Geodesic
Solution of the Cauchy problem for a parabolic equation with singular coefficients
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 47, Tome 477 (2018), pp. 35-53 
G. I. Bizhanova. Solution of the Cauchy problem for a parabolic equation with singular coefficients. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 47, Tome 477 (2018), pp. 35-53. http://geodesic.mathdoc.fr/item/ZNSL_2018_477_a2/
@article{ZNSL_2018_477_a2,
     author = {G. I. Bizhanova},
     title = {Solution of the {Cauchy} problem for a parabolic equation with singular coefficients},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {35--53},
     year = {2018},
     volume = {477},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2018_477_a2/}
}
TY  - JOUR
AU  - G. I. Bizhanova
TI  - Solution of the Cauchy problem for a parabolic equation with singular coefficients
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2018
SP  - 35
EP  - 53
VL  - 477
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2018_477_a2/
LA  - ru
ID  - ZNSL_2018_477_a2
ER  - 
%0 Journal Article
%A G. I. Bizhanova
%T Solution of the Cauchy problem for a parabolic equation with singular coefficients
%J Zapiski Nauchnykh Seminarov POMI
%D 2018
%P 35-53
%V 477
%U http://geodesic.mathdoc.fr/item/ZNSL_2018_477_a2/
%G ru
%F ZNSL_2018_477_a2

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

The paper is concerned with the Cauchy problem for the second order parabolic equation with the singular coefficients with respect to $t$ at the first order spatial derivatives. The solution of the problem is constructed in the explicit form. For it, it is defined a weighted Hölder space with the weight as a positive power of $t$. The existence, uniqueness, estimates of the solution are proved.

[1] M. Gevrey, “Sur les equations aux derivees partielles du type parabolique”, J. Math. Pur. Appl. Ser 6, 9:4 (1913), 305–471

[2] L. I. Kamynin, “K teorii Zhevre dlya parabolicheskikh potentsialov. V”, Differents. uravneniya, 7:3 (1972), 494–509 | MR

[3] L. I. Kamynin, “K teorii Zhevre dlya parabolicheskikh potentsialov. VI”, Differents. uravneniya, 8:6 (1972), 1015–1025 | MR | Zbl

[4] E. A. Baderko, “O reshenii pervoi kraevoi zadachi dlya parabolicheskikh uravnenii s pomoschyu potentsiala prostogo sloya”, DAN SSSR, 283:1 (1985), 11–13 | MR | Zbl

[5] E. A. Baderko, Reshenie metodom granichnykh integralnykh uravnenii zadach dlya lineinykh parabolicheskikh uravnenii proizvolnogo poryadka v negladkikh oblastyakh, Diss. dokt. fiz.-matem. nauk, MGU, M., 1992

[6] E. A. Baderko, “Kraevye zadachi dlya parabolicheskogo uravneniya i granichnye integralnye uravneniya”, Differents. uravneniya, 28:1 (1992), 17–23 | MR

[7] V. P. Mikhailov, “Teorema suschestvovaniya i edinstvennosti resheniya odnoi granichnoi zadachi dlya parabolicheskogo uravneniya v oblasti s osobymi tochkami na granitse”, Tr. MIAN SSSR, 91, 1967, 47–58 | Zbl

[8] G. Beitmen, A. Erdeii, Tablitsy integralnykh preobrazovanii. Preobrazovaniya Fure, Laplasa, Mellina, v. 1, Nauka, M., 1969

[9] O. A. Ladyzhenskaya, V.A. Solonnikov, N.N. Uraltseva, Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, Nauka, M., 1967 | MR