Geodesic
On uniqueness classes for degenerating parabolic equations
Sbornik. Mathematics, Tome 14 (1971) no. 4, pp. 453-469 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We study the uniqueness classes of a generalized solution of the Cauchy problem \begin{equation} u_t=\frac12\sum_{i,j=1}^na_{ij}(x)u_{x_ix_j}+\sum_{i=1}^na_i(x)u_{x_i}\equiv Lu,\quad u(0,x)=\varphi(x),\quad x\in\mathbf R^n,\ t\in[0,T], \end{equation} when the matrix $\bigl\{a_{ij}(x)\bigr\}$ is degenerate. A generalized solution is introduced with the help of an infinitesimal operator of a Markov process connected with the operator in (1). In the proof of the theorems we use probabilistic characteristics of this process. Bibliography: 11 titles. 
@article{SM_1971_14_4_a0,
     author = {I. M. Sonin},
     title = {On uniqueness classes for degenerating parabolic equations},
     journal = {Sbornik. Mathematics},
     pages = {453--469},
     year = {1971},
     volume = {14},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1971_14_4_a0/}
}
TY  - JOUR
AU  - I. M. Sonin
TI  - On uniqueness classes for degenerating parabolic equations
JO  - Sbornik. Mathematics
PY  - 1971
SP  - 453
EP  - 469
VL  - 14
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/SM_1971_14_4_a0/
LA  - en
ID  - SM_1971_14_4_a0
ER  - 
%0 Journal Article
%A I. M. Sonin
%T On uniqueness classes for degenerating parabolic equations
%J Sbornik. Mathematics
%D 1971
%P 453-469
%V 14
%N 4
%U http://geodesic.mathdoc.fr/item/SM_1971_14_4_a0/
%G en
%F SM_1971_14_4_a0
I. M. Sonin. On uniqueness classes for degenerating parabolic equations. Sbornik. Mathematics, Tome 14 (1971) no. 4, pp. 453-469. http://geodesic.mathdoc.fr/item/SM_1971_14_4_a0/

[1] S. D. Eidelman, Parabolicheskie sistemy, Nauka, Moskva, 1964 | MR | Zbl

[2] Todor Genchev, “V'erkhu zadachata na Koshi za obschoto ultraparabolichno uravnenie”, Izv. Matem. in-ta B'lg. AN, 8 (1964), 153–188

[3] G. N. Smirnova, “Zadachi Koshi dlya parabolicheskikh uravnenii, vyrozhdayuschikhsya na beskonechnosti”, Matem. sb., 70(112) (1966), 591–604 | MR | Zbl

[4] E. B. Dynkin, Markovskie protsessy, Fizmatgiz, Moskva, 1963 | MR

[5] A. V. Skorokhod, Issledovaniya po teorii sluchainykh protsessov, Kievskii un-t, Kiev, 1961 | Zbl

[6] M. I. Freidlin, “O faktorizatsii neotritsatelno opredelennykh simmetricheskikh matrits”, Teoriya veroyatn. i ee primeneniya, 8:2 (1968), 378–381

[7] M. I. Freidlin, “O stokhasticheskikh uravneniyakh K. Ito i vyrozhdayuschikhsya ellipticheskikh uravneniyakh”, Izv. AN SSSR, seriya matem., 26 (1962), 653–676 | MR | Zbl

[8] M. I. Freidlin, “Diffuzionnye protsessy i malyi parametr v ellipticheskikh uravneniyakh s razryvnymi koeffitsientami”, Izv. AN SSSR, seriya matem., 29 (1965), 1005–1036 | MR | Zbl

[9] I. I. Gikhman, A. V. Skorokhod, Stokhasticheskie differentsialnye uravneniya, Naukova Dumka, Kiev, 1968 | MR | Zbl

[10] A. S. Kalashnikov, “O lineinykh vyrozhdayuschikhsya parabolicheskikh uravneniyakh proizvolnogo poryadka s konechnoi oblastyu zavisimosti”, Matem. zametki, 6:3 (1969), 289–294 | Zbl

[11] M. I. Freidlin, “O postanovke granichnykh zadach dlya vyrozhdayuschikhsya ellipticheskikh uravnenii”, DAN SSSR, 170:2 (1966), 282 | MR | Zbl