Existence and uniqueness of bounded weak solutions for some nonlinear parabolic problems

In this paper we study a class of nonlinear parabolic problems with p(x,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p(x,t)$\end{document} growth conditions. We prove the existence and uniqueness of bounded solutions to such a problem, with less constraint to p(x,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p(x,t)$\end{document}. Our results are generalizations of the corresponding results in the constant exponent case.