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We establish the behavior of the solutions of the degenerate parabolic equation u t = ∇ · |∇u| p−2 ∇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.

We study the local behavior of the nodal sets of the solutions to elliptic quasilinear equations with nonlinear conductivity part,is assumed to be C α in case s = 0, and C 1,α (or higher) in case s > 0.Using geometric methods, we prove almost complete results (in analogy with standard PDEs) concerning the behavior of the nodal sets. More exactly, we show that the nodal sets, where solutions have (linear) nondegeneracy, are locally smooth graphs. Degenerate points are shown to have structures that follow the lines of arguments as that of the nodal sets for harmonic functions, and general PDEs.

In this paper, we establish local Hölder estimate for non-negative solutions of the singular equation (M.P) below, for m in the range of exponents ( n−2σ n+2σ , 1). Since we have trouble in finding the local energy inequality of v directly. we use the fact that the operator (−△) σ can be thought as the normal derivative of some extension v * of v to the upper half space, [CS], i.e., v is regarded as boundary value of v * the solution of some local extension problem. Therefore, the local Hölder estimate of v can be obtained by the same regularity of v * . In addition, it enables us to describe the behaviour of solution of non-local fast diffusion equation near their extinction time.1