Blow-up phenomena for p-Laplacian parabolic problems with Neumann boundary conditions

Abstract

In this paper, we deal with the blow-up and global solutions of the following p-Laplacian parabolic problems with Neumann boundary conditions: \t\t\t{(g(u))t=∇⋅(|∇u|p−2∇u)+k(t)f(u)in Ω×(0,T),∂u∂n=0on ∂Ω×(0,T),u(x,0)=u0(x)≥0in Ω‾,\\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}$$\\textstyle\\begin{cases} (g(u) )_{t} =\\nabla\\cdot ( {|\\nabla u|^{p-2}}\\nabla u )+k(t)f(u) & \\mbox{in } \\Omega\\times(0,T), \\\\ \\frac{\\partial{u}}{\\partial n}=0 &\\mbox{on } \\partial\\Omega\\times (0,T), \\\\ u(x,0)=u_{0}(x)\\geq0 & \\mbox{in } \\overline{\\Omega}, \\end{cases} $$\\end{document} where p>2 and Ω is a bounded domain in mathbb{R}^{n} (ngeq 2) with smooth boundary ∂Ω. By introducing some appropriate auxiliary functions and technically using maximum principles, we establish conditions to guarantee that the solution blows up in some finite time or remains global. In addition, the upper estimates of blow-up rate and global solution are specified. We also obtain an upper bound of blow-up time.