Blow-up in p-Laplacian heat equations with nonlinear boundary conditions

In this paper, we investigate the blow-up of solutions to the following p-Laplacian heat equations with nonlinear boundary conditions: h(u)t=∇·(|∇u|p∇u)+k(t)f(u)inΩ×(0,t∗),|∇u|p∂u∂n=g(u)on∂Ω×(0,t∗),u(x,0)=u0(x)≥0inΩ¯,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{\begin{array}{l@{\quad}l}\left(h(u)\right)_t =\nabla\cdot(|\nabla u|^{p}\nabla u)+k(t)f(u)\,&{\rm in }\, \Omega\times(0,t^{*}), \\ |\nabla u|^{p}\frac{\partial u}{\partial n}=g(u)\,\, &{\rm on}\, \partial\Omega\times(0,t^{*}), \\ u({\rm x},0)=u_{0}(x) \geq 0\, & {\rm in }\, \overline{\Omega},\end{array}\right.$$\end{document}where p≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p \geq 0}$$\end{document} and Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Omega}$$\end{document} is a bounded convex domain in RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}^{N}}$$\end{document}, N≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${N \geq 2}$$\end{document} with smooth boundary ∂Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\partial\Omega}$$\end{document}. By constructing suitable auxiliary functions and using a first-order differential inequality technique, we establish the conditions on the nonlinearities and data to ensure that the solution u(x, t) blows up at some finite time. Moreover, the upper and lower bounds for the blow-up time, when blow-up does occur, are obtained.