On the boundary value conditions of evolutionary p_{i}-Laplacian equation

Abstract

The initial boundary value problem of the evolutionary p_{i}-Laplacian equation ut=∑i=1N∂∂xi(ai(x)|uxi|pi−2uxi)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {u_{t}}=\\sum_{i=1}^{N} \\frac{\\partial}{\\partial x_{i}} \\bigl(a_{i}(x) \\vert u_{x_{i}} \\vert ^{p_{i}-2}u_{x_{i}} \\bigr) $$\\end{document} is considered, where a_{i}(x) is nonnegative but is with 0 measure degeneracy. The weak solutions do not belong to BV_{x}(Q_{T}), how to define the trace in a reasonable way? This is the main topic of this paper. A suitable new boundary value condition is quoted and the stability of weak solutions follows naturally.