Blow-up time estimates in porous medium equations with nonlinear boundary conditions

In this paper, we consider the blow-up problem of the following porous medium equations with nonlinear boundary conditions ut=Δum+k(t)f(u)inΩ×(0,t∗),∂u∂ν=g(u)on∂Ω×(0,t∗),u(x,0)=u0(x)inΩ¯,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} u_{t} =\Delta u^{m}+k(t)f(u) &{}\hbox { in } \Omega \times (0,t^{*}), \\ {}\displaystyle \frac{\partial u}{\partial \nu }=g(u) &{}\hbox { on } \partial \Omega \times (0,t^{*}), \\ {}\displaystyle u(x,0)=u_{0}(x) &{} \hbox { in } \overline{\Omega }, \end{array} \right. \end{aligned}$$\end{document}where m>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m>1$$\end{document}, Ω⊂Rn(n≥2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset \mathbb {R}^{n} \ (n\ge 2)$$\end{document} is a bounded convex domain with smooth boundary. Under appropriate assumptions on the data, a criterion is given to guarantee that solution u blows up at finite time, and an upper bound for blow-up time is derived. Moreover, a lower bound for blow-up time is also obtained.