Long time existence of Smooth solution for the porous medium equation in a bounded domain

In this paper, we are going to show the long time existence of the smooth solution for the porous medium equations in a smooth bounded domain: {equation} {cases} u_t=\La u^m\quad$\text{in }\mathrm{\Omega }\times [0,\mathrm{\infty })$ u(x,0)=u_0>0\quad$\text{in }\mathrm{\Omega }$ u(x,t)=0\quad\text{for $x\in \mathrm{\partial }\mathrm{\Omega }$} {cases} {equation} where $m>1$ is the permeability. The proof is based on the short time existence of $C_s^{2,\bar{\gamma }}$-smooth solution, the global $C_s^1$-estimate, the H"older estimate of divergence type degenerate equation with measurable coefficients and $C_s^{1,\bar{\gamma }}$-estimate of mixed type equation with Lipschitz coefficients.