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(0.1){ut=△umin Ω×[0,∞),u(x,0)=u0>0in Ω,u(x,t)=0for x∈∂Ω where m>1m>1 is the permeability. The proof is based on the short time existence of Cs2,γ¯-smooth solution, the global Cs1-estimate, the Hölder estimate of divergence type degenerate equation with measurable coefficients and Cs1,γ¯-estimate of mixed type equation with Lipschitz coefficients.