Existence and localization of weak solutions of nonlinear parabolic equations with variable exponent of nonlinearity

The aim of this paper is to study the existence and uniqueness of weak solutions of the initial Neumann problem for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${u_{t}={\rm div}(|\nabla u|^{p(x,t)-2}\nabla u+\vec{F}(x,t))}$$\end{document}. First, the authors construct suitable function spaces to which the solution belongs and then applies Galerkin’s approximation technique to prove the existence of weak solutions with necessary uniform estimates and a compactness argument. Second, the authors obtain the properties of extinction in finite time of weak solutions under suitable conditions by proving some energy estimates and applying a comparison principle.