Global existence and blow-up results for p-Laplacian parabolic problems under nonlinear boundary conditions

Abstract

This paper is devoted to studying the global existence and blow-up results for the following p-Laplacian parabolic problems: \t\t\t{(h(u))t=∇⋅(|∇u|p−2∇u)+f(u)in D×(0,t∗),∂u∂n=g(u)on ∂D×(0,t∗),u(x,0)=u0(x)≥0in D‾.\\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}$$\\textstyle\\begin{cases} (h(u) )_{t} =\\nabla\\cdot (|\\nabla u|^{p-2}\\nabla u )+f(u) &\\mbox{in } D\\times(0,t^{*}), \\\\ \\frac{\\partial u}{\\partial n}=g(u) &\\mbox{on } \\partial D\\times (0,t^{*}), \\\\ u(x,0)=u_{0}(x)\\geq0 & \\mbox{in } \\overline{D}. \\end{cases} $$\\end{document} Here p>2, the spatial region D in mathbb{R}^{N} (Ngeq2) is bounded, and ∂D is smooth. We set up conditions to ensure that the solution must be a global solution or blows up in some finite time. Moreover, we dedicate upper estimates of the global solution and the blow-up rate. An upper bound for the blow-up time is also specified. Our research relies mainly on constructing some auxiliary functions and using the parabolic maximum principles and the differential inequality technique.