Geodesic
Existence of a renormalized solution of a parabolic problem in anisotropic Sobolev--Orlicz spaces
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential Equations, Tome 163 (2019), pp. 39-64

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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.
Mots-clés : anisotropic parabolic equation, existence of solutions
Keywords: renormalized solution, nonpower nonlinearity, $N$-function. 
@article{INTO_2019_163_a3,
     author = {N. A. Vorobyev and F. Kh. Mukminov},
     title = {Existence of a renormalized solution of a parabolic problem in anisotropic {Sobolev--Orlicz} spaces},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {39--64},
     publisher = {mathdoc},
     volume = {163},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2019_163_a3/}
}
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N. A. Vorobyev; F. Kh. Mukminov. Existence of a renormalized solution of a parabolic problem in anisotropic Sobolev--Orlicz spaces. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential Equations, Tome 163 (2019), pp. 39-64. http://geodesic.mathdoc.fr/item/INTO_2019_163_a3/