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 Tr. Mat. Inst. Steklova, 2010, Volume 270, Pages 21–32 (Mi tm3011)

This article is cited in 18 scientific papers (total in 18 papers)

Existence theorems for solutions of parabolic equations with variable order of nonlinearity

Yu. A. Alkhutov, V. V. Zhikov

Vladimir State University for the Humanities, Vladimir, Russia

Abstract: We study the solvability of an initial-boundary value problem for second-order parabolic equations with variable order of nonlinearity. In the model case, the equation contains the $p$-Laplacian with a variable exponent $p(x,t)$. We prove that if the measurable exponent $p$ is separated from unity and infinity, then the problem has $W$- and $H$-solutions.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2010, 270, 15–26

Bibliographic databases:

UDC: 517.958
Received in October 2009

Citation: Yu. A. Alkhutov, V. V. Zhikov, “Existence theorems for solutions of parabolic equations with variable order of nonlinearity”, Differential equations and dynamical systems, Collected papers, Tr. Mat. Inst. Steklova, 270, MAIK Nauka/Interperiodica, Moscow, 2010, 21–32; Proc. Steklov Inst. Math., 270 (2010), 15–26

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Diening L., Naegele P., Ruzicka M., “Monotone Operator Theory for Unsteady Problems in Variable Exponent Spaces”, Complex Var. Elliptic Equ., 57:11, SI (2012), 1209–1231
2. Surnachev M.D., Zhikov V.V., “On Existence and Uniqueness Classes for the Cauchy Problem for Parabolic Equations of the P-Laplace Type”, Commun. Pure Appl. Anal, 12:4 (2013), 1783–1812
3. Yu. A. Alkhutov, V. V. Zhikov, “Existence and uniqueness theorems for solutions of parabolic equations with a variable nonlinearity exponent”, Sb. Math., 205:3 (2014), 307–318
4. E. R. Andriyanova, “Estimates of decay rate for solution to parabolic equation with non-power nonlinearities”, Ufa Math. J., 6:2 (2014), 3–24
5. E. R. Andriyanova, F. Kh. Mukminov, “Existence of solution for parabolic equation with non-power nonlinearities”, Ufa Math. J., 6:4 (2014), 31–47
6. Chai X., Li H., Niu W., “Large Time Behavior For P(X)-Laplacian Equations With Irregular Data”, Electron. J. Differ. Equ., 2015, 61
7. Zou W., Li J., “Existence and Uniqueness of Bounded Weak Solutions For Some Nonlinear Parabolic Problems”, Bound. Value Probl., 2015, 69
8. È. R. Andriyanova, F. Kh. Mukminov, “Existence and qualitative properties of a solution of the first mixed problem for a parabolic equation with non-power-law double nonlinearity”, Sb. Math., 207:1 (2016), 1–40
9. Winkert P., Zacher R., “Global a priori bounds for weak solutions to quasilinear parabolic equations with nonstandard growth”, Nonlinear Anal.-Theory Methods Appl., 145 (2016), 1–23
10. Giacomoni J., Tiwari S., Warnault G., “Quasilinear parabolic problem with p(x)-Laplacian: existence, uniqueness of weak solutions and stabilization”, NoDea-Nonlinear Differ. Equ. Appl., 23:3 (2016)
11. Erhardt A.H., “Existence of Solutions To Parabolic Problems With Nonstandard Growth and Irregular Obstacles”, Adv. Differ. Equat., 21:5-6 (2016), 463–504
12. Youssfi A., Azroul E., Lahmi B., “Nonlinear parabolic equations with nonstandard growth”, Appl. Anal., 95:12 (2016), 2766–2778
13. Niu W., Chai X., “Global attractors for nonlinear parabolic equations with nonstandard growth and irregular data”, J. Math. Anal. Appl., 451:1 (2017), 34–63
14. Erhardt A.H., “Compact embedding for p(x,?t)-Sobolev spaces and existence theory to parabolic equations with p(x,?t)-growth”, Rev. Mat. Complut., 30:1 (2017), 35–61
15. Erhardt A.H., “The Stability of Parabolic Problems With Nonstandard P (X, T)-Growth”, 5, no. 4, 2017, 50
16. Antontsev S., Kuznetsov I., Shmarev S., “Global Higher Regularity of Solutions to Singular P(X, T)-Parabolic Equations”, J. Math. Anal. Appl., 466:1 (2018), 238–263
17. Antontsev S., Shmarev S., “Higher Regularity of Solutions of Singular Parabolic Equations With Variable Nonlinearity”, Appl. Anal., 98:1-2, SI (2019), 310–331
18. Crispo F., Maremonti P., Ruzicka M., “Global l-R-Estimates and Regularizing Effect For Solutions to the P(T, X)-Laplacian Systems”, Adv. Differ. Equat., 24:7-8 (2019), 407–434
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