  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.
