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

• DOI: 10.1007/s00033-018-0993-y
• Publication Date: 2018-07-06T10:10:35Z
• Authors: Juntang Ding; Xuhui Shen
• Keywords: 0101 mathematics; 01 natural sciences

In this paper, we consider the blow-up problem of the following porous medium equations with nonlinear boundary conditions $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} u_{t} =\Delta u^{m}+k(t)f(u) &{}\hbox { in } \Omega \times (0,t^{*}), \\ {}\displaystyle \frac{\partial u}{\partial \nu }=g(u) &{}\hbox { on } \partial \Omega \times (0,t^{*}), \\ {}\displaystyle u(x,0)=u_{0}(x) &{} \hbox { in } \overline{\Omega }, \end{array} \right. \end{aligned}$$ where $$m>1$$ , $$\Omega \subset \mathbb {R}^{n} \ (n\ge 2)$$ 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.