math. Optim., 52 (2005), 143–165 | DOI | MR | Zbl

[25] Prüss J., Evolutionary Integral Equations amd Applications, Birkhauser Verlag, Basel–Boston–Berlin, 1993