Boundary behavior of the fast diffusion equation
Author:
Y. C. Kwong
Journal:
Trans. Amer. Math. Soc. 322 (1990), 263283
MSC:
Primary 35K55; Secondary 35B99
MathSciNet review:
1008697
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Abstract 
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Abstract: The fast diffusion equation , , is a degenerate nonlinear parabolic equation of which the existence of a unique continuous weak solution has been established. In this paper we are going to obtain a Lipschitz growth rate of the solution at the boundary of and estimate that in terms of the various data.
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G. Schroeder, Ph. D. Thesis, University of WisconsinMadison (to appear).
 [1]
 Benilan and M. G. Crandall, The continuous dependence on of the solution of , Indiana Univ. Math. J. 30 (1981). MR 604277 (83d:35071)
 [2]
 J. G. Berryman and C. Holland, Stability of the separable solution for fast diffusion, Arch. Rational Mech. Anal. 74 (1980). MR 588035 (81m:35065)
 [3]
 M. Bertsch and L. A. Peletier, Porous media type equations: An overview, Mathematical Institute Publication, no. 7, University of Leiden, 1983.
 [4]
 F. Chiarenza and R. Serapioni, A Harnack inequality for degenerate parabolic equations, Comm. Partial Differential Equations 9 (1984), 719749. MR 748366 (86c:35082)
 [5]
 M. G. Crandall and M. Pierre, Regularization effects for in , J. Funct. Anal. 45 (1982). MR 647071 (83g:34071)
 [6]
 D. G. Diaz and D. J. Diaz, Finite extinction time for a class of nonlinear parabolic equations, Comm Partial Differential Equations 4 (1979), 12131231. MR 546642 (80k:35043)
 [7]
 E. Di Benedetto, Continuity of weak solutions to a general porous medium equation, Indiana Univ. Math. J. 32 (1983). MR 684758 (85c:35010)
 [8]
 E. Di Benedetto and YaZhe Chen, On the local behavior of solutions of singular parabolic equations, Arch. Rational Mech. Anal. (to appear). MR 955531 (89k:35107)
 [9]
 M. A. Herrero and J. L. Vazquez, Asymptotic behaviour of the solutions of a strong nonlinear parabolic problem, Ann. Fac. Sci. Toulouse Math. (5) 3 (1981), 113127. MR 646311 (83e:35016)
 [10]
 J. Kazdan and F. Warner, Remarks on some quasilinear elliptic equations, Comm. Pure Appl. Math. 28 (1975), 567597. MR 0477445 (57:16972)
 [11]
 N. Krylov and M. Safonov, A certain property of solutions of parabolic equations with measurable coefficients, Math. USSR Izv. 16 (1981), 151164.
 [12]
 Y. C. Kwong, Extinction and interior and boundary regularity of plasma type equation with nonnegative initial data and homogeneous Dirichlet boundary condition, Proc. Amer. Math. Soc. 104 (1988). MR 962815 (90d:35143)
 [13]
 , Asymptotic behaviour of a plasma type equation at finite extinction, Arch. Rational Mech. Anal. 104 (1988).
 [14]
 J. Moser, A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math. 17 (1964), 101134. MR 0159139 (28:2357)
 [15]
 , On a pointwise estimate for parabolic differential equations, Comm. Pure Appl. Math. 24 (1971), 727740. MR 0288405 (44:5603)
 [16]
 S. Pohozaev, Eigen functions of the equation , Soviet Math. Dokl. 165 (1965), 14081411.
 [17]
 E. S. Sabanina, A class of nonlinear degenerate parabolic equations, Soviet Math. Dokl. 143 (1962), 495498.
 [18]
 P. E. Sacks, Continuity of solutions of a singular parabolic equation, Nonlinear Anal. 7 (1983), 387409. MR 696738 (84d:35081)
 [19]
 G. Schroeder, Ph. D. Thesis, University of WisconsinMadison (to appear).
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199010086970
PII:
S 00029947(1990)10086970
Article copyright:
© Copyright 1990
American Mathematical Society
