Geodesic
Boundary-value problems for a characteristically loaded hyperbolic-parabolic equation
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference on Mathematical Modelling in Applied Sciences — ICMMAS'19. Belgorod, August 20–24, 2019, Tome 195 (2021), pp. 127-138.

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In this paper, we consider analogs of the Tricomi problem, problems with displacement, and problems with integral conditions in the hyperbolic part for a certain characteristically loaded model equation of mixed hyperbolic-parabolic type. We find sufficient conditions of the existence of a unique solution. The uniqueness and existence of a solution are proved by the Tricomi method and the method of integral equations, respectively. We give an example, which shows that the violation of these conditions leads to nonuniqueness of the solution.
Keywords: loaded equation, mixed-type equation, hyperbolic-parabolic equation, Tricomi problem, nonlocal problem, problem with displacement, integral condition. 
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K. U. Khubiev. Boundary-value problems for a characteristically loaded hyperbolic-parabolic equation. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference on Mathematical Modelling in Applied Sciences — ICMMAS'19. Belgorod, August 20–24, 2019, Tome 195 (2021), pp. 127-138. http://geodesic.mathdoc.fr/item/INTO_2021_195_a14/

[1] Balkizov Zh. A., “Ob odnoi kraevoi zadache tipa zadachi Trikomi dlya uravneniya smeshannogo parabolo-giperbolicheskogo tipa vtorogo poryadka s tremya smescheniyami v giperbolicheskoi chasti oblasti”, Nauch. ved. BelGU. Ser. Mat. Fiz., 51:1 (2019), 5–14

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

[3] Dzhenaliev M. T., Ramazanov M. I., Nagruzhennye uravneniya kak vozmuscheniya differentsialnykh uravnenii, Almaty, 2010

[4] Eleev V. A., “O nekotorykh kraevykh zadachakh so smescheniem dlya odnogo uravneniya smeshannogo parabolo-giperbolicheskogo tipa”, Differ. uravn., 14:1 (1978), 22–29 | MR | Zbl

[5] Eleev V. A., “O nekotorykh kraevykh zadachakh dlya smeshannykh nagruzhennykh uravnenii vtorogo i tretego poryadka”, Differ. uravn., 30:2 (1994), 230–237 | MR | Zbl

[6] Zhegalov V. I., “Kraevaya zadacha dlya uravneniya smeshannogo tipa s granichnymi usloviyami na obeikh kharakteristikakh i s razryvami na perekhodnoi linii”, Uch. zap. Kazan. un-ta., 122:3 (1962), 3–16 | MR | Zbl

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

[8] Kozhanov A. I., “O razreshimosti prostranstvenno nelokalnykh kraevykh zadach dlya lineinykh giperbolicheskikh uravnenii vtorogo poryadka”, Dokl. RAN., 427:6 (2009), 747–749 | Zbl

[9] Kumykova S. K., “O nekotorykh kraevykh zadachakh so smescheniem dlya uravneniya Lavrenteva—Bitsadze”, Differ. uravn., 9:1 (1973), 106–114 | MR | Zbl

[10] Mirsaburov M., “Zadacha s nedostayuschim usloviem smescheniya dlya uravneniya Gellerstedta s singulyarnym koeffitsientom”, Izv. vuzov. Mat., 2018, no. 5, 52–63 | Zbl

[11] Nakhushev A. M., Uravneniya matematicheskoi biologii, Vysshaya shkola, M., 1995

[12] Nakhushev A. M., Drobnoe ischislenie i ego primenenie, Fizmatlit, M., 2003

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

[14] Nakhushev A. M., Nagruzhennye uravneniya i ikh primenenie, Nauka, M., 2012

[15] Nakhushev A. M., “O nekotorykh kraevykh zadachakh dlya smeshannykh nagruzhennykh uravnenii vtorogo i tretego poryadka”, Differ. uravn., 5:1 (1969), 44–59 | MR | Zbl

[16] Nakhushev A. M., “O zadache Darbu dlya odnogo vyrozhdayuschegosya nagruzhennogo integrodifferentsialnogo uravneniya vtorogo poryadka”, Differ. uravn., 12:1 (1976), 103–108 | MR | Zbl

[17] Nakhushev A. M., “Ob odnom priblizhennom metode resheniya kraevykh zadach dlya differentsialnykh uravnenii i ego prilozheniya k dinamike pochvennoi vlagi i gruntovykh vod”, Differ. uravn., 18:1 (1982), 72–81 | MR | Zbl

[18] Nakhushev A. M., “O nelokalnykh zadachakh so smescheniem i ikh svyazi s nagruzhennymi uravneniyami”, Differ. uravn., 21:1 (1985), 92–101 | MR | Zbl

[19] Ogorodnikov E. N., “Nekotorye kharakteristicheskie zadachi dlya sistem nagruzhennykh differentsialnykh uravnenii i ikh svyaz s nelokalnymi kraevymi zadachami”, Vestn. Samar. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki., 19 (2003), 22–28

[20] Pulkina L. S., “Kraevye zadachi dlya giperbolicheskogo uravneniya s nelokalnymi usloviyami I i II roda”, Izv. vuzov. Mat., 2012, no. 4, 74–83 | MR | Zbl

[21] Repin O. A., Kraevye zadachi so smescheniem dlya uravnenii giperbolicheskogo i smeshannogo tipov, Samara, 1992

[22] Sabitov K. B., “Nachalno-granichnaya zadacha dlya parabolo-giperbolicheskogo uravneniya s nagruzhennymi slagaemymi”, Izv. vuzov. Mat., 6 (2015), 31–42 | Zbl

[23] Sabitova Yu. K., “Zadacha Dirikhle dlya uravneniya Lavrenteva—Bitsadze s nagruzhennymi slagaemymi”, Izv. vuzov. Mat., 9 (2018), 42–58 | Zbl

[24] Tarasenko A. V., “O razreshimosti nelokalnoi zadachi dlya nagruzhennogo parabolo-giperbolicheskogo uravneniya”, Izv. vuzov. Mat., 2018, no. 3, 62–69 | Zbl

[25] Tikhonov A. N., Vasileva A. B., Sveshnikov A. G., Differentsialnye uravneniya, Nauka, M., 1980 | MR

[26] Khubiev K. U., “Analog zadachi Trikomi i zadacha so smescheniem dlya modelnogo nagruzhennogo uravneniya giperbolo-parabolicheskogo tipa”, Dokl. Adyg. (Cherkes.) Mezhdunar. akad. nauk., 10:2 (2008), 67–71

[27] Khubiev K. U., “Zadacha s integralnym usloviem v giperbolicheskoi chasti dlya kharakteristicheski nagruzhennogo giperbolo-parabolicheskogo uravneniya”, Mat. zametki SVFU., 23:4 (2016), 91–98 | MR | Zbl

[28] Khubiev K. U., “Analog zadachi Trikomi dlya kharakteristicheski nagruzhennogo uravneniya giperbolo-parabolicheskogo tipa s peremennymi koeffitsientami”, Ufim. mat. zh., 9:2 (2017), 94–103 | MR | Zbl

[29] Khubiev K. U., “Vnutrennekraevaya zadacha dlya uravneniya giperbolo-parabolicheskogo tipa s operatorom drobnoi diffuzii”, Nauch. ved. BelGU. Ser. Mat. Fiz., 49:27 (276) (2017), 52–56

[30] Khubiev K. U., “Zadachi so smescheniem dlya nagruzhennogo uravneniya giperbolo-parabolicheskogo tipa s operatorom drobnoi diffuzii”, Vestn. Udmurt. un-ta. Mat. Mekh. Kompyut. nauki., 28:1 (2018), 82–90 | MR | Zbl

[31] Khubiev K. U., “Kraevaya zadacha dlya nagruzhennogo uravneniya giperbolo-parabolicheskogo tipa s vyrozhdeniem poryadka v oblasti ego giperbolichnosti”, Itogi nauki tekhn. Sovr. mat. prilozh. Temat. obz., 149 (2018), 113–117 | MR

[32] Attaev A. Kh., “On a characteristic problem for a loaded hyperbolic equation”, Bull. Karaganda Univ. Math. Ser., 4 (96) (2019), 15–21 | DOI

[33] Baltaeva U., Agarwal P., “Boundary-value problems for the third-order loaded equation with noncharacteristic type-change boundaries”, Math. Meth. Appl. Sci., 41:9 (2018), 3307–3315 | DOI | MR | Zbl

[34] Kozhanov A. I., Pulkina L. S., “On the solvability of boundary value problems with a nonlocal boundary condition of integral form for multidimensional hyperbolic equations”, Differ. Equations., 42:9 (2006), 1233–1246 | DOI | MR | Zbl

[35] Moiseev E. I., Korzyuk V. I., Kozlovskaya I. S., “Classical solution of a problem with an integral condition for the one-dimensional wave equation”, Differ. Equations., 50:10 (2014), 1364–1377 | DOI | MR | Zbl

[36] Sadarangani K. B., Abdullaev O. Kh., “About a problem for loaded parabolic-hyperbolic type equation with fractional derivatives”, Int. J. Differ. Equations., 2016 (2016), 1–6 | DOI | MR | Zbl