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 Mat. Sb., 2018, Volume 209, Number 5, Pages 120–144 (Mi msb8921)

Existence of a renormalized solution to an anisotropic parabolic problem with variable nonlinearity exponents

F. Kh. Mukminovab

a Institution of Russian Academy of Sciences Institute of Mathematics with Computer Center, Ufa
b Ufa State Aviation Technical University

Abstract: The first boundary value problem is considered for a certain class of anisotropic parabolic equations with variable nonlinearity exponents in a cylindrical domain $( 0,T)\times\Omega$, where $\Omega$ is a bounded domain. The parabolic term in the equation has the form $(\beta(x,u))_t$ and is determined by the function $\beta(x,r)\in L_1(\Omega)$, where $r\in \mathbb R$, which only satisfies the Carathéodory condition and is increasing in $r$. The existence of a weak and a renormalized solution is proved.
Bibliography: 26 titles.

Keywords: anisotropic parabolic equation, renormalized solution, variable nonlinearity exponents, existence of a solution.

DOI: https://doi.org/10.4213/sm8921

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English version:
Sbornik: Mathematics, 2018, 209:5, 714–738

Bibliographic databases:

UDC: 517.954+517.956.45+517.958:531.72
MSC: 35K59

Citation: F. Kh. Mukminov, “Existence of a renormalized solution to an anisotropic parabolic problem with variable nonlinearity exponents”, Mat. Sb., 209:5 (2018), 120–144; Sb. Math., 209:5 (2018), 714–738

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb8921
• https://doi.org/10.4213/sm8921
• http://mi.mathnet.ru/eng/msb/v209/i5/p120

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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. Kozhevnikova L.M., “On Solutions of Anisotropic Elliptic Equations With Variable Exponent and Measure Data”, Complex Var. Elliptic Equ.
2. A. K. Gushchin, “The boundary values of solutions of an elliptic equation”, Sb. Math., 210:12 (2019), 1724–1752
3. F. Kh. Mukminov, “Existence of a Renormalized Solution to an Anisotropic Parabolic Problem for an Equation with Diffuse Measure”, Proc. Steklov Inst. Math., 306 (2019), 178–195
4. V. F. Vil'danova, “Existence and uniqueness of a weak solution of an integro-differential aggregation equation on a Riemannian manifold”, Sb. Math., 211:2 (2020), 226–257
5. A. K. Guschin, “Obobscheniya prostranstva nepreryvnykh funktsii; teoremy vlozheniya”, Matem. sb., 211:11 (2020), 54–71
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