The weak solutions of a doubly nonlinear parabolic equation related to the p(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p(x)$\end{document}-Laplacian

A nonlinear degenerate parabolic equation related to the p(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p(x)$\end{document}-Laplacian ut=div(b(x)|∇a(u)|p(x)−2∇a(u))+∑i=1N∂bi(u)∂xi+c(x,t)−b0a(u)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {u_{t}}= \operatorname{div} \bigl({b(x)} { \bigl\vert {\nabla a(u)} \bigr\vert ^{p(x) - 2}}\nabla a(u) \bigr)+\sum _{i=1}^{N}\frac{\partial b_{i}(u)}{ \partial x_{i}}+c(x,t) -b_{0}a(u) $$\end{document} is considered in this paper, where b(x)|x∈Ω>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$b(x)|_{x\in \varOmega }>0$\end{document}, b(x)|x∈∂Ω=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$b(x)|_{x \in \partial \varOmega }=0$\end{document}, a(s)≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a(s)\geq 0$\end{document} is a strictly increasing function with a(0)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a(0)=0$\end{document}, c(x,t)≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$c(x,t)\geq 0$\end{document} and b0>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$b_{0}>0$\end{document}. If ∫Ωb(x)−1p−−1dx≤c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\int _{\varOmega }b(x)^{-\frac{1}{p ^{-}-1}}\,dx\leq c$\end{document} and |∑i=1Nbi′(s)|≤ca′(s)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\vert \sum_{i=1}^{N}b_{i}'(s) \vert \leq c a'(s)$\end{document}, then the solutions of the initial-boundary value problem is well-posedness. When ∫Ωb(x)−(p(x)−1)dx<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\int _{\varOmega }b(x)^{-(p(x)-1)}\,dx<\infty $\end{document}, without the boundary value condition, the stability of weak solutions can be proved.