Lifespan estimate for a porous medium equation with gradient terms under nonlinear boundary conditions

In the present paper, we investigate the following porous media equations with gradient terms:$ \left\{ \begin{array}{ll} (g(u))_t = \Delta u^{m}+f(x, u, |\nabla u|^{2}, t), &x\in D, \ t\in(0, T^{*}), \\ \frac{\partial u}{\partial\nu} = h(u), &x\in\partial D, \ t\in(0, T^{*}), \\ u(x, 0) = u_{0}(x), &x\in\overline{D}. \end{array} \right. $Here $ m>1 $, $ D $ is a convex bounded domain in $ R^{n} \ (n\geq2) $ with smooth boundary $ \partial D $. We establish some suitable conditions to ensure that the solutions of the above problem blow up in a finite blow-up time $ T^* $. Furthermore, with the help of some first-order differential inequalities and some embedding theorems in Sobolev space, the upper and lower bounds of $ T^* $ are also given.