Abstract
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.
Similar content being viewed by others
References
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).
J. Moser, “On a pointwise estimate for parabolic differential equations,” Commun. Pure Appl. Math., 24, No. 5, 727–740 (1971).
S. N. Kruzhkov, “A priori estimates for generalized solutions of elliptic and parabolic equations,” Dokl. Akad. Nauk SSSR, 150, No. 4, 748–751 (1963).
S. N. Kruzhkov, “A priori estimates and specific properties of solutions of elliptic and parabolic equations,” Mat. Sb., 65, No. 4, 522–570 (1964).
S. N. Kruzhkov, “Boundary-value problems for degenerate elliptic equations of the second order,” Mat. Sb., 77, No. 3, 299–334 (1968).
D. G. Aronson and J. Serrin, “Local behavior of solutions of quasilinear parabolic equations,” Arch. Rat. Mech. Anal., 25, 81–122 (1967).
N. Trudinger, “On the regularity of generalized solutions of linear, non-uniformly elliptic equations,” Arch. Rat. Mech. Anal., 42, No. 1, 50–62 (1971).
N. Trudinger, “Pointwise estimates and quasilinear parabolic equations,” Commun. Pure Appl. Math., 21, 205–226 (1968).
A. V. Ivanov, “Estimates of the Hölder constant of generalized solutions of degenerate parabolic equations,” Zap. Nauch. Sem. LOMI, 152, 21–44 (1986).
A. V. Ivanov, “Hölder estimates for quasilinear parabolic equations with double degeneration,” Zap. Nauch. Sem. LOMI, 171, 70–105 (1989).
A. V. Ivanov, “Uniform Hölder estimates for generalized solutions of quasilinear parabolic equations permitting double degeneration,” Algebra Analiz, 3, No. 2, 139–179 (1991).
A. V. Ivanov, “Quasilinear parabolic equations permitting double degeneration,” Algebra Analiz, 4, No. 6, 114–130 (1992).
F. M. Chiarenza and R. P. Serapioni, “A Harnack inequality for degenerate parabolic equations,” Commun. Part. Diff. Equat., 9 (8), 719–749 (1984).
F. Chiarenza and R. Serapioni, “Pointwise estimates for degenerate parabolic equations,” Appl. Anal., 23, 287–299 (1987).
E. Di Benedetto and Friedman, “Hölder estimates for nonlinear degenerate parabolic system,” J. Reine Math., 357, 82–128 (1985).
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).
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).
G. R. Cirmi, “Problemi parabolici degeneri,” Rend. Circ. Math. Palermo. Ser. 2, 41, No. 2, 197–208 (1992).
I. M. Kolodii, “Qualitative properties of generalized solutions of degenerate elliptic equations,” Ukr. Mat. Zh., 27, No. 3, 320–328 (1975).
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).
I. M. Kolodii, “The Liouville theorem for generalized solutions of degenerate quasilinear parabolic equations,” Differents. Uravn., 21, No. 5, 841–854 (1985).
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).
Lu Ven’-tuan, “Embedding theorems for spaces of functions with partial derivatives summed with various indices,” Vestn. Leningrad. Univ., No. 7, 23–27 (1961).
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).
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).
Additional information
Lviv State University, Lviv. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 12, pp. 1624–1637, December, 1997.
Rights and permissions
About this article
Cite this article
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
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02513062