Geodesic
Spectral analysis of integrodifferential equations in a Hilbert space
Contemporary Mathematics. Fundamental Directions, Proceedings of the Seminar on Differential and Functional Differential Equations supervised by A. L. Skubachevskii (Peoples' Friendship University of Russia), Tome 62 (2016), pp. 53-71 
V. V. Vlasov; N. A. Rautian. Spectral analysis of integrodifferential equations in a Hilbert space. Contemporary Mathematics. Fundamental Directions, Proceedings of the Seminar on Differential and Functional Differential Equations supervised by A. L. Skubachevskii (Peoples' Friendship University of Russia), Tome 62 (2016), pp. 53-71. http://geodesic.mathdoc.fr/item/CMFD_2016_62_a3/
@article{CMFD_2016_62_a3,
     author = {V. V. Vlasov and N. A. Rautian},
     title = {Spectral analysis of integrodifferential equations in {a~Hilbert} space},
     journal = {Contemporary Mathematics. Fundamental Directions},
     pages = {53--71},
     year = {2016},
     volume = {62},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CMFD_2016_62_a3/}
}
TY  - JOUR
AU  - V. V. Vlasov
AU  - N. A. Rautian
TI  - Spectral analysis of integrodifferential equations in a Hilbert space
JO  - Contemporary Mathematics. Fundamental Directions
PY  - 2016
SP  - 53
EP  - 71
VL  - 62
UR  - http://geodesic.mathdoc.fr/item/CMFD_2016_62_a3/
LA  - ru
ID  - CMFD_2016_62_a3
ER  - 
%0 Journal Article
%A V. V. Vlasov
%A N. A. Rautian
%T Spectral analysis of integrodifferential equations in a Hilbert space
%J Contemporary Mathematics. Fundamental Directions
%D 2016
%P 53-71
%V 62
%U http://geodesic.mathdoc.fr/item/CMFD_2016_62_a3/
%G ru
%F CMFD_2016_62_a3

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

We investigate the correct solvability of initial-value problems for abstract integrodifferential equations with unbounded operator coefficients in a Hilbert space. We do spectral analysis of operator-functions describing symbols of such equations. These equations are an abstract form of linear integrodifferential partial derivative equations arising in the viscoelasticity theory and having some other important applications. We establish the localization and the spectrum structure of operator-functions describing symbols of these equations.

[1] V. V. Vlasov, “On solvability and properties of solutions of functional differential equations in a Hilbert space”, Math. Digest, 186:8 (1995), 67–92 | MR | Zbl

[2] V. V. Vlasov, “On solvability and estimates of solutions of functional differential equations in Sobolev spaces”, Proc. Math. Inst. Russ. Acad. Sci., 227, 1999, 109–121 | MR | Zbl

[3] V. V. Vlasov, “On correct solvability of abstract parabolic equations with aftereffect”, Rep. Russ. Acad. Sci., 415:2 (2007), 151–152

[4] V. V. Vlasov, Dzh. Vu, G. R. Kabirova, “Correct solvability and spectral properties of abstract hyperbolic equations with aftereffect”, Contemp. Math. Fundam. Directions, 35, 2010, 44–59 | MR | Zbl

[5] V. V. Vlasov, A. A. Gavrikov, S. A. Ivanov, D. Yu. Knyaz'kov, V. A. Samarin, A. S. Shamaev,, “Spectral properties of combined media”, Contemp. Probl. Math. Mech., 5:1 (2009), 134–155

[6] V. V. Vlasov, D. A. Medvedev, “Functional differential equations in Sobolev spaces and related questions of the spectral theory”, Contemp. Math. Fundam. Directions, 30, 2008, 3–173 | MR

[7] V. V. Vlasov, D. A. Medvedev, N. A. Rautian, Functional differential equations in Sobolev spaces and their spectral analysis, MSU, Moscow, 2011

[8] V. V. Vlasov, N. A. Rautian, “Correct solvability and spectral analysis of abstract hyperbolic integrodifferential equations”, Proc. Petrovskii Semin., 28, 2011, 75–114 | Zbl

[9] V. V. Vlasov, N. A. Rautian, “On properties of solutions of integrodifferential equations arising in the heat-mass exchange theory”, Proc. Moscow Math. Soc., 75, no. 2, 2014, 219–243

[10] V. V. Vlasov, N. A. Rautian, “Correct solvability and spectral analysis of integrodifferential equations arising in the viscoelasticity theory”, Contemp. Math. Fundam. Directions, 58, 2015, 22–42 | MR

[11] V. V. Vlasov, N. A. Rautian, “Correct solvability of Volterra integrodifferential equations in a Hilbert space”, Differ. Equ., 52:9 (2016), 1168–1177 | DOI | Zbl

[12] V. V. Vlasov, N. A. Rautian, A. C. Shamaev, “Solvability and spectral analysis of integrodifferential equations arising in thermal physics and acoustics”, Rep. Russ. Acad. Sci., 434:1 (2010), 12–15 | Zbl

[13] V. V. Vlasov, N. A. Rautian, A. C. Shamaev, “Spectral analysis and correct solvability of abstract integrodifferential equations arising in thermal physics and acoustics”, Contemp. Math. Fundam. Directions, 39, 2011, 36–65 | MR

[14] V. V. Vlasov, K. I. Shmatov, “Correct solvability of hyperbolic-type equations with delay in a Hilbert space”, Proc. Math. Inst. Russ. Acad. Sci., 243, 2003, 127–137 | MR | Zbl

[15] I. Ts. Gokhberg, M. G. Kreyn, Introduction to the Theory of Linear Nonselfadjoint Operators, Nauka, Moscow, 1965 | MR

[16] V. V. Zhikov, “On one extension and application of the two-scale convergence method”, Math. Digest, 191:7 (2000), 31–72 | DOI | MR | Zbl

[17] V. V. Zhikov, “On two-scale convergence”, Proc. Petrovskii Semin., 23, 2003, 149–187 | Zbl

[18] A. A. Il'yushin, B. E. Pobedrya, “Essentials of Mathematical Theory of Thermoviscoelasticity”, Nauka, Moscow, 1970

[19] Zh. P. Lions, E. Madzhenes, Nonhomogeneous Boundary-Value Problems and Their Applications, Mir, Moscow, 1971 | MR

[20] A. V. Lykov, Problem of Heat-Mass Exchange, Nauka i tekhnika, Minsk, 1976

[21] A. I. Miloslavskiy, Spectral properties of an operator pencil arising in viscoelasticity, Submitted to Ukr. NIINTI, 13.07.87, No. 1229-UK87, Khar'kov, 1987

[22] V. V. Palin, E. V. Radkevich, “Conservation laws and their hyperbolic regularizations”, Contemp. Probl. Math. Mech., 5:1 (2009), 88–115

[23] Yu. N. Rabotnov, Elements of Heritable Mechanics of Solid Bodies, Nauka, Moscow, 1977 | MR

[24] E. Sanches-Palensiya, Nonhomogeneous Media and Oscillation Theory, Mir, Moscow, 1984 | MR

[25] A. S. Shamaev, V. V. Shumilova, “Averaging of the acoustics equations for viscoelastic matter with channels filled with viscous compressible fluid”, Bull. Russ. Acad. Sci. Ser. Mech. Fluid Gas, 2011, no. 2, 92–103 | Zbl

[26] A. A. Shkalikov, “Strongly dumped operator pencils and solvability of corresponding operator-differential equations”, Math. Digest, 135(177):1 (1988), 96–118 | MR | Zbl

[27] A. A. Shkalikov, “Elliptic equations in a Hilbert space and related spectral problems”, Proc. Petrovskii Semin., 14, 1989, 140–224 | Zbl

[28] Desch W., Miller R. K., “Exponential stabilization of Volterra integrodifferential equations in Hilbert space”, J. Differ. Equ., 70 (1987), 366–389 | DOI | MR | Zbl

[29] Di Blasio G., “Parabolic Volterra equations of convolution type”, J. Integr. Equ. Appl., 6 (1994), 479–508 | DOI | MR | Zbl

[30] Di Blasio G., Kunisch K., Sinestari E., “$L^2$-regularity for parabolic partial integrodifferential equations with delays in the highest order derivatives”, J. Math. Anal. Appl., 102 (1984), 38–57 | DOI | MR | Zbl

[31] Di Blasio G., Kunisch K., Sinestari E., “Stability for abstract linear functional differential equations”, Israel J. Math., 50:3 (1985), 231–263 | DOI | MR | Zbl

[32] 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

[33] 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

[34] Kato T., Perturbation theory for linear operators, Springer, New York, 1966 | MR | Zbl

[35] Kopachevsky N. D., Krein S. G., Operator approach to linear problems of hydrodynamics, v. 2, Nonselfadjoint problems for viscous fluids, Birkhäuser, Berlin—Basel—Boston, 2003 | MR | Zbl

[36] Kunisch K., Mastinsek M., “Dual semigroups and structural operators for partial differential equations with unbounded operators acting on the delays”, Differ. Integr. Equ., 3:4 (1990), 733–756 | MR | Zbl

[37] Medvedev D. A., Vlasov V. V., Wu J., “Solvability and structural properties of abstract neutral functional differential equations”, Funct. Differ. Equ., 66:3–4 (2008), 249–272 | MR

[38] Miller R. K., “Volterra integral equation in Banach space”, Funkcial. Ekvac., 18 (1975), 163–194 | MR

[39] Miller R. K., “An integrodifferential equation for rigid heat conductors with memory”, J. Math. Anal. Appl., 66 (1978), 313–332 | DOI | MR | Zbl

[40] Miller R. K., Wheeler R. L., “Well-posedness and stability of linear Volterra interodifferential equations in abstract spaces”, Funkcial. Ekvac., 21 (1978), 279–305 | MR | Zbl

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

[42] Shapiro J., Composition operators and classical function theory, Springer, New York, 1993 | MR | Zbl

[43] Vlasov V. V., Wu J., “Solvability and spectral analysis of abstract hyperbolic equations with delay”, J. Funct. Differ. Equ., 16:4 (2009), 751–768 | MR | Zbl

[44] Wu J., “Semigroup and integral form of class of partial differential equations with infinite delay”, Differ. Integr. Equ., 4:6 (1991), 1325–1351 | MR | Zbl

[45] Wu J., Theory and applications of partial functional differential equations, Springer-Verlag, New York, 1996 | MR | Zbl