On a gradient maximum principle for some quasilinear parabolic equations on convex domains
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Abstract:
We establish a spatial gradient maximum principle for classical solutions to the initial and Neumann boundary value problem of some quasilinear parabolic equations on smooth convex domains.References
- Nicholas D. Alikakos and Rouben Rostamian, Gradient estimates for degenerate diffusion equations. I, Math. Ann. 259 (1982), no. 1, 53–70. MR 656651, DOI 10.1007/BF01456828
- Lawrence C. Evans, Partial differential equations, 2nd ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010. MR 2597943, DOI 10.1090/gsm/019
- Charles S. Kahane, A gradient estimate for solutions of the heat equation. II, Czechoslovak Math. J. 51(126) (2001), no. 1, 39–44. MR 1814630, DOI 10.1023/A:1013745503001
- Bernd Kawohl and Nikolai Kutev, Maximum and comparison principle for one-dimensional anisotropic diffusion, Math. Ann. 311 (1998), no. 1, 107–123. MR 1624275, DOI 10.1007/s002080050179
- Seonghak Kim and Baisheng Yan, Radial weak solutions for the Perona-Malik equation as a differential inclusion, J. Differential Equations 258 (2015), no. 6, 1889–1932. MR 3302525, DOI 10.1016/j.jde.2014.11.017
- Seonghak Kim and Baisheng Yan, Convex integration and infinitely many weak solutions to the Perona-Malik equation in all dimensions, SIAM J. Math. Anal. 47 (2015), no. 4, 2770–2794. MR 3369068, DOI 10.1137/15M1012220
- Seonghak Kim and Baisheng Yan, On Lipschitz solutions for some forward-backward parabolic equations, preprint.
- O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural′ceva, Linear and quasilinear equations of parabolic type, Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1968 (Russian). Translated from the Russian by S. Smith. MR 0241822, DOI 10.1090/mmono/023
- Gary M. Lieberman, Second order parabolic differential equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. MR 1465184, DOI 10.1142/3302
- Murray H. Protter and Hans F. Weinberger, Maximum principles in differential equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. MR 0219861
- Patrizia Pucci and James Serrin, The maximum principle, Progress in Nonlinear Differential Equations and their Applications, vol. 73, Birkhäuser Verlag, Basel, 2007. MR 2356201, DOI 10.1007/978-3-7643-8145-5
Additional Information
- Seonghak Kim
- Affiliation: Institute for Mathematical Sciences, Renmin University of China, Beijing 100872, People’s Republic of China
- Email: kimseo14@ruc.edu.cn
- Received by editor(s): February 12, 2015
- Received by editor(s) in revised form: May 13, 2016
- Published electronically: September 15, 2016
- Communicated by: Catherine Sulem
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 1203-1208
- MSC (2010): Primary 35B50, 35B65, 35K20, 35K59
- DOI: https://doi.org/10.1090/proc/13291
- MathSciNet review: 3589319