Global existence and decay of the inhomogeneous Muskat problem with Lipschitz initial data

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