Positive solutions of a nonlinear parabolic equation with double variable exponents

The well-posedness problem of a nonlinear parabolic equation with double variable exponents is studied in this paper. This kind of nonlinear parabolic equation includes the non-Newtonian fluids equation, the polytropic filtration equation and the so-called electro-rheological fluid equation. One of the important characteristics is that there is a diffusion coefficient a(x, t) in the equation. Unlike the usual assumption a(x,t)>a>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a(x,t)>a>0$$\end{document}, the paper only assumes a(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}$$a(x,t)\ge 0$$\end{document}. If a(x,t)|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}$$a(x,t)|_{x\in \partial \Omega }=0$$\end{document}, by choosing a(x, t) as a test function, the stability result of positive solutions can be established without the boundary value condition.