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Matematicheskie Zametki, 2010, Volume 87, Issue 2, Pages 179–200
DOI: https://doi.org/10.4213/mzm5256
(Mi mzm5256)
 

This article is cited in 28 scientific papers (total in 28 papers)

On the Property of Higher Integrability for Parabolic Systems of Variable Order of Nonlinearity

V. V. Zhikova, S. E. Pastukhovab

a Vladimir State Humanitarian University
b Moscow State Institute of Radio-Engineering, Electronics and Automation (Technical University)
References:
Abstract: We study a parabolic system of the form tu=divxA(x,t,xu) in a bounded cylinder QT=Ω×(0,T)Rn+1x,t. Here the matrix function A(x,t,ξ) is subject to the conditions of power growth in the variable ξ and coercitivity with variable exponent p(x,t). It is assumed that p(x,t) has a logarithmic modulus of continuity and satisfies the estimate
2nn+2<αp(x,t)β<.
For the weak solution of the system, estimates of the higher integrability of the gradient are obtained inside the cylinder QT. The method of a solution is based on a localization of a special kind and a local variant (adapted for parabolic problems) of Gehring's lemma with variable exponent of integrability proved in the paper.
Keywords: parabolic system of variable order of nonlinearity, higher integrability for parabolic systems, Cacciopolli's inequality, Sobolev–Poincaré inequalities, Hölder's reverse inequality, Gehring's lemma, Lebesgue space, Sobolev–Orlicz space, Orlicz space.
Received: 22.06.2008
English version:
Mathematical Notes, 2010, Volume 87, Issue 2, Pages 169–188
DOI: https://doi.org/10.1134/S0001434610010256
Bibliographic databases:
UDC: 517.95
Language: Russian
Citation: V. V. Zhikov, S. E. Pastukhova, “On the Property of Higher Integrability for Parabolic Systems of Variable Order of Nonlinearity”, Mat. Zametki, 87:2 (2010), 179–200; Math. Notes, 87:2 (2010), 169–188
Citation in format AMSBIB
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  • https://doi.org/10.4213/mzm5256
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  • This publication is cited in the following 28 articles:
    1. Cholmin Sin, “Global Higher Integrability for Symmetric p(x, t)-Laplacian System”, Mediterr. J. Math., 20:1 (2023)  crossref
    2. Rakesh Arora, Sergey Shmarev, “Existence and regularity results for a class of parabolic problems with double phase flux of variable growth”, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat., 117:1 (2023)  crossref
    3. Skrypnik I.I., Voitovych M.V., “On the Continuity of Solutions of Quasilinear Parabolic Equations With Generalized Orlicz Growth Under Non-Logarithmic Conditions”, Ann. Mat. Pura Appl., 201:3 (2022), 1381–1416  crossref  mathscinet  isi
    4. Igor I. Skrypnik, “Harnack's inequality for singular parabolic equations with generalized Orlicz growth under the non-logarithmic Zhikov's condition”, J. Evol. Equ., 22:2 (2022)  crossref
    5. Arora R., Shmarev S., “Strong Solutions of Evolution Equations With P(X, T)-Laplacian: Existence, Global Higher Integrability of the Gradients and Second-Order Regularity”, J. Math. Anal. Appl., 493:1 (2021), 124506  crossref  mathscinet  isi
    6. Sin Ch., “Local Higher Integrability For Unsteady Motion Equations of Generalized Newtonian Fluids”, Nonlinear Anal.-Theory Methods Appl., 200 (2020), 112029  crossref  mathscinet  isi  scopus
    7. Antontsev S., Shmarev S., “Global Estimates For Solutions of Singular Parabolic and Elliptic Equations With Variable Nonlinearity”, Nonlinear Anal.-Theory Methods Appl., 195 (2020), 111724  crossref  mathscinet  isi
    8. Igor I. Skrypnik, Mykhailo V. Voitovych, “B1 classes of De Giorgi, Ladyzhenskaya, and Ural'tseva and their application to elliptic and parabolic equations with nonstandard growth”, J Math Sci, 246:1 (2020), 75  crossref
    9. 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
    10. Igor Skrypnik, Mykhailo Voitovych, “\mathfrak{B}_{1} classes of De Giorgi, Ladyzhenskaya, and Ural'tseva and their application to elliptic and parabolic equations with nonstandard growth”, UMB, 16:3 (2019), 403  crossref
    11. Kim S., “Lipschitz Regularity For Viscosity Solutions to Parabolic P(X,T)-Laplacian Equations on Riemannian Manifolds”, NoDea-Nonlinear Differ. Equ. Appl., 25:4 (2018), 27  crossref  mathscinet  isi  scopus
    12. 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
    13. 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
    14. 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
    15. Erhardt A.H., “Higher Integrability For Solutions To Parabolic Problems With Irregular Obstacles and Nonstandard Growth”, J. Math. Anal. Appl., 435:2 (2016), 1772–1803  crossref  mathscinet  zmath  isi  scopus
    16. 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
    17. Tersenov A.S., “The one dimensional parabolic p(x)-Laplace equation”, NoDea-Nonlinear Differ. Equ. Appl., 23:3 (2016)  crossref  mathscinet  zmath  isi  scopus
    18. 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
    19. 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
    20. V. Bögelein, Q. Li, “Very weak solutions of degenerate parabolic systems with non-standard p(x,t)-growth”, Nonlinear Anal., 98 (2014), 190–225  crossref  mathscinet  zmath  isi  scopus
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