Regularity in Sobolev spaces for the fast diffusion and the porous medium equation

The degenerate parabolic equation ut=Δ(|u|m−1u),m>0 is considered in a cylinder Ω×(0,T) under homogeneous Dirichlet boundary values. The regularity of weak solutions for the fast diffusion equation (m<1) and the porous medium equation (m>1) are investigated. Regularity of u and |u|m−1u in weighted Sobolev spaces and in fractional order Nikolskii spaces are proved.