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: {u(t) = Delta u(m) in Omega x [0, infinity), u(x,0) = u(0) > 0 in Omega, (0.1) u(x, t) = 0 for x is an element of partial derivative Omega where m > 1 is the permeability. The proof is based on the short time existence of C-s(2,(gamma) over bar)-smooth solution, the global C-s(1)-estimate, the Holder estimate of divergence type degenerate equation with measurable coefficients and C-s(1,(gamma) over bar)-estimate of mixed type equation with Lipschitz coefficients. (C) 2014 Elsevier Inc. All rights reserved.