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Literature Cited
G. I. Barenblatt, “On self-similar motions of a compressible fluid in a porous medium,” Prikl. Mat. Mekh.,16, 679–698 (1952).
A. S. Kalashnikov, “Problems in the theory of degenerate parabolic equations,” Usp. Mat. Nauk,42, No. 2, 135–174 (1987).
S. N. Antontsev, “On the nature of perturbations described by solutions of multidimensional degenerate parabolic equations,” Dinamika Sploshnoi Sredy,40, 114–122 (1979).
S. N. Antontsev, “On localization of solutions of nonlinear degenerate elliptic and parabolic equations,” Dokl. Akad. Nauk SSSR,260, No. 6, 1289–1293 (1981).
A. M. Meiermanov and V. V. Pukhnachev, “Lagrangian coordinates in a Stefan problem,” Dinamika Sploshnoi Sredy,47, 90–111 (1980).
M. E. Gurtin, R. C. McCamy, and E. A. Socolovsky, “A coordinate transformation for the porous media equation that renders the free-boundary stationary,” Quart. Appl. Math.,42, 345–357 (1984).
S. N. Shmarev, “Qualitative properties of solutions of a free-boundary problem for degenerate filtration and heat conduction equations,” Author's Abstract of Candidate's Dissertation, Physics-Math. Sciences, 01.01.02, Novosibirsk (1987).
S. N. Shmarev, “Qualitative properties of generalized solutions of degenerate solutions of filtration theory equations,” Usp. Mat. Nauk,41, No. 4, 189–190 (1986).
M. A. Herero and J. L. Vazquez, “On the propagation properties of a nonlinear degenerate parabolic equation,” Commun. Part. Different. Equats.,7, No. 12, 1381–1402 (1982).
A. S. Kalashnikov, “On a nonlinear equation arising in the theory of nonstationary filtration,” Trudy Seminara im. I. G: Petrovskogo,4, 137–146 (1979).
O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural'tseva, Linear and Quasilinear Equations of Parabolic Type [in Russian], Nauka, Moscow (1987).
O. A. Oleinik, A. S. Kalashnikov, and Chou Yu Lin, “Cuachy problem and curve problems for an equation of nonstationary filtration type,” Izv. Akad. Nauk SSSR, Ser. Mat.,22, No. 5, 667–704 (1958).
B. F. Knerr, “The porous medium equation in one dimension,” Trans. Am. Math. Soc.,234, No. 2, 381–415 (1977).
S. N. Kruzhkov, “Nonlinear parabolic equations with two independent variables,” Trudy Mosk. Mat. Obshch.,16, 329–346 (1967).
A. Friedman, Partial Differential Equations of Parabolic Type [Russian translation], Mir, Moscow (1958).
A. G. Podgaev, “Compactness of some nonlinear sets,” Dokl. Akad. Nauk SSSR,285, No. 5, 1064–1066 (1985).
J.-L. Lions, Some Methods of Solving Nonlinear Boundary Value Problems [Russian translation], Mir, Moscow (1972).
N. D. Alikakos and L. G. Evans, “Continuity of the gradient for weak solutions of a degenerate parabolic equation,” J. Math. Pures Appl.,62, No. 3, 253–268 (1983).
M. Beretsch, M. F. Gurtin, and D. Hilhorst, “On degenerate diffusion equation of the form c(z)t=F(zx)x with application to population dynamics,” J. Different. Equat.,67, 56–89 (1987).
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Novosibirsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 32, No. 3, pp. 39–46, May–June, 1991.
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Grebenev, V.N. A transformation of coordinates in a problem with a free boundary. Sib Math J 32, 383–389 (1991). https://doi.org/10.1007/BF00970473
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DOI: https://doi.org/10.1007/BF00970473