The stability of the solutions of an equation related to the p-Laplacian with degeneracy on the boundary

The equation related to the p-Laplacian ut=div(ρα|∇u|p−2∇u)+∑i=1N∂bi(u)∂xi,(x,t)∈Ω×(0,T),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_{t}= \operatorname{div} \bigl(\rho^{\alpha} \vert \nabla u \vert ^{p - 2}\nabla u \bigr) + \sum_{i = 1}^{N} \frac{\partial b_{i}(u)}{\partial x_{i}},\quad (x,t) \in \Omega \times(0,T), $$\end{document} is considered, where ρ(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}$\rho(x) = \operatorname{dist} (x,\partial\Omega )$\end{document} is the distance function from the boundary. If α<p−1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha< p-1$\end{document}, the weak solution belongs to Hγ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H^{\gamma}$\end{document} for some γ>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\gamma>1$\end{document}, the Dirichlet boundary condition can be imposed as usual, the stability of the solutions is proved. If α≥p−1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha\geq p-1$\end{document}, the weak solution lacks the regularity to define the trace on the boundary. It is surprising that we can still prove the stability of the solutions without any boundary condition. In other words, when α≥p−1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha\geq p-1$\end{document}, the phenomenon that the solutions of the equation may be free from any limitations of the boundary condition is revealed.