  The inhomogeneous Muskat problem models the dynamics of an interface between
two fluids of differing characteristics inside a non-uniform porous medium. We
consider a porous media with a permeability jump across a horizontal boundary
away from an interface between two fluids of different viscosities and
densities. For initial data of explicit medium size, depending on the
characteristics of the fluids and porous media, we will prove the global
existence of a solution that is instantly analytic and decays in time to the
flat interface.
