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: u(t) = Delta(m)(u) (m > 1) (0.1) with zero boundary condition. On the fixed boundary partial derivative(Omega) over bar, the flux del(m)(u) is nonzero while the gradient of pressure, del(m-1)(u), is zero. As a consequence the parabolic equation above becomes degenerate. The proof is based on a definition of weighted space C-s(2,gamma) corresponding to the given degeneracy and the Schauder estimates in its linearized equation w(t) = x(n)(alpha)a(ij)(x)D(ij)w (0.2) in C-s(2,gamma) for 0 < alpha = m-1/m < 1. (C) 2009 Elsevier Inc. All rights reserved.