Abstract
In this paper, we prove the Hölder continuity of solutions of parabolic equations containing the p(x, t)-Laplacian. The degree p must satisfy the so-called logarithmic condition.
Similar content being viewed by others
References
V. V. Zhikov, “On Lavrentiev’s Phenomenon,” Russ. J. Math. Phys., 3, No. 2, 249–269 (1994).
V. V. Zhikov, “On some variational problems,” Russ. J. Math. Phys., 5, No. 1, 105–116 (1996).
X. Fan, A Class of De Giorgi Type and Holder Continuity of Minimizers of Variational with m(x)-Growth Condition, Lanzhou Univ. (1995).
Yu. A. Alkhutov, “Harnaack’s inequality and Hölder continuity of solutions of nonlinear elliptic equations with nonstandard condition of growth,” Differ. Uravn., 33, No. 12, 1651–1660 (1997).
O. V. Krasheninnikova, “On point-wise continuity of solutions of elliptic equations with nonstandard condition of growth,” Tr. Mat. Inst. Steklova, 236, 204–211 (2002).
E. Acerbi and G. Mingione, “Regularity results for a class of functionals with non-standard growth,” Arch. Ration. Mech. Anal., 156, 121–140 (2001).
Yu. A. Alkhutov and O. V. Krasheninnikova, “Continuity at boundary points of solutions of quasilinear elliptic equations with nonstandard condition of growth,” Izv. Ross. Akad. Nauk, Ser. Mat., 68, No. 6, 3–60 (2004).
M. Ru̇z̆hic̆ka, Electrorheological Fluids: Modeling and Mathematical Theory, Lect. Notes Math., Vol. 1748, Springer, Berlin (2000).
E. DiBenedetto, Degenerate Parabolic Equations, Springer, New York (1993).
E. DiBenedetto, M. J. Urbano, and V. Vespri, Current Issues on Singular and Degenerate Evolution Equations, Preprint No. 03 24, Pré-Publicações do Departamento de Matemática Universidade de Coimbra (2003).
N. S. Antontsev and V. V. Zhikov, “Higher integrability for parabolic equations of p(x, t)-Laplacian type,” Adv. Differen. Equ., 10, No. 9, 1053–1080 (2005).
E. Acerbi, G. Mingione, and G. A. Seregin, “Regularity results for parabolic systems related to a class of non-Newtonian fluids,” Ann. Inst. Henri Poincaré, Anal. Non Linéare, 21, No. 1, 25–60 (2004).
Yu. A. Alkhutov, S. N. Antontsev, and V. V. Zhikov, “Parabolic equations with variable order of nonlinearity,” Zbirnik Prats Inst. Mat. NAN Ukr., 6, No. 1, 23–50 (2009).
E. De Giorgi, “Sulla differenziabilitá e l’analicitá delle estremali degli integrali multipli regolari,” Mem. Acc. Sci. Torino, Cl. Sc. Fis. Mat. Nat., 3, No. 3, 25–43 (1957).
A. S. Agranovich and M. I. Vishik, “Elliptic problems with a parameter and parabolic problems of general type,” Usp. Mat. Nauk, 19, No. 3, 53–161 (1964).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 28, Part I, pp. 8–74, 2011.
Rights and permissions
About this article
Cite this article
Alkhutov, Y.A., Zhikov, V.V. Hölder continuity of solutions of parabolic equations with variable nonlinearity exponent. J Math Sci 179, 347–389 (2011). https://doi.org/10.1007/s10958-011-0599-9
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-011-0599-9