Geodesic
The first boundary value problem for a degenerate hyperbolic equation
Vladikavkazskij matematičeskij žurnal, Tome 18 (2016) no. 2, pp. 19-30 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Under certain hypothesis on the coefficients the condition is found for unique solvability of the first boundary value problem for a degenerate hyperbolic equation in the region. The uniqueness of the solution of the problem is proved by Tricomi method and existence by the method of integral equations. The solutions obtained with respect to the traces of the sought solution of integral equations are found and written out explicitly. It is shown that whenever the hypothesis of the theorem is violated, then the homogeneous problem corresponding to the problem under study has an infinite number of linearly independent solutions. 
@article{VMJ_2016_18_2_a2,
     author = {Zh. A. Balkizov},
     title = {The first boundary value problem for a~degenerate hyperbolic equation},
     journal = {Vladikavkazskij matemati\v{c}eskij \v{z}urnal},
     pages = {19--30},
     year = {2016},
     volume = {18},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMJ_2016_18_2_a2/}
}
TY  - JOUR
AU  - Zh. A. Balkizov
TI  - The first boundary value problem for a degenerate hyperbolic equation
JO  - Vladikavkazskij matematičeskij žurnal
PY  - 2016
SP  - 19
EP  - 30
VL  - 18
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/VMJ_2016_18_2_a2/
LA  - ru
ID  - VMJ_2016_18_2_a2
ER  - 
%0 Journal Article
%A Zh. A. Balkizov
%T The first boundary value problem for a degenerate hyperbolic equation
%J Vladikavkazskij matematičeskij žurnal
%D 2016
%P 19-30
%V 18
%N 2
%U http://geodesic.mathdoc.fr/item/VMJ_2016_18_2_a2/
%G ru
%F VMJ_2016_18_2_a2
Zh. A. Balkizov. The first boundary value problem for a degenerate hyperbolic equation. Vladikavkazskij matematičeskij žurnal, Tome 18 (2016) no. 2, pp. 19-30. http://geodesic.mathdoc.fr/item/VMJ_2016_18_2_a2/

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

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

[3] Gellerstedt S., “Sur un equation linearre aux derivees partieelles de type mixte”, Arkiv Mat., Astr. och Fysik, 25A:29 (1937), 1–25 | Zbl

[4] Kalmenov T. Sh., “O kharakteristicheskoi zadache Koshi dlya odnogo klassa vyrozhdayuschikhsya giperbolicheskikh uravnenii”, Dif. uravneniya, 9:1 (1973), 84–96 | MR | Zbl

[5] Nakhushev A. M., “K teorii kraevykh zadach dlya vyrozhdayuschikhsya giperbolicheskikh uravnenii”, Soobscheniya AN GSSR, 77:3 (1975), 545–548 | MR | Zbl

[6] Kumykova S. K., Nakhusheva F. B., “Ob odnoi kraevoi zadache dlya giperbolicheskogo uravneniya, vyrozhdayuschegosya vnutri oblasti”, Dif. uravneniya, 14:1 (1978), 50–64 | MR

[7] Kumykova S. K., “Kraevaya zadacha so smescheniem dlya vyrozhdayuschegosya vnutri oblasti giperbolicheskogo uravneniya”, Dif. uravneniya, 16:1 (1980), 93–104 | MR | Zbl

[8] Salakhitdinov M. S., Mirsaburov M., “O nekotorykh kraevykh zadachakh dlya giperbolicheskogo uravneniya, vyrozhdayuschegosya vnutri oblasti”, Dif. uravneniya, 17:1 (1981), 129–136 | MR | Zbl

[9] Smirnov M. M., Vyrozhdayuschiesya giperbolicheskie uravneniya, Vysheishaya shkola, Minsk, 1977, 160 pp. | MR

[10] Repin O. A., Kraevye zadachi so smescheniem dlya uravnenii giperbolicheskogo i smeshannogo tipov, Samarskii filial SGU, Samara, 1992, 161 pp. | MR

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

[12] Kalmenov T. Sh., K teorii nachalno-kraevykh zadach dlya differentsialnykh uravnenii, Tsikl nauchnykh rabot T. Sh. Kalmenova, IMMM, Almaty, 2013, 406 pp.

[13] Nakhushev A. M., “K teorii lineinykh kraevykh zadach dlya uravneniya vtorogo poryadka smeshannogo giperbolo-parabolicheskogo tipa”, Dif. uravneniya, 14:1 (1978), 66–73 | MR | Zbl

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

[15] Samko S. G., Kilbas A. A., Marichev O. I., Integraly i proizvodnye drobnogo poryadka i nekotorye ikh prilozheniya, Nauka i tekhnika, Minsk, 1987, 688 pp. | MR

[16] Nakhushev A. M., Drobnoe ischislenie i ego primenenie, Fizmatlit, M., 2003, 271 pp.