Dirichlet boundary value problem for a third order parabolic-hyperbolic equation with degenerating type and order in the hyperbolicity domain

Zh. A. Balkizov

Ufa mathematical journal, Tome 9 (2017) no. 2, pp. 25-39

Résumé

In the work we study an analogue of Tricomi equation for a third order parabolic-hyperbolic equation with smaller derivatives having multiple characteristics. Under certain conditions for the given functions and parameters involved in the considered equation, we prove unique solvability theorem for the studied problem. The uniqueness of the solution is proved by means of the generalized Tricomi method, while the existence is proved via the method of integral equations.

Détail

Keywords: Degenerate hyperbolic equation, equation with multiple characteristics, third order parabolic-hyperbolic equation, Dirichlet boundary value problem, Tricomi method, second kind integral Volterra equation, second kind integral Fredholm equation.
Mots-clés : analogue of Tricomi equation

Bibliographie

[1] Bitsadze A.V., Uravneniya smeshannogo tipa, Izdatelstvo AN SSSR, M., 1949, 155 pp.

[2] Lykov A.V., “Primenenie metodov termodinamiki neobratimykh protsessov k issledovaniyu teplo i massoobmena”, Inzhenerno-fizicheskii zhurnal, 9:3 (1955), 287–304

[3] Nakhushev A.M., Uravneniya matematicheskoi biologii, Vysshaya shkola, M., 301 pp.

[4] Nakhushev A.M., Drobnoe ischislenie i ego primenenie, Fizmatlit, M., 2003, 272 pp.

[5] Bers L., Matematicheskie voprosy dozvukovoi i okolozvukovoi gazovoi dinamiki, Inostrannaya literatura, M., 1951, 208 pp.

[6] Frankl F.I., Izbrannye trudy po gazovoi dinamike, Nauka, M., 1973, 711 pp.

[7] Kalmenov T. Sh., “Kriterii edinstvennosti resheniya zadachi Darbu dlya odnogo vyrozhdayuschegosya giperbolicheskogo uravneniya”, Differents. uravneniya, 7:1 (1971), 178–181 | Zbl

[8] Kalmenov T. Sh., “O zadache Darbu dlya odnogo vyrozhdayuschegosya giperbolicheskogo uravneniya”, Differents. uravneniya, 10:1 (1974), 59–68 | Zbl

[9] Kalmenov T. Sh., “Kriterii nepreryvnosti resheniya zadachi Gursa dlya odnogo vyrozhdayuschegosya giperbolicheskogo uravneniya”, Differents. uravneniya, 8:1 (1972), 41–55

[10] Smirnov M.M., Vyrozhdayuschiesya giperbolicheskie uravneniya, Vysheishaya shkola, Minsk, 1977, 150 pp.

[11] Repin O.A., Kraevye zadachi so smescheniem dlya uravnenii giperbolicheskogo i smeshannogo tipov, Izdatelstvo Saratovskogo universiteta, Saratov, 1992, 161 pp.

[12] Nakhushev A.M., Zadachi so smescheniem dlya uravnenii v chastnykh proizvodnykh, Nauka, M., 2006, 287 pp.

[13] Kalmenov T. Sh., K teorii nachalno-kraevykh zadach dlya differentsialnykh uravnenii, Tsikl nauchnykh rabot T. Sh. Kalmenova, Institut matematiki i matematicheskogo modelirovaniya, Almaty, 2013, 306 pp.

[14] Dzhuraev T. D., Kraevye zadachi dlya uravnenii smeshannogo i smeshanno-sostavnogo tipov, FAN, Tashkent, 1979, 238 pp.

[15] L. Cattabriga, “Un Problema al contorno per una equazione parabolica di ordine dispari”, Annali della Scuola Normale Superiore di Pisa, 13:2 (1959), 163–203 | MR | Zbl

[16] Irgashev Yu., “Nekotorye kraevye zadachi dlya uravnenii tretego poryadka s kratnymi kharakteristikami”, Sbornik nauchnykh trudov «Kraevye zadachi dlya differentsialnykh uravnenii i ikh prilozheniya», FAN, Tashkent, 1976, 17–31

[17] Dzhuraev T. D., Abdinazarov S., “Kraevye zadachi tipa zadachi Bitsadze-Samarskogo dlya uravnenii tretego poryadka s kratnymi kharakteristikami”, Izvestiya AN Uzbekskoi SSR, 1981, no. 1, 8–11

[18] Abdinazarov S., “Obschie kraevye zadachi dlya uravneniya tretego poryadka s kratnymi kharakteristikami”, Differents. uravneniya, 17:1 (1981), 3–12

[19] Gelfand I. I., “Nekotorye voprosy analiza i differentsialnykh uravnenii”, UMN, 14:3(87) (1959), 3–19

[20] Bitsadze A.V., Uravneniya smeshannogo tipa, Izdatelstvo AN SSSR, M., 1959, 164 pp.

[21] Bitsadze A.V., “K teorii odnogo klassa uravnenii smeshannogo tipa”, Nekotorye problemy matematiki i mekhaniki, Nauka, L., 1970, 112–119

[22] Repin O.A., Kumykova S.K., “Zadacha so smescheniem dlya uravneniya tretego poryadka s razryvnymi koeffitsientami”, Vestnik Samarskogo gosudarstvennogo tekhnicheskogo universiteta. Seriya fiziko-matematicheskie nauki, 2012, no. 4 (29), 17–25 | DOI

[23] Repin O.A., Kumykova S.K., “Nelokalnaya zadacha dlya uravneniya smeshannogo tipa tretego poryadka s obobschennymi operatorami drobnogo integro-differentsirovaniya proizvolnogo poryadka”, Vestnik Samarskogo gosudarstvennogo tekhnicheskogo universiteta. Seriya fiziko-matematicheskie nauki, 2011, no. 4(25), 25–36 | DOI

[24] Eleev V.A., Kumykova S.K., “Vnutrennekraevaya zadacha dlya uravneniya smeshannogo tipa tretego poryadka s kratnymi kharakteristikami”, Izvestiya Kabardino-Balkarskogo nauchnogo tsentra RAN, 2010, no. 5, 5–14

[25] Balkizov Zh.A., “Lokalnye i nelokalnye kraevye zadachi dlya uravneniya smeshannogo tipa tretego poryadka s operatorom Trikomi v giperbolicheskoi chasti”, Vestnik Samarskogo gosudarstvennogo tekhnicheskogo universiteta, 2008, no. 2(17), 21–28 | DOI

[26] Balkizov Zh.A., “Kraevye zadachi dlya uravneniya smeshannogo tipa tretego poryadka s operatorom Gellerstedta v giperbolicheskoi chasti”, Izvestiya Kabardino-Balkarskogo gosudarstvennogo universiteta, 1:1 (2011), 21–33

[27] Balkizov Zh.A., “Analog zadachi Trikomi dlya uravneniya parabolo-giperbolicheskogo tipa tretego poryadka s operatorom Gellerstedta v oblasti giperbolichnosti”, Doklady Adygskoi (Cherkesskoi) mezhdunarodnoi akademii nauk, 16:2 (2014), 20–27 | Zbl

[28] Balkizov Zh.A., “Nelokalnaya kraevaya zadacha dlya modelnogo uravneniya parabolo-giperbolicheskogo tipa tretego poryadka”, Doklady Adygskoi (Cherkesskoi) mezhdunarodnoi akademii nauk, 17:4 (2015), 9–20 | Zbl

[29] Zolina L.A., “O kraevoi zadache dlya modelnogo uravneniya giperbolo-parabolicheskogo tipa”, Zh. vychisl. matem. i matem. fiz., 6:6 (1966), 991–1001 | Zbl

[30] Bzhikhatlov Kh.G., Nakhushev A.M., “Ob odnoi kraevoi zadache dlya uravneniya smeshannogo parabolo-giperbolicheskogo tipa”, Doklady Akademii nauk SSSR, 183:2 (1968), 261–264 | Zbl

[31] Samko S.G., Kilbas A.A., Marichev O.I., Integraly i proizvodnye drobnogo poryadka i nekotorye ikh prilozheniya, «Nauka i tekhnika», Minsk, 1987, 588 pp.

[32] Alikhanov A.A., “Apriornye otsenki reshenii kraevykh zadach dlya uravnenii drobnogo poryadka”, Differents. uravneniya, 46:5 (2010), 658–664 | Zbl

[33] Nakhushev A.M., “K teorii lineinykh kraevykh zadach dlya uravneniya vtorogo poryadka smeshannogo giperbolo-parabolicheskogo tipa”, Differents. uravneniya, 14:1 (1978), 56–73

[34] E.M. Wright, “The generalized Bessel function of order greater than one”, Quart. J. Math. Oxford Ser., 11 (1940), 36–48 | DOI | MR

[35] Pskhu A.V., “Nachalnaya zadacha dlya lineinogo obyknovennogo differentsialnogo uravneniya drobnogo poryadka”, Matematicheskii sbornik, 202:4 (2011), 111–122 | DOI | Zbl