Boundedness of Solutions to a Degenerate Diffusion Equation

The diffusion equation with a bounded saturation range under the time derivative and with Robin boundary conditions is shown to admit a regular bounded solution provided that the saturation function and the permeability coefficient have controlled decay at infinity. The result remains valid even if Preisach hysteresis is present in the pressure-saturation relation. The method of proof is based on a Moser-Alikakos iteration scheme which is compatible with a generalized Preisach energy dissipation mechanism.