EVENTUAL C-INFINITY-REGULARITY AND CONCAVITY FOR FLOWS IN ONE-DIMENSIONAL POROUS-MEDIA

We study the regularity and the asymptotic behavior of the solutions of the initial value problem for the porous medium equation with m greater than 1 and, u//0 a continuous, nonnegative function. We prove that each moving part of the interface is a C** infinity -curve and that upsilon is C** infinity on each side of the moving interface (and up to it). We also prove that for solutions with compact support the pressure becomes a concave function of chi after a finite time. This fact implies sharp convergence rates for the solution and the interfaces as t yields infinity .