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Classical Solvability of the First Initial Boundary-Value Problem for a Nonlinear Strongly Degenerate Parabolic Equation

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We prove the existence of a classical solution global in time for the first initial boundary-value problem for a nonlinear strongly degenerate parabolic equation.

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  1. 1.

    M. Ughi, “A degenerate parabolic equation modelling the spread of an epidemic,” Ann. Mat. Pura Appl., 143, 385–400 (1986).

  2. 2.

    M. Bertsch and R. Dal Passo, “A numerical treatment of a superdegenerate equation with application to the porous medium equation,” Quart. Appl. Math., 48, 133–152 (1990).

  3. 3.

    B. V. Bazalii and N. V. Krasnoshchek, “On the regularity of a solution of a problem with free boundary for the equation v t = (v m)xx,” Algebra Analiz, 12, 100–130 (2000).

  4. 4.

    A. S. Kalashnikov, “On some problems in the qualitative theory of nonlinear degenerate second-order parabolic equations,” Usp. Mat. Nauk, 42, 135–176 (1987).

  5. 5.

    O. A. Oleinik and E. V. Radkevich, “Second-order equations with nonnegative characteristic form,” VINITI Series in Mathematical Analysis, VINITI, Moscow (1971).

  6. 6.

    O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type [in Russian], Nauka, Moscow (1967).

  7. 7.

    B. V. Bazalii and S. P. Degtyarev, “First boundary-value problem for degenerate parabolic equations,” Nelin. Granich. Zadachi, 3, 6–12 (1991).

  8. 8.

    E. Ya. Riekstyn’sh, Asymptotic Expansions of Integrals [in Russian], Vol. 1, Zinatne, Riga (1974).

  9. 9.

    I. S. Gradshtein and I. M. Ryzhik, Tables of Integrals, Sums, Series, and Products [in Russian], Fizmatgiz, Moscow (1963).

  10. 10.

    A. Friedman, Partial Differential Equations of Parabolic Type [Russian translation], Mir, Moscow (1968).

  11. 11.

    L. A. Caffarelli and A. Friedman, “Regularity of the free boundary for the one-dimensional flow of a gas in porous medium,” Amer. J. Math., 101, 1181–1193 (1979).

  12. 12.

    D. G. Aronson and J. L. Vazquez, “Eventual C regularity and concavity for flows in one-dimensional porous medium,” Arch. Ration. Mech. Anal., 99, 329–348 (1987).

  13. 13.

    D. G. Aronson, “The porous medium equation,” Lect. Notes Math., 1224, 1–46 (1986).

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 10, pp. 1299 – 1320, October, 2004.

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Bazalii, B.V., Krasnoshchek, N.V. Classical Solvability of the First Initial Boundary-Value Problem for a Nonlinear Strongly Degenerate Parabolic Equation. Ukr Math J 56, 1547–1573 (2004).

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  • Parabolic Equation
  • Classical Solution
  • Degenerate Parabolic Equation
  • Classical Solvability