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Estimate of the maximum of modulus of generalized solutions of the first boundary-value problem for degenerate parabolic equations

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We study parabolic equations of the divergent form with degeneration. We have obtained an estimate for the maximum of modulus of generalized solutions of the first boundary-value problem with a zero on the parabolic boundary.

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References

  1. J. Moser, “A new proof of de Giorgi’s theorem concerning the regularity problem for elliptic differential equations,” Commun. Pure Appl. Math., 13, No. 3, 457–468 (1960).

    Article  MATH  Google Scholar 

  2. J. Moser, “On a pointwise estimate for parabolic differential equations,” Commun. Pure Appl. Math., 24, No. 5, 727–740 (1971).

    Article  MATH  Google Scholar 

  3. S. N. Kruzhkov, “A priori estimates for generalized solutions of elliptic and parabolic equations,” Dokl. Akad. Nauk SSSR, 150, No. 4, 748–751 (1963).

    MathSciNet  Google Scholar 

  4. S. N. Kruzhkov, “A priori estimates and specific properties of solutions of elliptic and parabolic equations,” Mat. Sb., 65, No. 4, 522–570 (1964).

    MathSciNet  Google Scholar 

  5. S. N. Kruzhkov, “Boundary-value problems for degenerate elliptic equations of the second order,” Mat. Sb., 77, No. 3, 299–334 (1968).

    MathSciNet  Google Scholar 

  6. D. G. Aronson and J. Serrin, “Local behavior of solutions of quasilinear parabolic equations,” Arch. Rat. Mech. Anal., 25, 81–122 (1967).

    Article  MATH  MathSciNet  Google Scholar 

  7. N. Trudinger, “On the regularity of generalized solutions of linear, non-uniformly elliptic equations,” Arch. Rat. Mech. Anal., 42, No. 1, 50–62 (1971).

    Article  MATH  MathSciNet  Google Scholar 

  8. N. Trudinger, “Pointwise estimates and quasilinear parabolic equations,” Commun. Pure Appl. Math., 21, 205–226 (1968).

    Article  MATH  MathSciNet  Google Scholar 

  9. A. V. Ivanov, “Estimates of the Hölder constant of generalized solutions of degenerate parabolic equations,” Zap. Nauch. Sem. LOMI, 152, 21–44 (1986).

    MATH  Google Scholar 

  10. A. V. Ivanov, “Hölder estimates for quasilinear parabolic equations with double degeneration,” Zap. Nauch. Sem. LOMI, 171, 70–105 (1989).

    MATH  Google Scholar 

  11. A. V. Ivanov, “Uniform Hölder estimates for generalized solutions of quasilinear parabolic equations permitting double degeneration,” Algebra Analiz, 3, No. 2, 139–179 (1991).

    MATH  Google Scholar 

  12. A. V. Ivanov, “Quasilinear parabolic equations permitting double degeneration,” Algebra Analiz, 4, No. 6, 114–130 (1992).

    MATH  Google Scholar 

  13. F. M. Chiarenza and R. P. Serapioni, “A Harnack inequality for degenerate parabolic equations,” Commun. Part. Diff. Equat., 9 (8), 719–749 (1984).

    Article  MATH  MathSciNet  Google Scholar 

  14. F. Chiarenza and R. Serapioni, “Pointwise estimates for degenerate parabolic equations,” Appl. Anal., 23, 287–299 (1987).

    Article  MATH  MathSciNet  Google Scholar 

  15. E. Di Benedetto and Friedman, “Hölder estimates for nonlinear degenerate parabolic system,” J. Reine Math., 357, 82–128 (1985).

    Google Scholar 

  16. E. Di Benedetto, “On local behavior of solutions of degenerate parabolic equations with measurable coefficients,” Ann. Sci. Norm. Sup., 13, No. 3, 485–535 (1986).

    Google Scholar 

  17. F. Nicolosi, “Soluzio ni deloli dei problemi al contorno per operatori parabolici che possono degenerate,” Ann. Math. Pure Appl., 4, No. 125, 135–155 (1980).

    Article  Google Scholar 

  18. G. R. Cirmi, “Problemi parabolici degeneri,” Rend. Circ. Math. Palermo. Ser. 2, 41, No. 2, 197–208 (1992).

    Article  MATH  MathSciNet  Google Scholar 

  19. I. M. Kolodii, “Qualitative properties of generalized solutions of degenerate elliptic equations,” Ukr. Mat. Zh., 27, No. 3, 320–328 (1975).

    MathSciNet  Google Scholar 

  20. S. N. Kruzhkov and I. M. Kolodii, “A priori estimates and the Harnack inequality for generalized solutions of degenerate quasilinear parabolic equations,” Sib. Mat. Zh., 18, No. 3, 608–628 (1977).

    MATH  Google Scholar 

  21. I. M. Kolodii, “The Liouville theorem for generalized solutions of degenerate quasilinear parabolic equations,” Differents. Uravn., 21, No. 5, 841–854 (1985).

    MathSciNet  Google Scholar 

  22. I. M. Kolodii, “Estimate of the maximum of modulus of generalized solutions of the Dirichlet problem for elliptic equations of the divergent form,” Ukr. Mat. Zh., 47, No. 5, 635–648 (1995).

    Article  MathSciNet  Google Scholar 

  23. Lu Ven’-tuan, “Embedding theorems for spaces of functions with partial derivatives summed with various indices,” Vestn. Leningrad. Univ., No. 7, 23–27 (1961).

    Google Scholar 

  24. S. N. Kruzhkov and A. G. Korolev, “On the theory of embedding of anisotropic function spaces,” Dokl. Akad. Nauk SSSR, 285, No. 5, 1054–1057 (1985).

    MathSciNet  Google Scholar 

  25. S. N. Kruzhkov and I. M. Kolodii, “On the theory of embedding of anisotropic Sobolev spaces,” Usp. Mat. Nauk., 38, No. 2, 207–208 (1983).

    MathSciNet  Google Scholar 

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Lviv State University, Lviv. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 12, pp. 1624–1637, December, 1997.

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Kolodii, I.M. Estimate of the maximum of modulus of generalized solutions of the first boundary-value problem for degenerate parabolic equations. Ukr Math J 49, 1827–1845 (1997). https://doi.org/10.1007/BF02513062

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  • DOI: https://doi.org/10.1007/BF02513062

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