Geometrical properties of solutions of the Porous Medium Equation for large times
Ki-Ahm Lee, J. L. Vazquez · Indiana University Mathematics Journal · 2003
We establish the behavior of the solutions of the degenerate parabolic equation u t = Δ(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).