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

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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 29 scientific papers (total in 29 papers)

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

V. V. Zhikov^a, S. E. Pastukhova^b

^a Vladimir State Humanitarian University
^b Moscow State Institute of Radio-Engineering, Electronics and Automation (Technical University)

Full-text PDF (638 kB) Citations (29)

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DOI: https://doi.org/10.4213/mzm5256

Abstract: We study a parabolic system of the form

$\partial_tu=\operatorname{div}_xA(x,t,\nabla_xu)$ in a bounded cylinder

$Q_T=\Omega\times(0,T)\subset\mathbb R^{n+1}_{x,t}$ . Here the matrix function

$A(x,t,\xi)$ is subject to the conditions of power growth in the variable

$\xi$ 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

$\frac{2n}{n+2}<\alpha\le p(x,t)\le\beta<\infty.$
For the weak solution of the system, estimates of the higher integrability of the gradient are obtained inside the cylinder

$Q_T$ . 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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Linking options:

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This publication is cited in the following 29 articles:

Rakesh Arora, Sergey Shmarev, “Global gradient estimates for solutions of parabolic equations with nonstandard growth”, Journal of Mathematical Analysis and Applications, 2025, 129582
Cholmin Sin, “Global Higher Integrability for Symmetric p(x, t)-Laplacian System”, Mediterr. J. Math., 20:1 (2023)
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)
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
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)
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
Sin Ch., “Local Higher Integrability For Unsteady Motion Equations of Generalized Newtonian Fluids”, Nonlinear Anal.-Theory Methods Appl., 200 (2020), 112029
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
Igor I. Skrypnik, Mykhailo V. Voitovych, “$ {\mathfrak{B}}_1 $ 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
Antontsev S., Shmarev S., “Higher Regularity of Solutions of Singular Parabolic Equations With Variable Nonlinearity”, Appl. Anal., 98:1-2, SI (2019), 310–331
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
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
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
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
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
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
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
Tersenov A.S., “The one dimensional parabolic p(x)-Laplace equation”, NoDea-Nonlinear Differ. Equ. Appl., 23:3 (2016)
Erhardt A.H., “Existence of Solutions To Parabolic Problems With Nonstandard Growth and Irregular Obstacles”, Adv. Differ. Equat., 21:5-6 (2016), 463–504
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

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

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