Global boundedness of the weak solutions to componentwise coercive parabolic systems

We prove essential boundedness of the weak solutions to the Cauchy-Dirichlet problem for the quasilinear parabolic system ut-div(A(x,t,u,Du))=b(x,t,u,Du)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\textbf{u}}_t- \mathrm {div\,}\big ({\textbf{A}}(x,t,{\textbf{u}},D{\textbf{u}})\big )= {\textbf{b}}(x,t,{\textbf{u}},D{\textbf{u}}) $$\end{document}which is modeled on the p-Laplacian vectorial operator. The nonlinear terms are given by Carath & eacute;odory functions and support controlled growth with respect to u\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{u}}$$\end{document} and Du,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D{\textbf{u}},$$\end{document} while their dependence on (x, t) is expressed in terms of suitable Lebesgue scales. Our result is proved by assuming additionally componentwise coercivity of the system and appropriate componentwise control of the lower-order terms.