Spectral analysis and correct solvability of abstract integrodifferential equations arising in thermophysics and acoustics

V. V. Vlasov ; N. A. Rautian ; A. S. Shamaev

Contemporary Mathematics. Fundamental Directions, Partial differential equations, Tome 39 (2011), pp. 36-65.

Résumé

In the present paper, we study integrodifferential equations with unbounded operator coefficients in Hilbert spaces. The principal part of the equation is an abstract hyperbolic equation perturbed by summands with Volterra integral operators. These equations represent an abstract form of the Gurtin–Pipkin integrodifferential equation describing the process of heat conduction in media with memory and the process of sound conduction in viscoelastic media and arise in averaging problems in perforated media (the Darcy law). The correct solvability of initial-boundary problems for the specified equations is established in weighted Sobolev spaces on a positive semiaxis. Spectral problems for operator-functions are analyzed. Such functions are symbols of these equations. The spectrum of the abstract integrodifferential Gurtin–Pipkin equation is investigated.

Bibliographie

[1] Vlasov V. V., O nekotorykh prostranstvakh vektor-funktsii, golomorfnykh v ugle, dep. v VINITI, No 4177-81, M., 1981

[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., “O razreshimosti i otsenkakh reshenii funktsionalno-differentsialnykh uravnenii v prostranstvakh Soboleva”, Tr. mat. in-ta im. V. A. Steklova, 227, 1999, 109–121 | MR | Zbl

[4] Vlasov V. V., “O korrektnoi razreshimosti abstraktnykh parabolicheskikh uravnenii s posledeistviem”, Dokl. RAN, 415:2 (2007), 151–154 | MR | Zbl

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

[6] Vlasov V. V., Vu Dzh., Kabirova G. R., “Korrektnaya razreshimost i spektralnye svoistva abstraktnykh giperbolicheskikh uravnenii s posledeistviem”, Sovrem. mat. Fundam. napravl., 35, 2010, 44–59

[7] Vlasov V. V., Gavrikov A. A., Ivanov C. A., Knyazkov D. Yu., Samarin V. A., Shamaev A. S., “Spektralnye svoistva kombinirovannykh sred”, Sovrem. probl. mat. i mekh., 5:1 (2009), 134–155

[8] 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

[9] Vlasov V. V., Rautian N. A., “Korrektnaya razreshimost i spektralnyi analiz abstraktnykh giperbolicheskikh integrodifferentsialnykh uravnenii”, Tr. sem. im. I. G. Petrovskogo (to appear)

[10] 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

[11] Vlasov V. V., Shmatov K. I., “Korrektnaya razreshimost uravnenii giperbolicheskogo tipa s zapazdyvaniem v gilbertovom prostranstve”, Tr. Mat. in-ta im. V. A. Steklova, 243, 2003, 127–137 | MR | Zbl

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

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

[14] Iosida K., Funktsionalnyi analiz, Mir, M., 1967 | MR

[15] Kosmodemyanskii D. A., Shamaev A. S., “O nekotorykh spekralnykh zadachakh v poristykh sredakh, nasyschennykh zhidkostyu”, Sovrem. mat. Fundam. napravl., 17, 2006, 88–109 | MR

[16] Lions Zh. P., Madzhenes E., Neodnorodnye granichnye zadachi i ikh prilozheniya, M., 1971

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

[18] Markus A. S., Vvedenie v spektralnuyu teoriyu polinomialnykh operatornykh puchkov, Shtiintsa, Kishinev, 1986 | MR | Zbl

[19] Miloslavskii A. I., Spektralnye svoistva operatornogo puchka, voznikayuschego v vyazkouprugosti, dep. v Ukr. NIINTI, No 1229-UK87, Kharkov, 1987

[20] Mikhailov V. P., Differentsialnye uravneniya v chastnykh proizvodnykh, Nauka, M., 1976 | MR | Zbl

[21] Mikhlin C. G., Lineinye uravneniya v chastnykh proizvodnykh, Vysshaya shkola, M., 1977 | MR

[22] Palin V. V., Radkevich E. V., “Zakony sokhraneniya i ikh giperbolicheskie regulyarizatsii”, Sovrem. probl. mat. i mekh., 5:1 (2009), 88–115 | MR

[23] Sandrakov G. V., “Spektralnye svoistva odnorodnykh modelei diffuzii v silno neodnorodnykh sredakh”, Dokl. RAN. Ser. mat., 411:2 (2006), 167–170 | MR | Zbl

[24] Sandrakov G. V., “Mnogofaznye osrednennye modeli diffuzii dlya zadach s neskolkimi parametrami”, Izv. RAN. Ser. mat., 71:6 (2007), 119–182 | MR | Zbl

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

[26] Avdonin S. A., Ivanov S. A., Families of exponentials. Method of moments in contrallability problems for distributed parameter systems, Cambridge University Press, Cambridge, 1995 | MR | Zbl

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

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

[29] Di Blasio G., “Parabolic Volterra equations of convolution type”, J. Integral Equations 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. Rat. Mech. Anal., 31 (1968), 113–126 | DOI | MR | Zbl

[33] Ivanov S. A., “Wave type” spectrum of the Gurtin–Pipkin equation of the second order, arXiv: 1002.2831

[34] 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

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

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

[37] Kunisch K., Shappacher W., “Necessary conditions for partial differential equations with delay to generate $l_0$-semigroup”, J. Differential Equations, 50 (1983), 49–79 | DOI | MR | Zbl

[38] Kopachevsky N. D., Krein S. G., Operator approach to linear problems of hydrodynamics, v. 2, Nonself-adjoint problems for viscous fluids, Birhäuser Verlag, Basel, 2003 | MR | Zbl

[39] Medvedev D. A., Vlasov V. V., Wu J., “Solvability and structural properties of abstract neutral functional differential equations”, J. Funct. Differential Equations, 15:3–4 (2008), 249–272 | MR | Zbl

[40] Meirmanov A., “Acoustic and filtration properties of a thermoelastic porous medium: Biot's equations of thermo-poroelasticity”, Sb. Math., 199:3 (2008), 361–384 | DOI | MR | Zbl

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

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

[43] 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

[44] Nguetseng G., “A general convergence result for a functional related to the theory of homogenezation”, SIAM J. Math. Anal., 21:6 (1990), 1396–1414 | DOI | MR

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

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

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

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