Abstract
This paper provides a survey of results devoted to the study of integro-differential equations with unbounded operator coefficients in a Hilbert space. These equations are operator models of integro-differential partial differential equations arising in numerous applications: in the theory of viscoelasticity, in the theory of heat propagation in media with memory (Gurtin–Pipkin equations), and in averaging theory. The most interesting and profound results of the survey are devoted to the spectral analysis of operator functions that are symbols of the integro-differential equations under study.
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References
G. Amendola, M. Fabrizio, and J. M. Golden, Thermodynamics of Materials with Memory. Theory and Applications, Springer, New York–Dordrecht–Heidelberg–London (2012).
O. A. Andronova and N. D. Kopachevsky, “On linear problems with surface energy dissipation,” Sovrem. Mat. Fundam. Napravl., 29, 11–28 (2008).
T. Ya. Azizov, N. D. Kopachevsky, and L. D. Orlova, “Evolution and spectral problems generated by the problem of small motions of a viscoelastic fluid,” Tr. SPb. Mat. Ob-va, 6, 5–33 (1988).
T. Ya. Azizov, N. D. Kopachevsky, and L. D. Orlova, “Operator approach to the study of the Oldroyd hydrodynamic model,” Mat. Zametki, 65, No. 6, 924–928 (1999).
M. A. Biot, “Generalized theory of acoustic propagation in porous dissipative media,” J. Acoust. Soc. Am., 34, 1254–1264 (1962).
C. M. Dafermos, “Asymptotic stability in viscoelasticity,” Arch. Ration. Mech. Anal., 37, 297–308 (1970).
A. V. Davydov and Y. A. Tikhonov, “Study of Kelvin–Voigt models arising in viscoelasticity,” Differ. Equ., 54, No. 12, 1620–1635 (2018).
W. Desch, R. K. Miller, “Exponential stabilization of Volterra integro-differential equations in Hilbert space,” J. Differ. Equ., 70, 366–389 (1987).
P. L. Devis, “Hyperbolic integro-differential equations,” Proc. Am. Math. Soc., 47, 155–160 (1975).
P. L. Devis, “On the hyperbolicity of the equations of the linear theory of heat conduction for materials with memory,” SIAM J. Appl. Math., 30, 75–80 (1976).
G. Di Blasio, “Parabolic Volterra equations of convolution type,” J. Integral Equ. Appl., 6, 479–508 (1994).
G. Di Blasio, K. Kunisch, and E. Sinestrari, “L2-regularity for parabolic partial integro-differential equations with delays in the highest order derivatives,” J. Math. Anal. Appl., 102, 38–57 (1984).
K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, New York–Berlin–Heidelberg (1999).
A. Eremenko and S. Ivanov, “Spectra of the Gurtin–Pipkin type equations,” SIAM J. Math. Anal., 43, 2296–2306 (2011).
M. E. Gurtin and A. C. Pipkin, “General theory of heat conduction with finite wave speed,” Arch. Ration. Mech. Anal., 31, 113–126 (1968).
A. A. Il’yushin and B. E. Pobedrya, Fundamentals of the Mathematical Theory of Thermoviscoelasticity [in Russian], Nauka, Moscow (1970).
R. S. Ismagilov, N. A. Rautian, and V. V. Vlasov, “Examples of very unstable linear partial functional differential equations,” arXiv, 1402.4107v1.
S. A. Ivanov, “Wave type” spectrum of the Gurtin–Pipkin equation of the second order,” arXiv, 1002.2831.
S. Ivanov and L. Pandolfi, “Heat equations with memory: lack of controllability to the rest,” J. Math. Anal. Appl., 355, 1–11 (2009).
S. A. Ivanov and T. L. Sheronova, “Spectrum of the heat equation with memory,” arXiv, 0912.1818v1.
T. Kato, Perturbation Theory for Linear Operators [Russian translation], Mir, Moscow (1972).
N. D. Kopachevsky, “Cauchy problem for a linear integro-differential equation in a Hilbert space,” Uch. Zap. Tavr. Nats. Un-ta Im. V. I. Vernadskogo, 16, 139–152 (2001).
N. D. Kopachevsky, Volterra Integro-Differential Equations in Hilbert Space. Special Course of Lectures [in Russian], Bondarenko, Simferopol’ (2012).
N. D. Kopachevsky and S. G. Krein, Operator Approach to Linear Problems of Hydrodynamics. Vol. 2: Non-Self-Adjoint Problems for Viscous Fluids, Birkhäuser, Basel–Boston–Berlin (2003).
N. D. Kopachevsky, S. G. Kreyn, and Ngo Zuy Kan, Operator Methods in Linear Hydrodynamics. Evolution and Spectral Problems [in Russian], Nauka, Moscow (1989).
N. D. Kopachevsky, L. D. Orlova, and Yu. S. Pashkova, “Differential-operator and integrodifferential equations in the problem of small oscillations of hydrodynamic systems,” Uch. Zap. Simf. Gos. Un-ta, 41, No. 2, 96–108 (1995).
N. D. Kopachevsky and E. V. Semkina, “On second-order Volterra integro-differential equations unresolved with respect to the higher derivative,” Uch. Zap. Tavr. Nats. Un-ta Im. V. I. Vernadskogo. Ser. Fiz.-Mat. Nauki, 26, No. 1, 52–79 (2013).
N. D. Kopachevsky and E. V. Semkina, “Linear Volterra integro-differential second-order equations unresolved with respect to the highest derivative,” Eurasian Math. J., 4, No. 4, 64–87 (2013).
S. G. Kreyn, Linear Differential Equations in Banach Space [in Russian], Nauka, Moscow (1967).
J.-L. Lions and E. Magenes, Non-Homogeneous Boundary-Value Problems and Their Applications [Russian translation], Mir, Moscow (1971).
A. V. Lykov, “Some problematic issues of the theory of heat and mass transfer,” In: Problems of Heat and Mass Transfer, Nauka i tekhnika, Minsk, pp. 9–82 (1976).
R. K. Miller, “Volterra integral equations in a Banach space,” Funkcialaj Ekvac., 18, 163–194 (1975).
R. K. Miller, “An integrodifferencial equation for rigid heat conductors with memory,” J. Math. Anal., 66, 313–332 (1978).
R. K. Miller and R. L. Wheeler, “Well-posedness and stability of linear Volterra integro-differential equations in abstract spaces,” Funkcialaj Ekvac., 21, 279–305 (1978).
A. I. Miloslavskiy, “On the stability of some classes of evolution equations,” Sib. Mat. Zh., 26, 118–132 (1985).
A. I. Miloslavskiy, “Spectral properties of an operator beam arising in viscoelasticity,” Ukr. NIINTI, Khar’kov, 13.07.87, No. 1229-UK87, 53 (1987).
A. I. Miloslavskiy, “On the spectrum of instability of an operator beam,” Mat. Zametki, 49, No. 4, 88–94 (1991).
J. E. Munoz Rivera, M. G. Naso, and F. M. Vegni, “Asymptotic behavior of the energy for a class of weakly dissipative second-order systems with memory,” J. Math. Anal. Appl., 286, 692–704 (2003).
A. D. Myshkis and V. V. Vlasov, “On an analogy between the classifications of functional differential equations and partial differential equations,” Funct. Differ. Equ., 16, No. 3, 545–560 (2009).
L. Pandolfi, “The controllability of the Gurtin–Pipkin equations: a cosine operator approach,” Appl. Math. Optim., 52, 143–165 (2005).
A. Pazy, Semigroups of Linear Operators and Applications of Partial Differential Equations, Springer, New York etc. (1983).
R. Perez Ortiz and V. V. Vlasov, “Correct solvability of Volterra integro-differential equations in Hilbert space,” Electron. J. Qual. Theory Differ Equ., 31, 1–17 (2016).
J. Prüss, “On linear Volterra equations of parabolic type in Banach spaces,” Trans. Am. Math. Soc., 301, No. 2, 691–721 (1987).
J. Prüss, “Bounded solutions of Volterra equations,” SIAM J. Math. Anal., 19, No. 1, 133–149 (1988).
J. Prüss, Evolutionary Integral Equations and Applications, Birkhäuser, Basel–Baston–Berlin (1993).
Yu. N. Rabotnov, Elements of Hereditary Solid Body Mechanics [in Russian], Nauka, Moscow (1977).
G. V. Radzievskiy, “Asymptotics of the distribution of characteristic numbers of operator functions analytic in an angle,” Mat. Sb., 112, No. 3, 396–420 (1980).
N. A. Rautian, “On the boundedness of a class of integral operators of fractional type,” Mat. Sb., 200, No. 12, 81–106 (2009).
N. A. Rautian, “On representation of solutions of integro-differential equations with unbounded operator coefficients in a Hilbert space,” Proc. Int. Conf. Qualitative Theory of Differential Equations and Applications, MESI, Moscow, pp. 116–134 (2011).
N. A. Rautian, “On the structure and properties of solutions of integro-differential equations arising in thermophysics and acoustics,” Mat. Zametki, 90, No. 3, 474–477 (2011).
N. A. Rautian, “Semigroups generated by Volterra integro-differential equations,” Diff. Uravn., 56, No. 9, 1226–1244 (2020).
N. A. Rautian, “Well-posedness of Volterra integro-differential equations with fractional exponential kernels,” In: Differential and Difference Equations with Applications, Springer, Cham, pp. 517–533 (2020).
E. Sanchez-Palencia, Non-Homogeneous Media and Vibration Theory [Russian translation], Mir, Moscow (1984).
G. V. Sandrakov, “Multiphase averaged diffusion models for problems with multiple parameters,” Izv. RAN. Ser. Mat., 71, No. 6, 119–72 (2007).
J. Shapiro, Composition Operators and Classical Function Theory, Springer, New York (1993).
A. A. Shkalikov, “Strongly damped operator pencils and the solvability of the corresponding operator-differential equations,” Mat. Sb., 177, No. 1, 96–118 (1988).
A. L. Skubachevskii, “Elliptic problems with nonlocal conditions near the boundary,” Mat. Sb., 129, No. 2, 279–302 (1986).
A. L. Skubachevskii, Elliptic Functional Differential Equations and Applications, Birkh¨auser, Basel (1997).
A. L. Skubachevskii, “On a class of functional differential operators satisfying the Kato hypothesis,” Algebra i Analiz, 30, No. 2, 249–273 (2018).
V. V. Vlasov, A. A. Gavrikov, S. A. Ivanov, and D. Yu. Knyaz’kov, V. A. Samarin, A. S. Shamaev, “Spectral properties of combined media,” J. Math. Sci. (N.Y.), 164, No. 6, 948–963 (2010).
V. V. Vlasov, D. A. Medvedev, and N. A. Rautian, Functional Differential Equations in Sobolev Spaces and Their Spectral Analysis [in Russian], MGU, Moscow (2011).
V. V. Vlasov and R. Perez Ortiz, “Spectral analysis of integro-differential equations arising in the theory of viscoelasticity and thermophysics,” Mat. Zametki, 98, No. 4, 630–634 (2015).
V. V. Vlasov, R. Perez Ortiz, and N. A. Rautian, “Investigation of Volterra integro-differential equations with kernels depending on a parameter,” Diff. Uravn., 54, No. 3, 369–386 (2018).
V. V. Vlasov and N. A. Rautian, “Correct solvability and spectral analysis of abstract hyperbolic integro-differential equations,” Tr. Sem. Im. I. G. Petrovskogo, 28, 75–113 (2011).
V. V. Vlasov and N. A. Rautian, “Study of integro-differential equations arising in the theory of viscoelasticity,” Izv. Vuzov. Ser. Mat., 6, 56–60 (2012).
V. V. Vlasov and N. A. Rautian, “Spectral analysis and representation of solutions of abstract integro-differential equations,” Dokl. RAN, 454, No. 2, 141–144 (2014).
V. V. Vlasov and N. A. Rautian, “Properties of solutions of integro-differential equations arising in the theory of heat and mass transfer,” Tr. Mosk. Mat. Ob-va, 75, No. 2, 219–243 (2014).
V. V. Vlasov and N. A. Rautian, “Spectral analysis and representations of solutions of abstract integro-differential equations in Hilbert space,” In: Concrete Operators, Spectral Theory, Operators in Harmonic Analysis and Approximation, IWOTA 11, Sevilla, Spain, July 3–9, 2011, Birkhäuser/Springer, Basel, pp. 517–535 (2014).
V. V. Vlasov and N. A. Rautian, “Well-posedness and spectral analysis of integro-differential equations arising in viscoelasticity theory,” Sovrem. Mat. Fundam. Napravl., 58, 22–42 (2015).
V. V. Vlasov and N. A. Rautian, Spectral Analysis of Functional Differential Equations [in Russian], MAKS Press, Moscow (2016).
V. V. Vlasov and N. A. Rautian, “Correct solvability of Volterra integro-differential equations in a Hilbert space,” Diff. Uravn., 52, No. 9, 1168–1177 (2016).
V. V. Vlasov and N. A. Rautian, “Investigation of operator models arising in viscoelasticity theory,” Sovrem. Mat. Fundam. Napravl., 64, No. 1, 60–73 (2018).
V. V. Vlasov and N. A. Rautian, “Spectral analysis of integro-differential equations in Hilbert spaces,” J. Math. Sci. (N.Y.), 239, No. 6, 771–787 (2019).
V. V. Vlasov and N. A. Rautian, “Correct solvability and representation of solutions of integro-differential equations arising in the theory of viscoelasticity,” Diff. Uravn., 55, No. 4, 574–587 (2019).
V. V. Vlasov and N. A. Rautian, “Spectral analysis and representation of solutions of integro-differential equations with fractional-exponential kernels,” Tr. Mosk. Mat. Ob-va, 80, No. 2, 197–220 (2019).
V. V. Vlasov and N. A. Rautian, “A study of operator models arising in problems of hereditary mechanics,” J. Math. Sci. (N.Y.), 244, No. 2, 170–182 (2020).
V. V. Vlasov and N. A. Rautian, “Properties of semigroups generated by Volterra integro-differential equations,” Diff. Uravn., 56, No. 8, 1122–1126 (2020).
V. V. Vlasov and N. A. Rautian, “Investigation of Volterra integro-differential equations with kernels representable by Stieltjes integrals,” Diff. Uravn., to be published (2021).
V. V. Vlasov, N. A. Rautian, and A. S. Shamaev, “Solvability and spectral analysis of integro-differential equations arising in thermophysics and acoustics,” Dokl. RAN, 434, No. 1, 12–15 (2010).
V. V. Vlasov, N. A. Rautian, and A. S. Shamaev, “Spectral analysis and correct solvability of abstract integro-differential equations arising in thermophysics and acoustics,” Sovrem. Mat. Fundam. Napravl., 39, 36–65 (2011).
V. V. Vlasov, N. A. Rautian, and A. S. Shamaev, “Analysis of operator models arising in problems of hereditary mechanics,” Sovrem. Mat. Fundam. Napravl., 45, 43–61 (2012).
V. V. Vlasov and J. Wu, “Solvability and spectral analysis of abstract hyperbolic equations with delay,” Funct. Differ. Equ., 16, No. 4, 751–768 (2009).
J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer, New York (1996).
D. A. Zakora, “Oldroyd model for compressible fluids,” Sovrem. Mat. Fundam. Napravl., 61, 41–66 (2016).
D. A. Zakora and N. D. Kopachevsky, “On a spectral problem associated with a second-order integro-differential equation,” Uch. Zap. Tavr. Nats. Un-ta Im. V. I. Vernadskogo, No. 2, 2–18 (2004).
V. V. Zhikov, “An extension and application of the two-scale convergence method,” Mat. Sb., 191 No. 7, 31–72 (2000).
V. V. Zhikov, “On two-scale convergence,” Tr. Sem. Im. I. G. Petrovskogo, 23, 149–187 (2003).
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Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 67, No. 2, Dedicated to the memory of Professor N. D. Kopachevsky, 2021.
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Vlasov, V.V., Rautian, N.A. Investigation of Integro-Differential Equations by Methods of Spectral Theory. J Math Sci 278, 55–81 (2024). https://doi.org/10.1007/s10958-024-06905-8
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DOI: https://doi.org/10.1007/s10958-024-06905-8