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 Mat. Sb., 2010, Volume 201, Number 7, Pages 67–98 (Mi msb7567)

The solvability of the first initial-boundary problem for parabolic and degenerate parabolic equations in domains with a conical point

S. P. Degtyarev

Institute of Applied Mathematics and Mechanics, Ukraine National Academy of Sciences

Abstract: The first initial-boundary problem for second-order parabolic and degenerate parabolic equations is investigated in a domain with a conical or angular point. The means of attack is already known and uses weighted classes of smooth or integrable functions. Sufficient conditions for a unique solution to exist and for coercive estimates for the solution to be obtained are formulated in terms of the angular measure of the solid angle and the exponent of the weight. It is also shown that if these conditions fail to hold, then the parabolic problem has elliptic properties, that is, it can have a nonzero kernel or can be nonsolvable, and, in the latter case, it is not even a Fredholm problem. A parabolic equation and an equation with some degeneracy or a singularity at a conical point are considered.
Bibliography: 49 titles.

Keywords: parabolic equation, irregular domain, coercive estimate, spectral properties.

DOI: https://doi.org/10.4213/sm7567

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English version:
Sbornik: Mathematics, 2010, 201:7, 999–1028

Bibliographic databases:

Document Type: Article
UDC: 517.954+517.956.8+517.956.4
MSC: Primary 35K20; Secondary 35K65

Citation: S. P. Degtyarev, “The solvability of the first initial-boundary problem for parabolic and degenerate parabolic equations in domains with a conical point”, Mat. Sb., 201:7 (2010), 67–98; Sb. Math., 201:7 (2010), 999–1028

Citation in format AMSBIB
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This publication is cited in the following articles:
1. Kheloufi A., Sadallah B.-Kh., “On the regularity of the heat equation solution in non-cylindrical domains: Two approaches”, Appl. Math. Comput., 218:5 (2011), 1623–1633
2. Kheloufi A., “Resolutions of parabolic equations in non-symmetric conical domains”, Electron. J. Differential Equations, 2012 (2012), 116, 14 pp.
3. A. Kheloufi, “Existence and uniqueness results for parabolic equations with Robin type boundary conditions in a non-regular domain of $\mathbb R^3$”, Appl. Math. Comput., 220 (2013), 756–769
4. A. Kheloufi, Boubaker-Khaled Sadallah, “Study of the heat equation in a symmetric conical type domain of $\mathbb R^{N+1}$”, Math. Methods Appl. Sci., 37:12 (2014), 1807–1818
5. Ferroudj Boulkouane, Arezki Kheloufim, “On a second order linear parabolic equation with variable coefficients in a non-regular domain of $\mathbb{R}^{3}$”, Zhurn. SFU. Ser. Matem. i fiz., 11:4 (2018), 416–429
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