Resumen de Geometrical properties of solutions of the Porous Medium Equation for large times

Ki-Ahm Lee, Juan Luis Vázquez

• We establish the behavior of the solutions of the degenerate parabolic equation ut = ∆(u m), m > 1, posed in the whole space when the initial data are nonnegative, continuous and compactly supported. We prove that after a finite time the pressure v = u m−1 becomes a concave function in the space variable which converges to all orders of differentiability to a truncated parabolic shape, so-called Barenblatt profile. In particular, the support of the solution is a convex subset of R N which converges to a ball. Estimates are optimal. The results are extended to the heat equation (log-concavity) and fast diffusion (pressure-convexity).