Abstract
One considers singular parabolic “equations” of the form ∂β(u)/∂t−Δu∋0,where
sign u is a multivalued function, equal to -I for u<0, to 1 for u>0, and to the segment [-I,I] for u=0. Such a class of equations contains, in particular, the model for the two-phase Stefan problem, the porous medium equation, and the plasma equation. For the bounded generalized solutions u(x,t) of the indicated equations (without the assumption ∂u/∂εL2one has established a qualified local estimate of the modulus of continuity.
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Literature cited
L. A. Caffarelli and L. C. Evans, “Continuity of the temperature in the two-phase Stefan problem,” Arch. Rational Mech. Anal.,81, 199–220 (1983).
E. Di Benedetto, “Continuity of weak solutions to certain singular parabolic equations,” Ann. Mat. Pura Appl., Ser. 4,130, 131–176 (1982).
E. DiBenedetto, “Continuity of weak solutions to a general porous medium equation,” Indiana Univ. Math. J.,32, No. 1, 83–118 (1983).
W. P. Ziemer, “Interior and boundary continuity of weak solutions of degenerate parabolic equations,” Trans. Am. Math. Soc.,221, 733–748 (1982).
P. E. Sacks, “Continuity of solutions of a singular parabolic equation,” Nonlinear Anal.,7, No. 4, 387–409 (1983).
J. Moser, “A Harnack inequality for parabolic differential equations,” Commun. Pure Appl. Math.,17, 101–134 (1964).
J. Moser, “On a pointwise estimate for parabolic differential equations,” Commun. Pure Appl. Math.,24, 727–740 (1971).
N. S. Trudinger, “Pointwise estimates and quasilinear parabolic equations,” Commun. Pure Appl. Math.,21, 205–226 (1968).
O. A. Ladyzhenskaya (O. A. Ladyzenskaja), V. A. Solonnikov, and N. N. Ural'tseva (N. N. Uralceva), Linear and Quasilinear Equations of Parabolic Type, Am. Math. Soc., Providence (1968).
A. V. Ivanov, “Quasilinear degenerate and nonuniformly elliptic and parabolic equations of second order,” Tr. Mat. Inst. Akad. Nauk SSSR,160 (1982).
E. DiBenedetto and A. Friedman, “Regularity of solutions of nonlinear degenerate parabolic systems,” J. Reine Angew. Math.,349, 83–128 (1984).
D. G. Aronson and J. Serrin, “Local behavior of solutions of quasilinear parabolic equations,” Arch. Rational Mech. Anal.,25, 81–122 (1967).
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Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Ins'tituta im. V. A. Steklova AN SSSR, Vol. 147, pp. 49–71, 1985.
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Ivanov, A.V. Estimate of the modulus of continuity of the generalized solutions of certain singular parabolic equations. J Math Sci 37, 823–837 (1987). https://doi.org/10.1007/BF01387721
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DOI: https://doi.org/10.1007/BF01387721