Hölder regularity for degenerate parabolic equations with variable exponents
Abstract
In this paper, we discuss a class of degenerate parabolic equations with variable exponents. By using the Steklov average and Young's inequality, we establish energy and logarithmicestimates for solutions to these equations. Then based on the intrinsic scaling method, we provethat local weak solutions are locally continuous.
Downloads
References
E. Acerbi, G. Mingione and G.A. Seregin, Regularity results for parabolic systems related to a class of non-Newtonian fluids, Ann. Inst. H. Poincar ́e Anal. Non Lin‘eaire, 21 (2004), 25-60. DOI: https://doi.org/10.1016/j.anihpc.2002.11.002
S. Antontsev and S. Shmarev, Evolution PDEs with Nonstansdard Growth Conditions. Atlantis Press, Amsterdam, 2015. DOI: https://doi.org/10.2991/978-94-6239-112-3
S. Antontsev, J. I. Dıaz, and S. Shmarev, Energy methods for free boundary problems: Applications to nonlinear PDEs and fluid mechanics. Progress in Nonlinear Differential Equations and their Applications, vol. 48 (2002) Birkh ̈auser, Boston. DOI: https://doi.org/10.1115/1.1483358
S. Antontsev and S. Shmarev, Parabolic equations with anisotropic nonstandard growth conditions, Internat. Ser. Numer. Math., vol. 154 (2007) Birkh ̈auser, Basel, 33-44 . DOI: https://doi.org/10.1007/978-3-7643-7719-9_4
Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math. 66 (2006), 1383-1406. DOI: https://doi.org/10.1137/050624522
E. DiBenedetto, Degenerate Parabolic Equations. Springer-Verlag, New York, 1993. DOI: https://doi.org/10.1007/978-1-4612-0895-2
E. DiBenedetto, J.M. Urbano, V. Vespri, Current issues on singular and degenerate evolution equations. in: Handbook of Differential Equations, Evolutionary Equations, vol. 1, (2004), pp. 169-286. DOI: https://doi.org/10.1016/S1874-5717(04)80005-7
S. Ouaro, A. Ouedraogo, Nonlinear parabolic problems with variable exponent and L1-data. Electron. J. Differ. Equ. Paper No. 109, 16 p. (2017).
K. Rajagopal, M. Ruzicka, Mathematical modelling of electro-rheological fluids, Contin. Mech. Thermodyn. 13 (2001) 59-78. DOI: https://doi.org/10.1007/s001610100034
M. Ruzicka, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics, vol. 1748, Springer, Berlin, 2000. DOI: https://doi.org/10.1007/BFb0104029
J.M, Urbano. The method of intrinsic scaling. A systematic approach to regularity for degenerate and singular PDEs. Lecture Notes in Mathematics, vol. 1930. Springer, Berlin (2008).12. P. Wittbold, A. Zimmermann, Existence and uniqueness of renormalized solutions to nonlinear elliptic equations with variable exponents and L1-data, Nonlinear Analysis TMA, 72 (2010), 2990-3008. DOI: https://doi.org/10.1016/j.na.2009.11.041
V.V. Zhikov, On the density of smooth functions in Sobolev-Orlicz spaces, Zap. Nauchn. Sem. S. Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 310 (2004), 67-81.
C. Zhang, S. Zhou., Renormalized and entropy solutions for nonlinear parabolic equations with variable exponents and L1-data, J. Differ. Equations, 248 (2010), 1376-1400
Copyright (c) 2021 Boletim da Sociedade Paranaense de Matemática
This work is licensed under a Creative Commons Attribution 4.0 International License.
When the manuscript is accepted for publication, the authors agree automatically to transfer the copyright to the (SPM).
The journal utilize the Creative Common Attribution (CC-BY 4.0).
