Geodesic
Three-dimensional initial boundary value problem for a parabolic-hyperbolic equation with a degenerate parabolic part
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 4 (2023), pp. 51-64.

Voir la notice de l'article provenant de la source Math-Net.Ru

An initial-boundary value problem is studied for an inhomogeneous equation of mixed parabolic-hyperbolic type in three variables in a rectangular parallelepiped. A criterion for the uniqueness of a solution is established. The solution is constructed as the sum of an orthogonal series. When justifying the convergence of the series, the problem of small denominators of two natural arguments arose. Estimates on the separation of small denominators from zero with the corresponding asymptotics are established. These estimates made it possible to substantiate the convergence of the constructed series in the class of regular solutions of this equation. The stability of the solution with respect to the boundary function is established.
Keywords: equation of mixed parabolic-hyperbolic type, initial-boundary value problem, uniqueness, series, small denominators, stability.
Mots-clés : existence 
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S. N. Sidorov. Three-dimensional initial boundary value problem for a parabolic-hyperbolic equation with a degenerate parabolic part. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 4 (2023), pp. 51-64. http://geodesic.mathdoc.fr/item/IVM_2023_4_a4/

[1] Ladyzhenskaya O.A., Stupyalis L., “Ob uravneniyakh smeshannogo tipa”, Vestn. LGU. Ser. matem., mekhan. i astr., 19:4 (1965), 38–46 | Zbl

[2] Stupyalis L., “Nachalno-kraevye zadachi dlya uravnenii smeshannogo tipa”, Tr. MIAN SSSR, 127, 1975, 115–145 | MR | Zbl

[3] Kapustin N.Yu., “Zadacha Trikomi dlya parabolo-giperbolicheskogo uravneniya s vyrozhdayuscheisya giperbolicheskoi chastyu. I”, Differents. uravneniya, 23:1 (1987), 72–78 | MR | Zbl

[4] Kapustin N.Yu., “Zadacha Trikomi dlya parabolo-giperbolicheskogo uravneniya s vyrozhdayuscheisya giperbolicheskoi chastyu. II”, Differents. uravneniya, 24:8 (1988), 1379–1386 | MR | Zbl

[5] Kapustin N.Yu., Moiseev E.I., “O spektralnoi zadache iz teorii parabolo-giperbolicheskogo uravneniya teploprovodnosti”, Dokl. RAN, 352:4 (1997), 451–454 | MR | Zbl

[6] Sabitov K.B., “Zadacha Trikomi dlya uravneniya smeshannogo parabolo-giperbolicheskogo tipa v pryamougolnoi oblasti”, Matem. zametki, 86:2 (2009), 273–279 | DOI | Zbl

[7] Sabitov K.B., Pryamye i obratnye zadachi dlya uravnenii smeshannogo parabolo-giperbolicheskogo tipa, Nauka, M., 2016 | MR

[8] Sabitov K.B., “Nachalno-granichnaya i obratnye zadachi dlya neodnorodnogo uravneniya smeshannogo parabolo-giperbolicheskogo uravneniya”, Matem. zametki, 102:3 (2017), 415–435 | DOI | MR | Zbl

[9] Sidorov S.N., “Nelokalnaya zadacha dlya vyrozhdayuschegosya parabolo-giperbolicheskogo uravneniya”, Dokl. AMAN, 14:3 (2012), 34–44

[10] Sabitov K.B., Sidorov S.N., “Ob odnoi nelokalnoi zadache dlya vyrozhdayuschegosya parabolo-giperbolicheskogo uravneniya”, Differents. uravneniya, 50:3 (2014), 356–365 | DOI | Zbl

[11] Sidorov S.N., “Nelokalnye zadachi dlya uravneniya smeshannogo parabolo-giperbolicheskogo tipa so stepennym vyrozhdeniem”, Izv. vuzov. Matem., 12 (2015), 55–64

[12] Sabitov K.B., Sidorov S.N., “Nachalno-granichnaya zadacha dlya neodnorodnykh vyrozhdayuschikhsya uravnenii smeshannogo paraboliko-giperbolicheskogo tipa”, Itogi nauki i tekhn. Ser. Sovrem. matem. i ee pril. Temat. obz., 137, 2017, 26–60

[13] Sabitov K.B., Sidorov S.N., “Initial-Boundary Problem for a Three-Dimensional Inhomogeneous Equation of Parabolic-Hyperbolic Typ”, Lobachevskii J. Math., 41:11 (2020), 2257–2268 | DOI | MR | Zbl

[14] Aldashev S.A., “Zadacha Dirikhle dlya odnogo klassa vyrozhdayuschikhsya mnogomernykh giperbolo-parabolicheskikh uravnenii”, Izv. Sarat. un-ta. Nov. ser. Ser. Matem. Mekhan. Informatika, 17:3 (2017), 244–254 | MR | Zbl

[15] Aldashev S.A., “Kriterii odnoznachnoi razreshimosti spektralnoi zadachi Dirikhle dlya odnogo klassa mnogomernykh giperbolo-parabolicheskikh uravnenii”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-matem. nauki, 22:2 (2018), 225–235 | DOI | Zbl

[16] Aldashev S.A., Kanapyanova Z.N., “Korrektnost smeshannoi zadachi dlya vyrozhdayuschikhsya trekhmernykh giperbolo-parabolicheskikh uravnenii”, Vestn. SamU. Estestvennonauchn. ser., 25:4 (2019), 7–13 | MR | Zbl

[17] Aldashev S.A., “Korrektnost smeshannoi zadachi dlya mnogomernogo giperbolo-parabolicheskogo uravneniya”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-matem. nauki, 24:3 (2020), 574–582 | DOI | Zbl

[18] Sabitov K.B., Safin E.M., “Obratnaya zadacha dlya uravneniya smeshannogo parabolo-giperbolicheskogo tipa”, Matem. zametki, 87:6 (2010), 907–918 | DOI

[19] Sabitov K.B., Sidorov S.N., “Obratnaya zadacha dlya vyrozhdayuschegosya parabolo-giperbolicheskogo uravneniya s nelokalnym granichnym usloviem”, Izv. vuzov. Matem., 2015, no. 1, 46–59 | Zbl

[20] Sidorov S.N., “Obratnye zadachi dlya uravneniya smeshannogo parabolo-giperbolicheskogo tipa s vyrozhdayuscheisya parabolicheskoi chastyu”, Sib. elektron. matem. izv., 16 (2019), 144–157 | Zbl

[21] Sidorov S.N., “Obratnye zadachi dlya vyrozhdayuschegosya smeshannogo parabolo-giperbolicheskogo uravneniya po nakhozhdeniyu somnozhitelei pravykh chastei, zavisyaschikh ot vremeni”, Ufimsk. matem. zhurn., 11:1 (2019), 72–86 | Zbl

[22] Bukhshtab A.A., Teoriya chisel, Lan, SPb., 2008

[23] Khinchin A.Ya., Tsepnye drobi, Nauka, M., 1978 | MR

[24] Zigmund A., Trigonometricheskie ryady, v. 1, Mir, M., 1965