Regularity of viscosity solutions of a degenerate parabolic equation

Authors:
Yun-Guang Lu and Liwen Qian

Journal:
Proc. Amer. Math. Soc. **130** (2002), 999-1004

MSC (2000):
Primary 35K55; Secondary 35K65, 35D10

DOI:
https://doi.org/10.1090/S0002-9939-01-06313-4

Published electronically:
November 9, 2001

MathSciNet review:
1873772

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study the Cauchy problem for the nonlinear degenerate parabolic equation of second order

and present regularity results for the viscosity solutions.

**[1]**M. Ughi,*A Degenerate Parabolic Equation Modelling the Spread of an Epidemic*, Ann. Mat. Pura Appl.**143**(1986), 385-400. MR**88g:35105****[2]**M. Bertsch, R.D. Passo, and M. Ughi,*Discontinuous ``viscosity" Solution of A Degenerate Parabolic Equation*, Transactions of The American Mathematical Society**320**(1990), 2, 779-798. MR**90m:35086****[3]**M. Bertsch and M. Ughi,*Positivity properties of viscosity solutions of a degenerate parabolic equation*, Nonlinear Anal. TMA.**14**(1990), 7, 571-592. MR**92a:35006****[4]**A. Friedman,*Partial Differential Equations of Parabolic Type*, Englewood Cliffs, N.J., Prentice-Hall Inc., 1964. MR**31:6062****[5]**Yunguang Lu,*Hölder Estimates of Solutions of Biological Population Equations*, Applied Mathematical Letters**13**(2000), 123-126. MR**2001b:35182****[6]**Yunguang Lu,*Hölder estimates of solutions to some doubly nonlinear degenerate parabolic equations*, Commun. PDE.**24(5, 6)**(1999), 895-913. MR**2000c:35105****[7]**Liwen Qian and Wentao Fan,*Hölder estimates of Solutions for Some Degenerate Parabolic Equations*, Acta Math. Sci.**4**(1999). MR**2000j:35156****[8]**P.Z. Mkrtychyan,*A degenerate quasilinear parabolic equation that arises in the theory of nonstationary filtration*, Izv. Akad. Nauk Armyan. SSSR Mat**24**(1989), 103-116 (English transl. in Soviet J. Contemp. Math.**24**(1989), 1-13). MR**90i:35134****[9]**P.Z. Mkrtychyan,*Estimation of the gradient of a solution and the classical solvability of the first initial-boundary value problem for a class of quasilinear nonuniformly parabolic equations*, Izv. Akad. Nauk Armyan. SSSR Mat**24**(1989), 293-299 (English transl. in Soviet J. Contemp. Math.**24**(1989), 85-89). MR**90k:35141****[10]**B. H. Gilding,*Hölder continuity of solutions of parabolic equations*, J. London Math. Soc.**13**(1976), 103-106. MR**53:3501**

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Additional Information

**Yun-Guang Lu**

Affiliation:
Departamento de Matematicas y Estadistica, Universidad Nacional de Colombia, Bogota, Colombia

Email:
yglu@matematicas.unal.edu.co

**Liwen Qian**

Affiliation:
Department of Computational Science, National University of Singapore, Singapore 117543

Address at time of publication:
Singapore-MIT Alliance, National University of Singapore, Singapore 119260

Email:
qianlw@cz3.nus.edu.sg, smaqlw@nus.edu.sg

DOI:
https://doi.org/10.1090/S0002-9939-01-06313-4

Keywords:
Degenerate parabolic equation,
viscosity solution,
Lipschitz continuity,
maximum principle

Received by editor(s):
November 1, 1998

Received by editor(s) in revised form:
April 10, 2000

Published electronically:
November 9, 2001

Communicated by:
Suncica Canic

Article copyright:
© Copyright 2001
American Mathematical Society