Boundary value problem with displacement for a third-order parabolic-hyperbolic equation

Zh. A. Balkizov   ; A. G. Ezaova   ; L. V. Kanukoeva

Vladikavkazskij matematičeskij žurnal, Tome 23 (2021) no. 2, pp. 5-18

Résumé

A boundary value problem with a shift is investigated for an inhomogeneous third order equation of parabolic-hyperbolic type when one of the boundary conditions is a linear combination of values of the sought function on independent characteristics. The following results are obtained in this work: the inequality of the characteristics $AC$ and $BC$, which bound the hyperbolic part $\Omega_{1}$ of the domain $\Omega$, as carriers of the data of the Tricomi problem for $0\le x\le\pi n$, $n \in N$ and the solvability of the Tricomi problem with data on the characteristic $BC$ in this case, in general, does not imply the solvability of the Tricomi problem with data on the characteristic $AC$; necessary and sufficient conditions for the existence and uniqueness of a regular solution of the problem are found. Under certain requirements for given functions, the solution to the problem is written out explicitly. It is shown that if the necessary conditions for the given functions found in the work are violated, the homogeneous problem corresponding to the problem has an infinite set of linearly independent solutions, and the set of solutions to the corresponding inhomogeneous problem can exist only with an additional requirement for the given functions.

Bibliographie

[1] Dzhuraev T. D., Kraevye zadachi dlya uravnenii smeshannogo i smeshanno-sostavnogo tipov, FAN, Tashkent, 1979, 239 pp. | MR

[2] Nakhushev A. M., Uravneniya matematicheskoi biologii, Vysshaya shkola, M., 1995, 301 pp.

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

[4] Zhegalov V. I., “Kraevaya zadacha dlya uravneniya smeshannogo tipa s granichnymi usloviyami na obeikh kharakteristikakh i s razryvami na linii perekhoda”, Uchenye zap. Kazanskogo un-ta, 122, no. 3, 1962, 3–16 | Zbl

[5] Nakhushev A. M., “O nekotorykh kraevykh zadachakh dlya giperbolicheskikh uravnenii i uravnenii smeshannogo tipa”, Dif. uravneniya, 5:1 (1969), 44–59 | Zbl

[6] Nakhushev A. M., “Novaya kraevaya zadacha dlya odnogo vyrozhdayuschegosya giperbolicheskogo uravneniya”, Dokl. AN SSSR, 187:4 (1969), 736–739 | Zbl

[7] Nakhushev A. M., “O zadache Darbu dlya vyrozhdayuschikhsya giperbolicheskikh uravnenii”, Dif. uravneniya, 7:1 (1971), 49–56 | MR | Zbl

[8] Bers L., Matematicheskie voprosy dozvukovoi i okolozvukovoi gazovoi dinamiki, Izd-vo inostr. lit-ry, M., 1961, 208 pp. | MR

[9] Frankl F. I., Izbrannye trudy po gazovoi dinamike, Nauka, M., 1973, 712 pp. | MR

[10] Salakhitdinov M. S., Uravneniya smeshanno-sostavnogo tipa, FAN, Tashkent, 1974, 156 pp. | MR

[11] Repin O. A., Kraevye zadachi so smescheniem dlya uravnenii giperbolicheskogo i smeshannogo tipa, Izd-vo Samarskogo filiala Saratovskogo gos. un-ta, Samara, 1992, 161 pp. | MR

[12] Kalmenov T. Sh., Kraevye zadachi dlya lineinykh uravnenii v chastnykh proizvodnykh giperbolicheskogo tipa, Gylaya, Shymkent, 1993, 328 pp.

[13] Zhegalov V. I., Mironov A. N., Differentsialnye uravneniya so starshimi chastnymi proizvodnymi, Kazanskoe matem. ob-vo, Kazan, 2001, 226 pp.

[14] Repin O. A., Kilbas A. A., Marichev O. I., Kraevye zadachi dlya uravnenii v chastnykh proizvodnykh s razryvnymi koeffitsientami, Izd-vo Samarskogo gos. ekonom. un-ta, Samara, 2008, 275 pp.

[15] Pulkina L. S., Zadachi s neklassicheskimi usloviyami dlya giperbolicheskikh uravnenii, Izd-vo Samarskogo un-ta, Samara, 2012, 194 pp.

[16] Sabitov K. B., K teorii uravnenii smeshannogo tipa, Fizmatlit, M., 2014, 304 pp.

[17] Sabitov K. B., Pryamye i obratnye zadachi dlya uravnenii parabolo-giperbolicheskogo tipa, Gilem, Ufa, 2015, 240 pp. | MR

[18] Nakhusheva Z. A., Nelokalnye kraevye zadachi dlya osnovnykh i smeshannogo tipov differentsialnykh uravnenii, KBNTs RAN, Nalchik, 2011, 196 pp.

[19] Ezaova A. G., Lesev V. N., Kozhanov A. I., “Nelokalnaya zadacha s drobnymi proizvodnymi dlya uravneniya tretego poryadka”, Mat. zametki SVFU, 26:1 (2019), 14–22 | DOI

[20] Balkizov Zh. A., “Kraevaya zadacha so smescheniem dlya modelnogo uravneniya parabolo-giperbolicheskogo tipa tretego poryadka”, Vestn. KRAUNTs. Fiz.-mat. nauki, 2018, no. 3(23), 19–26 | DOI | MR | Zbl

[21] 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. vedomosti Belgorodskogo gos. un-ta. Ser. Matematika. Fizika, 51:1 (2019), 5–14 | DOI

[22] Balkizov Zh. A., “On a boundary value problem for a third-order parabolic-hyperbolic type equation with a displacement boundary condition in its hyperbolicity domain”, Vestn. Sam. gos. tekhn. un-ta. Ser. fiz.-mat. nauki, 24:2 (2020), 211–225 | DOI | MR | Zbl

[23] Balkizov Zh. A., “Pervaya kraevaya zadacha so smescheniem dlya uravneniya parabolo-giperbolicheskogo tipa vtorogo poryadka”, Vestn. Dagestanskogo gos. un-ta. Ser. 1. Estestv. nauki, 35:1 (2020), 13–20 | DOI

[24] Ezaova A. G., “Odnoznachnaya razreshimost odnoi zadachi tipa zadachi Bitsadze — Samarskogo dlya uravneniya s razryvnymi koeffitsientami”, Vladikavk. mat. zhurn., 20:4 (2018), 50–58 | DOI | MR | Zbl

[25] Tikhonov A. N., Samarskii A. A., Uravneniya matematicheskoi fiziki, Nauka, M., 1977, 736 pp. | MR