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 Mat. Sb. (N.S.), 1985, Volume 128(170), Number 4(12), Pages 451–473 (Mi msb2170)

Theorems on the complete set of isomorphisms in the $L_2$-theory of generalized solutions of boundary value problems for a Petrovskii parabolic equation

N. V. Zhitarashu

Abstract: The general boundary value problem is studied for a parabolic equation in spaces of insufficiently smooth and generalized functions. Starting from Green's formula, the generalized solution of a boundary value problem is defined, and two families (scales) of spaces are constructed in which the boundary value problem is studied: the spaces of solutions $\widetilde{\mathscr H}^s(\Omega)$, and the sapces of right-hand sides $\mathscr K^s(\Omega)$. It is proved that the closure with respect to continuity of the boundary value problem operator establishes an isomorphism of the spaces $\widetilde{\mathscr H}^s(\Omega)$ and $\mathscr K^s(\Omega)$ for $-\infty<s<\infty$.
Bibliography: 35 titles.

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English version:
Mathematics of the USSR-Sbornik, 1987, 56:2, 447–471

Bibliographic databases:

UDC: 517.946
MSC: 35K20, 35D05

Citation: N. V. Zhitarashu, “Theorems on the complete set of isomorphisms in the $L_2$-theory of generalized solutions of boundary value problems for a Petrovskii parabolic equation”, Mat. Sb. (N.S.), 128(170):4(12) (1985), 451–473; Math. USSR-Sb., 56:2 (1987), 447–471

Citation in format AMSBIB
\Bibitem{Zhi85} \by N.~V.~Zhitarashu \paper Theorems on the complete set of isomorphisms in the $L_2$-theory of generalized solutions of boundary value problems for a~Petrovskii parabolic equation \jour Mat. Sb. (N.S.) \yr 1985 \vol 128(170) \issue 4(12) \pages 451--473 \mathnet{http://mi.mathnet.ru/msb2170} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=820397} \zmath{https://zbmath.org/?q=an:0609.35045} \transl \jour Math. USSR-Sb. \yr 1987 \vol 56 \issue 2 \pages 447--471 \crossref{https://doi.org/10.1070/SM1987v056n02ABEH003046} 

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This publication is cited in the following articles:
1. N. V. Zhitarashu, “The $L_2$-theory of generalized solutions of general linear model parabolic boundary value problems”, Math. USSR-Izv., 31:2 (1988), 273–305
2. Roitberg YA., “Boundary-Value and Mixed Problems for General Hyperbolic Systems in a Complete Scale of Sobolev Type Spaces”, 318, no. 4, 1991, 820–824
3. Zhitarashu N., Eidel'man S., “L-P-Theory on a Class of Non-Local Parabolic Boundary Value Problems”, Dokl. Akad. Nauk, 356:4 (1997), 445–448
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