On the stability of the equation with a partial boundary value condition

The degenerate parabolic equation with a convection term is considered. Let Ω be a bounded domain with C2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C^{2}$\end{document} smooth boundary and d(x)=dist(x,∂Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$d(x)=\operatorname{dist}(x, \partial\Omega)$\end{document} be the distance function from the boundary. If Δd≤0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Delta d\leq0$\end{document} when x is near to the boundary, then the stability of the entropy solutions is proved independent of the boundary value conditions. The degeneracy of the convection term on the boundary can take place of the usual boundary value condition.