Geodesic
On boundary value problems for a class of ultraparabolic equations, and their applications
Sbornik. Mathematics, Tome 61 (1988) no. 2, pp. 529-544 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Let $\lambda_i(t)\ge\alpha>0$, and let $L$ be a strictly elliptic operator of second order in space variables $x$, with coefficients depending only on $x=(x_1,\dots,x_m)$. Using potentials, solutions of some initial-boundary value problems for the ultraparabolic equation $\sum^n_{i=1}\lambda_i(x)\frac{\partial u}{\partial t_i}=L(u)$ are constructed. These solutions belong to special Hölder spaces $H^{P,P/2}_{x\lambda}$ depending on the vector $\lambda=(\lambda_1,\dots,\lambda_n)$. By means of these notions the first boundary value problem for the equation $\sum^n_{i=1}\lambda_i\frac{\partial u}{\partial t_i}=u_{xx}\operatorname{sgn}x$ is studied in a domain containing the hyperplane $x=0$. Necessary and sufficient conditions for the existence of a solution of this problem in the spaces $H^{P,P/2}_{x\lambda}$ are given. Bibliography: 14 titles. 
@article{SM_1988_61_2_a15,
     author = {S. A. Tersenov},
     title = {On boundary value problems for a~class of ultraparabolic equations, and their applications},
     journal = {Sbornik. Mathematics},
     pages = {529--544},
     year = {1988},
     volume = {61},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1988_61_2_a15/}
}
TY  - JOUR
AU  - S. A. Tersenov
TI  - On boundary value problems for a class of ultraparabolic equations, and their applications
JO  - Sbornik. Mathematics
PY  - 1988
SP  - 529
EP  - 544
VL  - 61
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/SM_1988_61_2_a15/
LA  - en
ID  - SM_1988_61_2_a15
ER  - 
%0 Journal Article
%A S. A. Tersenov
%T On boundary value problems for a class of ultraparabolic equations, and their applications
%J Sbornik. Mathematics
%D 1988
%P 529-544
%V 61
%N 2
%U http://geodesic.mathdoc.fr/item/SM_1988_61_2_a15/
%G en
%F SM_1988_61_2_a15
S. A. Tersenov. On boundary value problems for a class of ultraparabolic equations, and their applications. Sbornik. Mathematics, Tome 61 (1988) no. 2, pp. 529-544. http://geodesic.mathdoc.fr/item/SM_1988_61_2_a15/

[1] Piskunov N. S., “Kraevye zadachi dlya uravnenii elliptiko-parabolicheskogo tipa”, Matem. sb., 7:3 (1940), 385 | Zbl

[2] Ganchev N. S., “Ob ultraparabolicheskikh uravneniyakh”, DAN SSSR, 151:2 (1963), 265–268

[3] Ilin A. M., “Ob odnom klasse ultraparabolicheskikh uravnenii”, DAN SSSR, 159:6 (1964), 1214–1217 | MR

[4] Vladimirov V. S., Drozhzhinov Yu. N., “Obobschennaya zadacha Koshi dlya ultraparabolicheskikh uravnenii”, Izv. AN SSSR. Ser. matem., 31 (1967), 1341–1360 | Zbl

[5] Drozhzhinov Yu. N., “Stabilizatsiya resheniya obobschennoi zadachi Koshi dlya ultraparabolicheskikh uravnenii”, Izv. AN SSSR. Ser. matem., 33:2 (1969), 368–372

[6] Ladyzhenskaya O. A., Solonnikov V. A., Uraltseva I. I., Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, Nauka, M., 1967

[7] Muskhelishvili N. I., Singulyarnye integralnye uravneniya, Nauka, M., 1968 | MR | Zbl

[8] Khartman F., Obyknovennye differentsialnye uravneniya, Mir, M., 1970 | MR | Zbl

[9] Tersenov S. A., Parabolicheskie uravneniya s menyayuschimsya napravleniem vremeni, Nauka, Novosibirsk, 1985 | MR | Zbl

[10] Akhmedov Kh. Kh., Razreshimost nekotorykh kraevykh zadach dlya pryamo i obratno parabolicheskikh uravnenii, Preprint No 78, IM SO AN SSSR, Novosibirsk, 1984

[11] Konisov A. K., Pervaya kraevaya zadacha dlya odnogo parabolicheskogo uravneniya s menyayuschimsya napravleniem vremeni, Preprint No 79, IM SO AN SSSR, Novosibirsk, 1984

[12] Tablitsa integralnykh preobrazovanii, T. 2, Nauka, M., 1970

[13] Tersenov S. A., Razreshimost kraevykh zadach dlya uravneniya $u_t=u_{xx}\sgn x$ v klassakh $H_{xt}^{p,p/2}$, Preprint No 74, IM SO AN SSSR, Novosibirsk, 1984

[14] Sin Donkha Z., Ob odnoi kraevoi zadache dlya ultraparabolicheskogo uravneniya peremennogo tipa, Dep. v VINITI, 1984, No 3771-84