L infty loc estimates for local solutions of degenerate parabolic equations

Local sup-estimates for solutions of the porous media equation (or its quasi-linear variations) are employed in several papers to study the behavior (local or at infinity) of such solutions. We are interested here in local sup-estimates which are completely independent of initial or boundary data. To our knowledge quantitative estimates of this kind are not present in the literature in a sharp form. In this paper we prove a sharp estimate, coinciding with known results in the linear case. The estimate will have the form of an interpolation inequality with a free parameter.