  In this work we study the inhomogeneous Muskat problem, \emph{i.e.} the
evolution of an internal wave between two different fluids in a porous medium
with discontinuous permeability. In particular, under precise conditions on the
initial datum and the physical quantities of the problem, our results ensure
the decay of the solutions towards the equilibrium state in the Lipschitz norm.
In addition, we establish the global existence and decay of Lipschitz
solutions.
