Large-time geometric properties of solutions of the evolution p-Laplacian equation

We establish the behavior of the solutions of the degenerate parabolic equation ut = ∇ · ( |∇u|∇u ) , p > 2, posed in the whole space with nonnegative, continuous and compactly supported initial data. We prove a nonlinear concavity estimate for the pressure away from the maximum point. The estimate has important geometric consequences: it implies that the support of the solution becomes convex for large times and converges to a ball. In dimension one, we know also that the pressure itself eventually becomes p-concave. In several dimensions we prove concavity but for a small neighborhood of the maximum point.