Consider the equation \t\t\tut=div(dα|∇u|p−2∇u)+∂bi(u,x,t)∂xi,(x,t)∈Ω×(0,T),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$${u_{t}} = \\operatorname{div} \\bigl(d^{\\alpha} \\vert \\nabla u \\vert ^{p - 2}\\nabla u\\bigr) + \\frac{\\partial b_{i}(u,x,t)}{\\partial{x_{i}}},\\quad (x,t) \\in\\Omega \\times(0,T), $$\\end{document} where Ω is a bounded domain, d(x) is the distance function from the boundary ∂Ω. Since the nonlinearity, the boundary value condition cannot be portrayed by the Fichera function. If alpha< p-1, a partial boundary value condition is portrayed by a new way, the stability of the weak solutions is proved by this partial boundary value condition. If alpha>p-1, the stability of the weak solutions may be proved independent of the boundary value condition.