This work is focused on the analysis of nonlinear flows of slightly compressible fluids in porous media not adequately described by Darcy's law. We study a class of generalized nonlinear momentum equations which covers all three well-known Forchheimer equations, the so-called two-term, power, and three-term laws. The generalized Forchheimer equation is inverted to a nonlinear Darcy equation with implicit permeability tensor depending on the pressure gradient. This results in a degenerate parabolic equation for the pressure. Two classes of boundary conditions are considered, given pressure and given total flux. In both cases they are allowed to be unbounded in time. The uniqueness, Lyapunov and asymptotic stabilities, and other long-time dynamical features of the corresponding initial boundary value problems are analyzed. The results obtained in this paper have clear hydrodynamic interpretations and can be used for quantitative evaluation of engineering parameters. Some numerical simulations are also included.