Properties of solutions of integro-differential equations arising in heat and~mass transfer theory

V. V. Vlasov ; N. A. Rautian

Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 75 (2014) no. 2, pp. 219-243.

Résumé

The aim of the present paper is to study the asymptotic behavior of solutions of integro-differential equations on the basis of spectral analysis of their symbols. To this end, we obtain representations of strong solutions of these equations in the form of a sum of terms corresponding to the real and nonreal parts of the spectrum of the operator functions that are the symbols of these equations. These representations are new for the class of integro-differential equations considered in the paper.

Bibliographie

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

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

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

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

[5] Prüss J., Evolutionary integral equations and applications, Monographs in Mathematics, 87, Birkhäuser Verlag, Basel, 1993 | DOI | MR | Zbl

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

[7] Wu J., Theory and applications of partial functional differential equations, Appl. Math. Sci., 119, Springer-Verlag, New York, 1996 | MR

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

[9] Vlasov V. V., “O razreshimosti i otsenkakh reshenii funktsionalno-differentsialnykh uravnenii v prostranstvakh Soboleva”, Tr. MIAN, 227, 1999, 109–121 | MR | Zbl

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

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

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

[13] Vlasov V. V., Shmatov K. I., “Korrektnaya razreshimost uravnenii giperbolicheskogo tipa s posledeistviem v gilbertovom prostranstve”, Tr. MIAN, 243, 2003, 127–137 | MR | Zbl

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

[15] Vlasov V. V., Medvedev D. A., “Funktsionalno-differentsialnye uravneniya v prostranstvakh Soboleva i svyazannye s nimi voprosy spektralnoi teorii”, Sovr. matem. Fundam. napravleniya, 30, 2008, 3–173

[16] Vlasov V. V., Vu Dzh., Kabirova G. R., “Korrektnaya razreshimost i spektralnye svoistva abstraktnykh giperbolicheskikh uravnenii s posledeistviem”, Sovr. matem. Fundam. napravleniya, 35, 2010, 44–59 | MR | Zbl

[17] Miller R. K., “Volterra integral equations in a Banach space”, Funkcial. Ekvac., 18 (1975), 163–193 | MR | Zbl

[18] Lions J. L., Magenes E., Non-homogeneous boundary value problems and applications, Springer-Verlag, New York, 1972

[19] Gurtin M. E., Pipkin A. C., “A general theory of heat conduction with finite wave speeds”, Arch. Rat. Mech. Anal., 31 (1968), 113–126 | DOI | MR | Zbl

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

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

[22] Rautian N. A., “O strukture i svoistvakh reshenii integrodifferentsialnykh uravnenii, voznikayuschikh v teplofizike i akustike”, Matem. zametki, 90:3 (2011), 470–473 | DOI | MR | Zbl

[23] Ivanov S., Sheronova T., Spectrum of the heat equation with memory, arXiv: 0912.1818v1[math.CV]

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

[25] 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

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

[27] Vlasov V. V., Rautian N. A., Shamaev A. S., “Spektralnyi analiz i korrektnaya razreshimost abstraktnykh integrodifferentsialnykh uravnenii, voznikayuschikh v teplofizike i akustike”, Sovr. matem. Fundam. napravleniya, 39, 2011, 36–65

[28] Vlasov V. V., Rautian N. A., “Correct solvability of integro-differential equations arising in the theory of heat transfer and acoustics”, Func. Diff. Equations, 19:1–2 (2012), 207–224 | MR

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

[30] Eremenko A., Ivanov S., “Spectra of the Gurtin–Pipkin type equations”, SIAM J. Math. Anal., 43:5 (2011), 2296–2306 | DOI | MR | Zbl

[31] Vlasov V. V., Rautian N. A., Shamaev A. S., “Issledovanie operatornykh modelei, voznikayuschikh v zadachakh nasledstvennoi mekhaniki”, Sovr. matem. Fundam. napravleniya, 45, 2012, 43–61

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

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

[34] Vlasov V. V., Rautian N. A., “Spectral analysis and representations of solutions of abstract integrodifferential equations in Hilbert space”, Operator Theory: Advances and Applications, 236, 2014, 519–537 | MR

[35] Vlasov V. V., Medvedev D. A., Rautian N. A., Funktsionalno-differentsialnye uravneniya v prostranstvakh Soboleva i ikh spektralnyi analiz, Sovremennye problemy matematiki i mekhaniki, VIII, no. 1, Izdatelstvo MGU, M., 2011

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

[37] Fedoryuk M. V., Asimptotika, integraly i ryady, Nauka, M., 1987 | MR

[38] Bellman R., Kuk K., Differentsialno-raznostnye uravneniya, Mir, M., 1967 | MR | Zbl