Far Eastern Mathematical Journal

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On Faedo-Galerkin methods and monotony in a non-cylindrical domain for a degenerate quasi-linear equation

Podgaev A. G., Istomina N. E.

2014, issue 1, P. 73-89

Abstract

In this article a monotony method for nonstationary equations adapt to noncylindrical domains. Existence theorems are proved. A family of basic functions constructed. These functions have a smooth parameter and a completeness property for every one

Keywords: non-cylindrical domain, monotony method, family of basic functions, quasi-linear equation

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References

[1] M. Gevrey, “Les equations paraboliques”, J. de Math., 9. (1913), 187–235. [2] I. G. Petrowsky, “Zur ersten Randwertaufgabe der Warmeleitungsgleichung”, Compositio math, 1 (1935), 389–419. [3] V. P. Mikhailov, “O zadache Dirikhle i pervoi smeshannoi zadache dlia parabolicheskogo uravneniia”, Dokl. AN SSSR, 140:2 (1961), 303–306. [4] V. P. Mikhailov, “O zadache Dirikhle dlia parabolicheskogo uravneniia”, Mat. sbornik, 61(103):1 (1963), 40–64. [5] S. G. Krein, G. I. Laptev, “Abstraktnaia skhema rassmotreniia parabolicheskikh zadach v netsilindricheskikh oblastiakh”, Differentsial'nye uravneniia, 5:8 (1969), 1458–1469. [6] R. Benabidallah, J. Ferreira, “On hyperbolic – parabolic equations with nonlinearity of Kirchhoff – Carrier type in domains with moving boundary”, Nonlinear Analysis, 37 (1999), 269–287. [7] J. Ferreira, N. A. Lar’kin, “Global solvability of a mixed problem for a nonlinear hyperbolic – parabolic equation in noncylindrical domains”, Portugaliae Mathematica, 53:4 (1996), 381–395. [8] P. V. Vinogradova, A. G. Zarubin, “O metode Galerkina dlia kvazilineinykh parabolicheskikh uravnenii v netsilindricheskoi oblasti”, Dal'nevostochnyi mat. zhurnal., 3:1 (2002), 3–17. [9] Zh.-L. Lions, Nekotorye metody resheniia nelineinykh kraevykh zadach, Mir, M, 1972, 588 s. [10] S. N. Glazatov, “Nekotorye zadachi dlia dvazhdy nelineinykh parabolicheskikh uravnenii v netsilindricheskikh oblastiakh”, Dif. i integr. uravneniia mat. modeli, Tezisy dokladov mezhd. nauch. konferentsii, Cheliabinskii gos. un-t, Cheliabinsk, 2002, 31. [11] N. E. Istomina, A. G. Podgaev, “O razreshimosti zadachi dlia kvazilineinogo vyrozhdaiushchegosia parabolicheskogo uravneniia v oblasti s netsilindricheskoi granitsei”, Dal'nevostochnyi matematicheskii zhurnal, 1:1 (2000), 63–73. [12] E. G. Agapova, Issledovanie razreshimosti zadach dlia nestatsionarnykh vyrozhdaiushchikhsia na reshenii nelineinykh uravnenii, Dis. . . . kand. f.-m. nauk: 01.01.02., KhGTU, Khabarovsk, 2000. [13] Kh. Gaevskii, K. Greger, K. Zakharias, Nelineinye operatornye uravneniia i operatorno-differentsial'nye uravneniia, Mir, M., 1978, 336 s. [14] Iu.A. Dubinskii, “Kvazilineinye ellipticheskie i parabolicheskie uravneniia liubogo poriadka”, Uspekhi matematicheskikh nauk, XXIII:1(139) (1968), 45–90. [15] G. V. Demidenko, Vvedenie v teoriiu sobolevskikh prostranstv, Uchebnoe posobie, Novosib. un-t., Novosibirsk, 1995, 111 s. [16] N. E. Istomina, Razvitie metoda monotonnosti na sluchai parabolicheskogo uravneniia v netsilindricheskoi oblasti, Preprint № 6 IPM DVO RAN, Dal'nauka, Vladivostok, 2001, 40 s. [17] N. E. Istomina, A. G. Podgaev, “Teorema edinstvennosti dlia nelineinogo parabolicheskogo uravneniia v netsilindricheskoi oblasti”, Mat. zametki IaGU, 10:1 (2003), 27–33. [18] A. G. Podgaev, A. Z. Sin, “Ob odnom obobshchenii lemmy Vishika – Dubinskogo i neravenstva Gronuolla”, Elektronnoe nauchnoe izdanie "Uchenye zametki TOGU", 4:4 (2013), 2113–2118. [19] O. A. Ladyzhenskaia, Kraevye zadachi matematicheskoi fiziki, Nauka, M, 1973, 408 s. [20] A. N. Kolmogorov, S. V. Fomin, Elementy teorii funktsii i funktsional'nogo analiza, Nauka, M, 1968, 496 s.

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