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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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\by Yu.~A.~Alkhutov, V.~V.~Zhikov
\paper Existence theorems for solutions of parabolic equations with variable order of nonlinearity
\inbook Differential equations and dynamical systems
\bookinfo Collected papers
\serial Tr. Mat. Inst. Steklova
\yr 2010
\vol 270
\pages 21--32
\publ MAIK Nauka/Interperiodica
\publaddr Moscow
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    Citing articles on Google Scholar: Russian citations, English citations
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    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  crossref  mathscinet  zmath  isi  elib  scopus
    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  crossref  mathscinet  zmath  isi  elib  scopus
    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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    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  mathnet  crossref  elib
    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  mathnet  crossref
    6. Chai X., Li H., Niu W., “Large Time Behavior For P(X)-Laplacian Equations With Irregular Data”, Electron. J. Differ. Equ., 2015, 61  mathscinet  zmath  isi  elib
    7. Zou W., Li J., “Existence and Uniqueness of Bounded Weak Solutions For Some Nonlinear Parabolic Problems”, Bound. Value Probl., 2015, 69  crossref  mathscinet  zmath  isi  elib  scopus
    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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    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  crossref  mathscinet  zmath  isi  scopus
    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)  crossref  mathscinet  zmath  isi  scopus
    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  mathscinet  zmath  isi  elib
    12. Youssfi A., Azroul E., Lahmi B., “Nonlinear parabolic equations with nonstandard growth”, Appl. Anal., 95:12 (2016), 2766–2778  crossref  mathscinet  zmath  isi  elib  scopus
    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  crossref  mathscinet  zmath  isi  scopus
    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  crossref  mathscinet  zmath  isi  scopus
    15. Erhardt A.H., “The Stability of Parabolic Problems With Nonstandard P (X, T)-Growth”, 5, no. 4, 2017, 50  crossref  zmath  isi  scopus
    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  crossref  mathscinet  zmath  isi  scopus
    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  crossref  mathscinet  isi  scopus
    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  isi
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