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, Trudy Mat. Inst. Steklova, 270, MAIK Nauka/Interperiodica, Moscow, 2010, 21–32; Proc. Steklov Inst. Math., 270 (2010), 15

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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2010, Volume 270, Pages 21–32 (Mi tm3011)

This article is cited in 31 scientific papers (total in 31 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

Full-text PDF (210 kB) Citations (31)

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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

$p$ -Laplacian with a variable exponent

p (x, t)

$p(x,t)$ . We prove that if the measurable exponent

p

$p$ is separated from unity and infinity, then the problem has

W

$W$ - and

H

$H$ -solutions.

Received in October 2009

English version:
Proceedings of the Steklov Institute of Mathematics, 2010, Volume 270, Pages 15–26
DOI: https://doi.org/10.1134/S0081543810030028

Bibliographic databases:

Document Type: Article

UDC: 517.958

Language: Russian

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, Trudy 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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Linking options:

https://www.mathnet.ru/eng/tm3011

https://www.mathnet.ru/eng/tm/v270/p21

This publication is cited in the following 31 articles:

Mokhtar Naceri, “Variable exponents anisotropic nonlinear elliptic systems with Lp′→(⋅)-data”, Applicable Analysis, 2024, 1
Rakesh Arora, Sergey Shmarev, “Optimal global second-order regularity and improved integrability for parabolic equations with variable growth”, Advances in Nonlinear Analysis, 13:1 (2024)
Peter Kogut, Yaroslav Kohut, “Optimal Sparse Control Formulation for Reconstruction of Noise-Affected Images”, Axioms, 12:12 (2023), 1073
M. D. Surnachev, “Harnack's Inequality of Weak Type for the Parabolic $p (x)$ -Laplacian”, Math. Notes, 111:1 (2022), 161–165
Mikhail Surnachev, “On the weak Harnack inequality for the parabolic p ( x )-Laplacian”, ASY, 130:1-2 (2022), 127
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
S. N. Antontsev, I. V. Kuznetsov, S. A. Sazhenkov, “A shock layer arising as the source term collapses in the $p(\boldsymbol{x})$ -Laplacian equation”, Probl. anal. Issues Anal., 9(27):3 (2020), 31–53
Arora R., Giacomoni J., Warnault G., “Doubly Nonlinear Equation Involving P(X)-Homogeneous Operators: Local Existence, Uniqueness and Global Behaviour”, J. Math. Anal. Appl., 487:2 (2020), 124009
Arumugam G., Tyagi J., “Nonnegative Solutions to Reaction-Diffusion System With Cross-Diffusion and Nonstandard Growth Conditions”, Math. Meth. Appl. Sci., 43:10 (2020), 6576–6597
Ding M., Zhang Ch., Zhou Sh., “Global Boundedness and Holder Regularity of Solutions to General P(X, T)-Laplace Parabolic Equations”, Math. Meth. Appl. Sci., 43:9 (2020), 5809–5831
Arumugam G., Erhardt A.H., “Existence of Weak Solutions to a Certain Homogeneous Parabolic Neumann Problem Involving Variable Exponents and Cross-Diffusion”, J. Elliptic Parabol. Equat., 6:2 (2020), 685–709
Arumugam G., Erhardt A.H., “Existence and Uniqueness of Weak Solutions to Parabolic Problems With Nonstandard Growth and Cross Diffusion”, Electron. J. Differ. Equ., 2020, 123
Gurusamy Arumugam, Andre H. Erhardt, “Existence and uniqueness of weak solutions to parabolic problems with nonstandard growth and cross diffusion”, ejde, 2020:01-132 (2020), 123
Antontsev S., Shmarev S., “Higher Regularity of Solutions of Singular Parabolic Equations With Variable Nonlinearity”, Appl. Anal., 98:1-2, SI (2019), 310–331
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
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
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
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
Erhardt A.H., “The Stability of Parabolic Problems With Nonstandard P (X, T)-Growth”, 5, no. 4, 2017, 50
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

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