Investigation of integrodifferential equations by methods of spectral theory

V. V. Vlasov ; N. A. Rautian

Contemporary Mathematics. Fundamental Directions, Dedicated to the memory of Professor N. D. Kopachevsky, Tome 67 (2021) no. 2, pp. 255-284.

Résumé

This paper provides a survey of results devoted to the study of integrodifferential equations with unbounded operator coefficients in a Hilbert space. These equations are operator models of integrodifferential 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 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 integrodifferential equations under study.

Bibliographie

[1] Azizov T. Ya., Kopachevskii N. D., Orlova L. D., “Evolyutsionnye i spektralnye zadachi, porozhdaemye problemoi malykh dvizhenii vyazkouprugoi zhidkosti”, Tr. SPb. Mat. ob-va, 6 (1988), 5–33

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

[3] Andronova O. A., Kopachevskii N. D., “O lineinykh zadachakh s poverkhnostnoi dissipatsiei energii”, Sovrem. mat. Fundam. napravl., 29, 2008, 11–28

[4] Vlasov V. V., Medvedev D. A., Rautian N. A., Funktsionalno-differentsialnye uravneniya v prostranstvakh Soboleva i ikh spektralnyi analiz, MGU, M., 2011

[5] Vlasov V. V., Perez Ortiz R., “Spektralnyi analiz integrodifferentsialnykh uravnenii, voznikayuschikh v teorii vyazkouprugosti i teplofizike”, Mat. zametki, 98:4 (2015), 630–634 | Zbl

[6] Vlasov V. V., Perez Ortiz R., Rautian N. A., “Issledovanie volterrovykh integro-differentsialnykh uravnenii s yadrami, zavisyaschimi ot parametra”, Diff. uravn., 54:3 (2018), 369–386 | Zbl

[7] Vlasov V. V., Rautian N. A., “Korrektnaya razreshimost i spektralnyi analiz abstraktnykh giperbolicheskikh integrodifferentsialnykh uravnenii”, Tr. sem. im. I. G. Petrovskogo, 28, 2011, 75–113 | Zbl

[8] Vlasov V. V., Rautian N. A., “Issledovanie integrodifferentsialnykh uravnenii, voznikayuschikh v teorii vyazkouprugosti”, Izv. vuzov. Ser. Mat., 6 (2012), 56–60 | Zbl

[9] Vlasov V. V., Rautian N. A., “Spektralnyi analiz i predstavlenie reshenii abstraktnykh integrodifferentsialnykh uravnenii”, Dokl. RAN, 454:2 (2014), 141–144 | Zbl

[10] Vlasov V. V., Rautian N. A., “O svoistvakh reshenii integrodifferentsialnykh uravnenii, voznikayuschikh v teorii teplomassoobmena”, Tr. Mosk. mat. ob-va, 75, no. 2, 2014, 219–243 | Zbl

[11] Vlasov V. V., Rautian N. A., “Korrektnaya razreshimost i spektralnyi analiz integrodifferentsialnykh uravnenii, voznikayuschikh v teorii vyazkouprugosti”, Sovrem. mat. Fundam. napravl., 58, 2015, 22–42

[12] Vlasov V. V., Rautian N. A., Spektralnyi analiz funktsionalno-differentsialnykh uravnenii, MAKS Press, M., 2016

[13] Vlasov V. V., Rautian N. A., “Korrektnaya razreshimost volterrovykh integro-differentsialnykh uravnenii v gilbertovom prostranstve”, Diff. uravn., 52:9 (2016), 1168–1177 | Zbl

[14] Vlasov V. V., Rautian N. A., “Issledovanie operatornykh modelei, voznikayuschikh v teorii vyazkouprugosti”, Sovrem. mat. Fundam. napravl., 64, no. 1, 2018, 60–73

[15] Vlasov V. V., Rautian N. A., “Korrektnaya razreshimost i predstavlenie reshenii integro-differentsialnykh uravnenii, voznikayuschikh v teorii vyazkouprugosti”, Diff. uravn., 55:4 (2019), 574–587 | Zbl

[16] Vlasov V. V., Rautian N. A., “Spektralnyi analiz i predstavlenie reshenii integro-differentsialnykh uravnenii s drobno-eksponentsialnymi yadrami”, Tr. Mosk. mat. ob-va, 80, no. 2, 2019, 197–220 | Zbl

[17] Vlasov V. V., Rautian N. A., “O svoistvakh polugrupp, porozhdaemykh volterrovymi integro-differentsialnymi uravneniyami”, Diff. uravn., 56:8 (2020), 1122–1126 | Zbl

[18] Vlasov V. V., Rautian N. A., “Issledovanie volterrovykh integro-differentsialnykh uravnenii s yadrami, predstavimymi integralami Stiltesa”, Diff. uravn., 2021 (to appear)

[19] 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 | Zbl

[20] 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

[21] Vlasov V. V., Rautian N. A., Shamaev A. C., “Issledovanie operatornykh modelei, voznikayuschikh v zadachakh nasledstvennoi mekhaniki”, Sovrem. mat. Fundam. napravl., 45, 2012, 43–61

[22] Zhikov V. V., “Ob odnom rasshirenii i primenenii metoda dvukhmasshtabnoi skhodimosti”, Mat. sb., 191:7 (2000), 31–72 | Zbl

[23] Zhikov V. V., “O dvukhmasshtabnoi skhodimosti”, Tr. sem. im. I. G. Petrovskogo, 23, 2003, 149–187 | Zbl

[24] Zakora D. A., “Model szhimaemoi zhidkosti Oldroita”, Sovrem. mat. Fundam. napravl., 61, 2016, 41–66

[25] Zakora D. A., Kopachevskii N. D., “O spektralnoi zadache, svyazannoi s integro-differentsialnym uravneniem vtorogo poryadka”, Uch. zap. Tavr. nats. un-ta im. V. I. Vernadskogo, 2004, no. 2, 2–18

[26] Ilyushin A. A., Pobedrya B. E., Osnovy matematicheskoi teorii termovyazkouprugosti, Nauka, M., 1970

[27] Kato T., Teoriya vozmuschenii lineinykh operatorov, Mir, M., 1972

[28] Kopachevskii N. D., “Zadacha Koshi dlya lineinogo integro-differentsialnogo uravneniya v gilbertovom prostranstve”, Uch. zap. Tavr. nats. un-ta im. V. I. Vernadskogo, 16 (2001), 139–152

[29] Kopachevskii N. D., Volterrovy integrodifferentsialnye uravneniya v gilbertovom prostranstve. Spetsialnyi kurs lektsii, «Bondarenko O. A.», Simferopol, 2012

[30] Kopachevskii N. D., Krein S. G., Ngo Zui Kan, Operatornye metody v lineinoi gidrodinamike. Evolyutsionnye i spektralnye zadachi, Nauka, M., 1989

[31] Kopachevskii N. D., Orlova L. D., Pashkova Yu. S., “Differentsialno-operatornye i integro-differentsialnye uravneniya v probleme malykh kolebanii gidrodinamicheskikh sistem”, Uch. zap. Simf. gos. un-ta, 41:2 (1995), 96–108

[32] Kopachevskii N. D., Semkina E. V., “Ob integro-differentsialnykh uravneniyakh Volterra vtorogo poryadka, nerazreshennykh otnositelno starshei proizvodnoi”, Uch. zap. Tavr. nats. un-ta im. V. I. Vernadskogo. Ser. fiz.-mat. nauki, 26:1 (2013), 52–79

[33] Krein S. G., Lineinye differentsialnye uravneniya v banakhovom prostranstve, Nauka, M., 1967

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

[35] Lykov A. V., “Nekotorye problemnye voprosy teorii teplomassoperenosa”, Problemy teplo- i massoperenosa, Nauka i tekhnika, Minsk, 1976, 9–82

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

[37] Miloslavskii A. I., Spektralnye svoistva operatornogo puchka, voznikayuschego v vyazkouprugosti, Dep. v Ukr. NIINTI, No 1229-UK87, Kharkov, 13.07.87, 53

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

[39] Rabotnov Yu. N., Elementy nasledstvennoi mekhaniki tverdykh tel, Nauka, M., 1977

[40] Radzievskii G. V., “Asimptotika raspredeleniya kharakteristicheskikh chisel operator-funktsii, analiticheskikh v ugle”, Mat. sb., 112:3 (1980), 396–420 | Zbl

[41] Rautian N. A., “Ob ogranichennosti odnogo klassa integralnykh operatorov drobnogo tipa”, Mat. sb., 200:12 (2009), 81–106 | Zbl

[42] Rautian N. A., Sb. tr. mezhd. konf., MESI, 2011

[43] Rautian N. A., “O strukture i svoistvakh reshenii integrodifferentsialnykh uravnenii, voznikayuschikh v teplofizike i akustike”, Mat. zametki, 90:3 (2011), 474–477

[44] Rautian N. A., “Polugruppy, porozhdaemye volterrovymi integro-differentsialnymi uravneniyami”, Diff. uravn., 56:9 (2020), 1226–1244 | Zbl

[45] Sandrakov G. V., “Mnogofaznye osrednennye modeli diffuzii dlya zadach s neskolkimi parametrami”, Izv. RAN. Ser. Mat., 71:6 (2007), 119–72

[46] Sanches Palensiya E., Neodnorodnye sredy i teoriya kolebanii, Mir, M., 1984

[47] Skubachevskii A. L., “Ellipticheskie zadachi s nelokalnymi usloviyami v blizi granitsy”, Mat. sb., 129:2 (1986), 279–302

[48] Skubachevskii A. L., “Ob odnom klasse funktsionalno-differentsialnykh operatorov, udovletvoryayuschikh gipoteze Kato”, Algebra i analiz, 30:2 (2018), 249–273

[49] Shkalikov A. A., “Silno dempfirovannye puchki operatorov i razreshimost sootvetstvuyuschikh operatorno-differentsialnykh uravnenii”, Mat. sb., 177:1 (1988), 96–118 | Zbl

[50] Amendola G., Fabrizio M., Golden J. M., Thermodynamics of Materials with Memory. Theory and Applications, Springer, New York—Dordrecht—Heidelberg—London, 2012 | Zbl

[51] Biot M. A., “Generalized theory of acoustic propagation in porous dissipative media”, J. Acoust. Soc. Am., 34 (1962), 1254–1264 | DOI

[52] Dafermos C. M., “Asymptotic stability in viscoelasticity”, Arch. Ration. Mech. Anal., 37 (1970), 297–308 | DOI | Zbl

[53] Davydov A. V., Tikhonov Y. A., “Study of Kelvin—Voigt models arising in viscoelasticity”, Differ. Equ., 54:12 (2018), 1620–1635 | DOI | Zbl

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

[55] Devis P. L., “Hyperbolic integrodifferential equations”, Proc. Am. Math. Soc., 47 (1975), 155–160 | DOI

[56] Devis P. L., “On the hyperbolicity of the equations of the linear theory of heat conduction for materials with memory”, SIAM J. Appl. Math., 30 (1976), 75–80 | DOI

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

[58] Di Blasio G., Kunisch K., Sinestrari 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 | Zbl

[59] Engel K.-J., Nagel R., One-parameter semigroup for linear evolution equations, Springer, New York—Berlin—Heidelberg, 1999

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

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

[62] Ismagilov R. S., Rautian N. A., Vlasov V. V., Examples of very unstable linear partial functional differential equations, arXiv: 1402.4107v1

[63] Ivanov S. A., «Wave type» spectrum of the Gurtin—Pipkin equation of the second order, arXiv: 1002.2831

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

[65] Ivanov S. A., Sheronova T. L., Spectrum of the heat equation with memory, arXiv: 0912.1818v1

[66] Kopachevsky N. D., Krein S. G., Operator approach to linear problems of hydrodynamics, v. 2, Non-self-adjoint problems for viscous fluids, Birkhäuser, Basel—Boston—Berlin, 2003 | Zbl

[67] Kopachevsky N. D., Syomkina E. V., “Linear Volterra integro-differential second-order equations unresolved with respect to the highest derivative”, Eurasian Math. J., 4:4 (2013), 64–87 | Zbl

[68] Miller R. K., “Volterra integral equations in a Banach space”, Funkcialaj Ekvac., 18 (1975), 163–194

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

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

[71] Munoz 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 | Zbl

[72] Myshkis A. D., Vlasov V. V., “On an analogy between the classifications of functional differential equations and partial differential equations”, Funct. Differ. Equ., 16:3 (2009), 545–560 | Zbl

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

[74] Pazy A., Semigroups of linear operators and applications of partial differential equations, Springer, New York etc., 1983 | Zbl

[75] Perez Ortiz R., Vlasov V. V., “Correct solvability of Volterra integrodifferential equations in Hilbert space”, Electron. J. Qual. Theory Differ Equ., 31 (2016), 1–17 | DOI

[76] Prüss J., “On linear Volterra equations of parabolic type in Banach spaces”, Trans. Am. Math. Soc., 301:2 (1987), 691–721 | DOI | Zbl

[77] Prüss J., “Bounded solutions of Volterra equations”, SIAM J. Math. Anal., 19:1 (1988), 133–149 | DOI | Zbl

[78] Prüss J., Evolutionary integral equations and applications, Birkhäuser, Basel—Baston—Berlin, 1993 | Zbl

[79] Rautian N. A., “Well-posedness of Volterra integro-differential equations with fractional exponential kernels”, Differential and Difference Equations with Applications, Springer, Cham, 2020, 517–533 | DOI | Zbl

[80] Shapiro J., Composition Operators and Classical Function Theory, Springer, New York, 1993 | Zbl

[81] Skubachevskii A. L., Elliptic Functional Differential Equations and Applications, Birkhäuser, Basel, 1997 | Zbl

[82] Vlasov V. V., Gavrikov A. A., Ivanov S. A., Knyaz'kov D. Yu., Samarin V. A., Shamaev A. S., “Spectral properties of combined media”, J. Math. Sci. (N.Y.), 164:6 (2010), 948–963 | DOI | Zbl

[83] Vlasov V. V., Rautian N. A., “Spectral analysis and representations of solutions of abstract integro-differential equations in Hilbert space”, Concrete operators, spectral theory, operators in harmonic analysis and approximation, IWOTA 11 (Sevilla, Spain, July 3–9, 2011), Birkhäuser/Springer, Basel, 2011, 517–535

[84] Vlasov V. V., Rautian N. A., “Spectral analysis of integrodifferential equations in Hilbert spaces”, J. Math. Sci. (N.Y.), 239:6 (2019), 771–787 | DOI | Zbl

[85] Vlasov V. V., Rautian N. A., “A study of operator models arising in problems of hereditary mechanics”, J. Math. Sci. (N.Y.), 244:2 (2020), 170–182 | DOI | Zbl

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

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