Hölder estimates of solutions to a degenerate diffusion equation

Author:
Yunguang Lu

Journal:
Proc. Amer. Math. Soc. **130** (2002), 1339-1343

MSC (2000):
Primary 35K55, 35K65, 35D10, 35K15

DOI:
https://doi.org/10.1090/S0002-9939-01-06312-2

Published electronically:
December 20, 2001

MathSciNet review:
1879955

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper is concerned with the Hölder estimates of weak solutions of the Cauchy problem for the general degenerate parabolic equations

with the initial data , where the diffusion function can be a constant on a nonzero measure set, such as the equations of two-phase Stefan type. Some explicit Hölder exponents of the composition function with respect to the space variables are obtained by using the maximum principle.

**1.**S.B. Angenent and D.G. Aronson, The focusing problem for the radially symmetric porous medium equation, Commun. in P.D.E., 20(1995), 1217-1240. MR**96c:35074****2.**H. Brezis and M.G. Crandall, Uniqueness of solution of the initial value problem for , J. Pure Appl. Math., 58(1979), 153-163. MR**80e:35029****3.**L.A. Caffarelli and L.C. Evans, Continuity of the temperature in the two-phase Stefan problem, Arch. Rat. Mech. Anal., 83(1983),199-220. MR**84g:35070****4.**L.A. Caffarelli, J.L. Vazquez and N.I. Wolanski, Lipschitz continuity of solutions and interfaces of the N-dimensional porous medium equation, Indiana Univ. Math. Journal, 36(1987), 373-401. MR**88k:35221****5.**E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, 1993 MR**94h:35130****6.**E. DiBenedetto and V. Vespri, On the Singular Equation , Arch. Rat. Mech. Anal., 132(1995), 247-309. MR**96m:35179****7.**A. Friedman, Partial Differential Equations of Parabolic Type, Englewood Cliffs,N.J.: Prentice-Hall Inc. 1964. MR**31:6062****8.**W. Jäger and Y.G. Lu, Global regularity of solutions for general degenerate parabolic equations in 1-D, J. Diff. Equs., 140(1997), 365-377. MR**98j:35112****9.**A.S. Kalashnikov, Some problems of the qualitative theory of non-linear degenerate second-order parabolic equations, Russian Math. Surveys, 42, 169-222(1987). MR**88h:35054****10.**Y.G. Lu, Hölder Estimates of Solutions of Some Doubly Nonlinear Degenerate Parabolic Equations, Commun. in P.D.E., 24(1999), 895-914. MR**2000c:35105****11.**Y.G. Lu, Hölder Estimates of Solutions of Biological Population Equations, Appl. Math. Letters, 13(2000), 123-126. MR**2001b:35182****12.**Y.G. Lu and W. Jäger, Regularity of Solutions to Nonlinear Reaction-Diffusion-Convection Equations with Degenerate Diffusion, J. Diff. Equations, 170(2001), No.1, 1-21. MR**2001j:35159****13.**Y. G. Lu, Hölder Estimates of Solutions for General Degenerate Parabolic Equation, to appear in Applicable Analysis.**14.**Y. G. Lu and L.W. Qian, Regularity of viscosity solutions of a degenerate parabolic equation, to appear in Proc. AMS.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
35K55,
35K65,
35D10,
35K15

Retrieve articles in all journals with MSC (2000): 35K55, 35K65, 35D10, 35K15

Additional Information

**Yunguang Lu**

Affiliation:
Departamento de Matemáticas y Estadística, Universidad Nacional de Colombia, Bogotá, Colombia – and – Department of Mathematics, University of Science & Technology of China, Hefei, People’s Republic of China

DOI:
https://doi.org/10.1090/S0002-9939-01-06312-2

Keywords:
Degenerate parabolic equation,
H\"older solution,
maximum principle

Received by editor(s):
April 12, 2000

Published electronically:
December 20, 2001

Communicated by:
Suncica Canic

Article copyright:
© Copyright 2001
American Mathematical Society