Maximum estimates for generalized Forchheimer flows in heterogeneous porous media

This article continues our previous study of generalized Forchheimer flows in heterogeneous porous media. Such flows are used to account for deviations from Darcy's law. In heterogeneous media, the derived nonlinear partial differential equation for the pressure can be singular and degenerate in the spatial variables, in addition to being degenerate for large pressure gradient. Here we obtain the estimates for the $L^\infty$-norms of the pressure and its time derivative in terms of the initial and the time-dependent boundary data. They are established by implementing De Giorgi's iteration in the context of weighted norms with the weights specifically defined by the Forchheimer equation's coefficient functions. With these weights, we prove suitable weighted parabolic Poincar\'e-Sobolev inequalities and use them to facilitate the iteration. Moreover, local in time $L^\infty$-bounds are combined with uniform Gronwall-type energy inequalities to obtain long-time $L^\infty$-estimates.