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Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz., 2019, Volume 163, Pages 39–64 (Mi into450)  

Existence of a renormalized solution of a parabolic problem in anisotropic Sobolev–Orlicz spaces

N. A. Vorobyeva, F. Kh. Mukminovbc

a Mavlyutov Institute of Mechanics, Ufa Centre of the Russian Academy of Sciences
b Ufa State Aviation Technical University
c Institution of Russian Academy of Sciences Institute of Mathematics with Computer Center, Ufa

Abstract: We consider the first mixed problem for a certain class of anisotropic parabolic equations of the form
$$ (\beta(x,u))'_t-\operatorname{div} a(t,x,u,\nabla u) -b(t,x,u,\nabla u)=\mu $$
where $\mu$ is a measure and the coefficients contain noonpower nonlinearities in the cylindrical domain $D^T=(0,T)\times\Omega$, where $\Omega\subset \mathbb{R}^n$ is a bounded domain. We prove the existence of a renormalized solution of the problem for $g_t=0$ and a function $\beta(x,r)$, which increases with respect to $r$ and satisfies the Carathéodory condition.

Keywords: anisotropic parabolic equation, renormalized solution, nonpower nonlinearity, existence of solutions, $N$-function

Funding Agency Grant Number
Russian Foundation for Basic Research 18-01-00428а
This work was supported by the Russian Foundation for Basic Research (project No. 18-01-00428а).


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Bibliographic databases:
UDC: 517.954, 517.956.45, 517.958:531.72
MSC: 35K20, 35K55, 35K65

Citation: N. A. Vorobyev, F. Kh. Mukminov, “Existence of a renormalized solution of a parabolic problem in anisotropic Sobolev–Orlicz spaces”, Differential Equations, Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz., 163, VINITI, Moscow, 2019, 39–64

Citation in format AMSBIB
\Bibitem{VorMuk19}
\by N.~A.~Vorobyev, F.~Kh.~Mukminov
\paper Existence of a renormalized solution of a parabolic problem in anisotropic Sobolev--Orlicz spaces
\inbook Differential Equations
\serial Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz.
\yr 2019
\vol 163
\pages 39--64
\publ VINITI
\publaddr Moscow
\mathnet{http://mi.mathnet.ru/into450}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=4014975}


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