Existence of solutions for some quasilinear parabolic systems in Orlicz spaces

In this work, we prove an existence theorem for quasilinear parabolic problems of the form ∂u∂t-div(σ(x,t,Du)+Φ(x,t,u))=finQ,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{\partial u}{\partial t}-\text {div}\big (\sigma (x,t,Du)+\varPhi (x,t,u)\big )=f\quad \text {in}\;Q, \end{aligned}$$\end{document}where f belongs to W-1,xEM¯(Q;Rm)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W^{-1,x}E_{\overline{M}}(Q;\mathbb {R}^m)$$\end{document}. The function σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma$$\end{document} and the lower term Φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varPhi$$\end{document} satisfy some conditions which will be used to prove the needed result through the theory of Young measures.