Geodesic
Degenerating parabolic equations with a variable direction of evolution
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 16 (2019), pp. 718-731.

Voir la notice de l'article provenant de la source Math-Net.Ru

The aim of the paper is to study the solvability in the classes of regular solutions of boundary value problems for differential equations $$ \varphi(t)u_t-\psi(t)\Delta u+c(x,t)u=f(x,t)\quad (x\in\Omega\subset \mathbb{R}^n,\quad 0). $$ A feature of these equations is that the function $\varphi (t)$ in them can arbitrarily change the sign on the segment $[0, T]$, while the function $\psi (t)$ is nonnegative for $t \in [0, T]$. For the problems under consideration, we prove existence and uniqueness theorems.
Keywords: degenerate parabolic equations, boundary value problems, regular solutions, uniqueness.
Mots-clés : variable direction of evolution, existence 
@article{SEMR_2019_16_a85,
     author = {A. I. Kozhanov and E. E. Macievskaya},
     title = {Degenerating parabolic equations with a variable direction of evolution},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {718--731},
     publisher = {mathdoc},
     volume = {16},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2019_16_a85/}
}
TY  - JOUR
AU  - A. I. Kozhanov
AU  - E. E. Macievskaya
TI  - Degenerating parabolic equations with a variable direction of evolution
JO  - Sibirskie èlektronnye matematičeskie izvestiâ
PY  - 2019
SP  - 718
EP  - 731
VL  - 16
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SEMR_2019_16_a85/
LA  - ru
ID  - SEMR_2019_16_a85
ER  - 
%0 Journal Article
%A A. I. Kozhanov
%A E. E. Macievskaya
%T Degenerating parabolic equations with a variable direction of evolution
%J Sibirskie èlektronnye matematičeskie izvestiâ
%D 2019
%P 718-731
%V 16
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SEMR_2019_16_a85/
%G ru
%F SEMR_2019_16_a85
A. I. Kozhanov; E. E. Macievskaya. Degenerating parabolic equations with a variable direction of evolution. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 16 (2019), pp. 718-731. http://geodesic.mathdoc.fr/item/SEMR_2019_16_a85/

[1] G. Fichera, “On a Unified Theory of Boundary Value Problems for Elliptic-Parabolic Equations of Second Orde”, Boundary Problems. Differential Equations, Univ. of Wisconsin Press. Madison, Wisconsin, 1960, 97–120 | MR | Zbl

[2] O.A. Oleinik, E.V. Radkevich, Second-Order Equations with Nonnegative Characteristic Form, Business Media, 2012 | MR

[3] V.N. Vragov, Kraevye Zadachi dlya Neklassicheskih Uravneniy Matematicheskoy Fiziki, Novosibirskiy Universitet, Novosibirsk, 1983

[4] I.E. Egorov, V.E. Fedorov, Neklassicheskie Uravneniya Matematicheskoy Fiziki Vysokogo Poryadka, Vychisl. Tsentr SO RAN, Novosibirsk, 1995

[5] I.E. Egorov, V.E. Fedorov, I.M. Tikhonova, E.S. Efimova, Metod Galerkina dlya Neklassicheskih Uravneniy Matematicheskoy Fiziki, VIII, Mezhdunarodnaya Konferentsyya po Matematicheskomu Modelirovaniyu, Tezisy dokl. (Yakutsk, 2017), 11

[6] I.E. Egorov, V.E. Fedorov, I.M. Tikhonova, E.S. Efimova, “The Galerkin method for nonclassical equations of mathematical physics”, AIP Conference Proceedings, 1907, 2017, 020011 | DOI

[7] A.I. Kozhanov, “Kraevye Zadachi dlya Ultraparabolicheskih i Kvasiultraparabolicheskih Uravneniy s Menyayuschimsya Napravleniem Evolutsyi”, Itogi Nauki i Tehniki. Sovremennaya Matematika i ee Prilozheniya. Tematicheskiye Obzory, 149, 2018, 56–63

[8] A.I. Kozhanov, G.A. Lukina, “Nonlocal Boundary Value Problems with Partially Integral Conditions for Degenerate Differential Equations with Multiple Characteristics”, Journal of Mathematical Sciences, 237:4 (2019), 549–562 | DOI | MR

[9] S.S. Sobolev, Some Applications of Functional Analysis in Mathematical Physics, Translations of Mathematical Monographs, 90, American Mathematical Society, Providence, RI, 1991 | MR | Zbl

[10] O.A. Ladyzhenskaia, V. A. Solonnikov, N.N. Ural'tseva, Linear and Quasi-linear Equations of Parabolic Type, Translations of Mathematical Monographs, 23, American Mathematical Society, Providence, RI, 1968 | DOI | MR | Zbl

[11] O.V. Besov, V.P. Il'in, S.M. Nikol'skii, Integral Representations of Functions and Imbedding Theorems, Scripta Series in Mathematics, V.H. Winston Sons, Washington, D.C., 1979 | MR | Zbl

[12] O.A. Ladyzhenskaia, N.N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Mathematics in Science and Engineering, 46, Academic Press, New York–London, 1968 | MR | Zbl

[13] V.A. Trenogin, Funktsyanalnyi Analiz, Nauka, M., 1980 | Zbl

[14] H. Triebel, Interpolation Theory. Function Spaces. Differential Operators, Deutscher Verlag des Wissenschaften, Berlin, 1978 | MR | Zbl

[15] V.B. Shakhmurov, “Separable anisotropic differential operators and applications”, J. of Math. Analysis and Applications, 327:2 (2007), 1182–1201 | DOI | MR | Zbl