Smooth solution for the porous medium equation in a bounded domain

In this paper, we show the short time existence of the smooth solution for the porous medium equations in a smooth bounded domain:equation(0.1)ut=Δum(m>1) with zero boundary condition. On the fixed boundary ∂Ω¯, the flux ∇um∇um is nonzero while the gradient of pressure, ∇um−1∇um−1, is zero. As a consequence the parabolic equation above becomes degenerate. The proof is based on a definition of weighted space Cs2,γ corresponding to the given degeneracy and the Schauder estimates in its linearized equationequation(0.2)wt=xnαaij(x)Dijw in Cs2,γ for 0<α=m−1m<1.