  In this paper we study the "stiff pressure limit" of the porous medium
equation, where the initial density is a bounded, integrable function with a
sufficient decay at infinity. Our particular model, introduced by
Perthame-Quiros-Vazquez, describes the growth of a tumor zone with a
restriction on the maximal cell density. In a general context, this extends
previous results of Caffarelli-Vazquez and Kim who restrict the initial data to
be the characteristic function of a compact set. In the limit a Hele-Shaw type
problem is obtained, where the interface motion law reflects the acceleration
effect of the presence of a positive cell density on the expansion of the
maximal density (tumor) zone.
