Parabolic equations with unknown time-dependent coefficients

A. I. Kozhanov

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 57 (2017) no. 6, pp. 961-972

Résumé

The solvability of inverse problems of finding the coefficients of a parabolic equation together with solving this equation is studied. In these problems, certain additional conditions on the boundary are used as overdetermination conditions. Existence and uniqueness theorems for regular solutions of such problems are proven.

Bibliographie

[1] Prilepko A. I., Orlovsky D. G., Vasin I. A., Methods for solving inverse problems in mathematical physics, Marcel Dekker, New York, 1999 | MR

[2] Kabanikhin S. I., Obratnye i nekorrektnye zadachi, Sibirskoe knizhnoe izdatelstvo, Novosibirsk, 2009

[3] Cannon J. R., Lin Y., “Determination of a parameter in some quasi-linear parabolic differential equations”, Inverse Problems, 4 (1988), 35–45 | DOI | MR | Zbl

[4] Kamynin V. L., Franchini E., “Ob odnoi obratnoi zadache dlya parabolicheskogo uravneniya vysokogo poryadka”, Matematicheskie zametki, 64:5 (1998), 680–691 | DOI | Zbl

[5] Kamynin V. L., Saroldi M., “Nelineinaya obratnaya zadacha dlya parabolicheskogo uravneniya vysokogo poryadka”, Zh. vychisl. matem. i matem. fiz., 38:10 (1998), 1683–1691 | Zbl

[6] Ablabekov B. S., Obratnye zadachi dlya psevdoparabolicheskikh uravnenii, Ilim, Bishkek, 2001

[7] Anikonov Yu.E., Inverse problems for kinetic and other evolution equations, VSP, Utrecht, 2001 | MR

[8] Belov Yu.Ya., Inverse problems for partial differential equations, VSP, Utrecht, 2002 | MR | Zbl

[9] Ivanchov M., Inverse problems for equations of parabolic type, Mathematical studies. Monograph Series, 10, 2003 | MR | Zbl

[10] Kozhanov A. I., “Parabolicheskie uravneniya s neizvestnym koeffitsientom, zavisyaschim ot vremeni”, Zh. vychisl. matem. i matem. fiz., 45:12 (2005), 2168–2184 | Zbl

[11] Kozhanov A. I., “Obratnye zadachi dlya uravneniya s kratnymi kharakteristikami: sluchai neizvestnogo koeffitsienta, zavisyaschego ot vremeni”, Dokl. Adygskoi (Cherkesskoi) Mezhdunarodnoi Akademii nauk, 8:1 (2005), 38–49

[12] Valitov I. R., Kozhanov A. I., “Obratnye zadachi dlya giperbolicheskikh uravnenii: sluchai neizvestnykh koeffitsientov, zavisyaschikh ot vremeni”, Vestnik NGU. Ser. Matematika, mekhanika, informatika, 6:1 (2006), 3–18

[13] Kozhanov A. I., “O razreshimosti obratnoi zadachi nakhozhdeniya starshego koeffitsienta v uravnenii sostavnogo tipa”, Vestnik Yuzhno-Uralskogo gosudarstvennogo universiteta. Matematicheskoe modelirovanie i programmirovanie, 2008, no. 1(15), 27–36

[14] Kozhanov A. I., “O razreshimosti obratnykh zadach vosstanovleniya koeffitsientov v uravneniyakh sostavnogo tipa”, Vestnik NGU. Ser. Matematika, mekhanika, informatika, 8:3 (2008), 81–99

[15] Kozhanov A. I., “O razreshimosti koeffitsientnykh obratnykh zadach dlya nekotorykh uravnenii sobolevskogo tipa”, Nauchnye vedomosti Belgorodskogo universiteta, 5(76):18 (2010), 88–97 | Zbl

[16] Pavlov S. S., “Nelineinye obratnye zadachi dlya mnogomernykh giperbolicheskikh uravnenii s integralnym pereopredeleniem”, Matem. zametki YaGU, 19:2 (2011), 128–154

[17] Telesheva L. A., “O razreshimosti obratnoi zadachi dlya parabolicheskogo uravneniya vysokogo poryadka s neizvestnym koeffitsientom pri proizvodnoi po vremeni”, Matem. zametki YaGU, 18:2 (2011), 180–201 | Zbl

[18] Kozhanov A. I., “O razreshimosti nekotorykh nelokalnykh i svyazannykh s nimi obratnykh zadach dlya parabolicheskikh uravnenii”, Matem. zametki YaGU, 18:2 (2011), 64–78 | Zbl

[19] Slodicka M., “Determination of a solely time-dependent source in a semilinear parabolic problem by meand of boundary measurements”, J. Comput. and Appl. Math., 289 (2015), 433–440 | DOI | MR | Zbl

[20] Safiullova R. R., “Obratnaya zadacha dlya giperbolicheskogo uravneniya vtorogo poryadka s neizvestnym koeffitsientom, zavisyaschim ot vremeni”, Vestnik Yuzhno-Uralskogo gosudarstvennogo un-ta. Matematicheskoe modelirovanie i programmirovanie, 6:4 (2013), 73–86

[21] Namsaraeva G. V., “O razreshimosti obratnykh zadach dlya psevdoparabolicheskikh uravnenii”, Matematicheskie zametki YaGU, 20:2 (2013), 111–137 | Zbl

[22] Megraliev Ya. T., “Obratnaya zadacha dlya uravneniya Bussineska-Lyava s dopolnitelnym integralnym usloviem”, Sib. zh. industr. mat., 16:1 (2013), 75–83 | Zbl

[23] Pyatkov S. G., Safonov E. I., “Ob opredelenii funktsii istochnika v matematicheskikh modelyakh konvektsii-diffuzii”, Matematicheskie zametki SVFU, 21:2 (2014), 117–130

[24] Shergin S. N., Pyatkov S. G., “O nekotorykh klassakh obratnykh zadach dlya psevdoparabolicheskikh uravnenii”, Matem. zametki SVFU, 21:2 (2014), 106–116 | Zbl

[25] Dzhenaliev M. T., K teorii lineinykh kraevykh zadach dlya nagruzhennykh differentsialnykh uravnenii, Institut teoreticheskoi i prikladnoi matematiki, Almaty, 1995

[26] Nakhushev A. M., Nagruzhennye uravneniya i ikh primeneniya, Nauka, M., 2012

[27] Vladimirov V. S., Uravneniya matematicheskoi fiziki, Nauka, M., 1976

[28] Ladyzhenskaya O. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya ellipticheskogo tipa, Nauka, M., 1973

[29] Ladyzhenskaya O. A., Solonnikov V. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, Nauka, M., 1967

[30] Trenogin V. A., Funktsionalnyi analiz, Nauka, M., 1980

[31] Sobolev S. L., Nekotorye primeneniya funktsionalnogo analiza v matematicheskoi fizike, Nauka, M., 1988