Geodesic
Solvability quasi-linear parabolic equation in domains with picewise monotone boundary
Dalʹnevostočnyj matematičeskij žurnal, Tome 13 (2013) no. 2, pp. 250-272.

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We investigate the existence of regular solutions for the quasilinear parabolic equation in non-cylindrical domain with boundary of class $W^1_2$. The equation can degenerate to the decision. Approximate solutions are constructed using the projection method of the family of projectors depending on the time parameter. We prove that a limit of these solutions will be the solution of the problem. To justify the existence of the limit methods are used for the functions of the compact scale of Banach spaces. 
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A. G. Podgaev; K. V. Lisenkov. Solvability quasi-linear parabolic equation in domains with picewise monotone boundary. Dalʹnevostočnyj matematičeskij žurnal, Tome 13 (2013) no. 2, pp. 250-272. http://geodesic.mathdoc.fr/item/DVMG_2013_13_2_a6/

[1] K. V. Lisenkov, “Proektsionyi metod resheniya zadachi dlya kvazilineinogo parabolicheskogo uravneniya v netsilindricheskoi oblasti s granitsei klassa $W_2^1$”, Dalnevost. matem. zhurn., 12:1 (2012), 48–59 | MR | Zbl

[2] N. E. Istomina, A. G. Podgaev, “O razreshimosti zadachi dlya kvazilineinogo vyrozhdayuschegosya parabolicheskogo uravneniya v oblasti s netsilindricheskoi granitsei”, Dalnevost. matem. zhurn., 1:1 (2000), 63–73

[3] I. G. Petrovskii, Composito mathematica, 1935

[4] P. V. Vinogradova, A. G. Zarubin, “O metode Galerkina dlya kvazilineinykh parabolicheskikh uravnenii v netsilindricheskoi oblasti”, Dalnevost. matem. zhurn., 3:1 (2002), 3–17

[5] 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 | MR | Zbl

[6] A. I. Kozhanov, N. A. Lar'kin, “On solvability of boundary-value problems for the wave equation with a nonlinear dissipation in noncylindrical domains”, Siberian Mathematical Journal, 42:6 (2001), 1062–1081 | DOI | MR | Zbl

[7] A. I. Kozhanov, N. A. Larkin, “O razreshimosti kraevykh zadach dlya silno nelineinykh uravnenii vyazkouprugosti v netsilindrichekikh oblastyakh”, Matematicheskie zametki YaGU, 6:1 (1999), 36 | MR | Zbl

[8] S. N. Glazatov, “O nekotorykh zadachakh dlya dvazhdy nelineinykh parabolicheskikh uravnenii i uravnenii peremennogo tipa”, Matem. tr., 3:2 (2000), 71–110 | MR | Zbl

[9] R. G. Zainullin, “Ob odnom analiticheskom podkhode k resheniyu odnomernoi zadachi perenosa tepla so svobodnymi granitsami”, Izv. Vyssh. Ucheb. Zav. Matematika, 2 (2008), 24–31 | MR

[10] E. A. Baderko, “O razreshimosti granichnykh zadach dlya parabolicheskikh uravnenii vysokogo poryadka v oblastyakh s krivolineinymi bokovymi granitsami”, Differentsialnye uravneniya, 12:10 (1976), 1780–1792 | MR

[11] Yu. T. Silchenko, “Odna kraevaya zadacha dlya oblasti s podvizhnoi granitsei”, Izv. Vyssh. Ucheb. Zav. Matematika, 3 (1998), 44–46 | MR | Zbl

[12] D. C. Antonopoulou, “Discontinuous Galerkin methods for the linear Schrodinger equation in non-cylindrical domains”, Numer. Math., 115:2 (2010), 585–608 | DOI | MR | Zbl

[13] P. Jamet, “Galerkin-type approximations which are discontinuous in time for parabolic equations in a variable domain”, SIAM J.Numer. Anal., 15:5 (1978), 912–928 | DOI | MR | Zbl

[14] N. A. Dragieva, “Primenenie metoda Galerkina k resheniyu volnovogo uravneniya v oblasti s podvizhnymi granitsami”, Zhurn. Vych. Mat. i Mat. Fiz., 15:4 (1975), 946–956 | MR | Zbl

[15] L. F. Voevodin, “Chislennoe modelirovanie rosta ledyanogo pokrova v vodoeme”, Sib. zhurn. ind. matem., 9:1 (2006), 47–54 | Zbl

[16] R. A. Mustafaev, “Reshenie obobschennym metodom integralnykh sootnoshenii odnoi nestatsionarnoi zadachi filtratsii s podvizhnoi granitsei”, Zhurn. Vych. Mat. i Mat. Fiz., 48:2 (2008), 282–287 | MR | Zbl

[17] A. G. Podgaev, “Ob otnositelnoi kompaktnosti mnozhestva abstraktnykh funktsii iz shkaly banakhovykh prostranstv”, Sib. matem. zhurn., 34:2 (1993), 135–-145 | MR | Zbl

[18] Yu. A. Dubinskii, “Nelineinye ellipticheskie i parabolicheskie uravneniya”, Itogi nauki i tekhn. Ser. Sovrem. probl. mat. Nov. dostizh., 9 (1976), 1–130

[19] Zh. L. Lions, Nekotorye metody resheniya nelineinykh kraevykh zadach, Mir, M., 1972, 587 pp. | MR

[20] O. A. Ladyzhenskaya, V. A. Solonnikov, N. N. Uraltseva, Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, Nauka, M., 1967, 736 pp. | MR