We study the regularity of the free boundary for solutions of the porous medium equation u t = Δ u m u_{t}=\Delta u^{m} , m > 1 m >1 , on R 2 × [ 0 , T ] {\mathcal {R}}^{2} \times [0,T] , with initial data u 0 = u ( x , 0 ) u^{0}=u(x,0) nonnegative and compactly supported. We show that, under certain assumptions on the initial data u 0 u^{0} , the pressure f = m u m − 1 f=m\, u^{m-1} will be smooth up to the interface Γ = ∂ { u > 0 } \Gamma = \partial \{ u >0 \} , when 0 > t ≤ T 0>t\leq T , for some T > 0 T >0 . As a consequence, the free-boundary Γ \Gamma is smooth.

We provide the classification of locally conformally flat gradient Yamabe solitons with positive sectional curvature. We first show that locally conformally flat gradient Yamabe solitons with positive sectional curvature have to be rotationally symmetric and then give the classification and asymptotic behavior of all radially symmetric gradient Yamabe solitons. We also show that any eternal solutions to the Yamabe flow with positive Ricci curvature and with the scalar curvature attaining an interior space-time maximum must be a steady Yamabe soliton.

We study the compact noncollapsed ancient convex solutions to Mean Curvature Flow in R n+1 with O(1) × O(n) symmetry. We show they all have unique asymptotics as t → −∞ and we give precise asymptotic description of these solutions. In particular, solutions constructed by White, and Haslhofer and Hershkovits have those asymptotics (in the case of those particular solutions the asymptotics was predicted and formally computed by Angenent [2]).

We consider a one-parameter family of strictly convex hypersurfaces in R n+1 moving with speed −K α ν, where ν denotes the outward-pointing unit normal vector and α ≥ 1 n+2 . For α > 1 n+2 , we show that the flow converges to a round sphere after rescaling. In the affine invariant case α = 1 n+2 , our arguments give an alternative proof of the fact that the flow converges to an ellipsoid after rescaling.

We study the extinction behavior of solutions to the fast diffusion equation ut = ∆u m on R N × (0, T ), in the range of exponents m ∈ (0, N−2 N ), N > 2. We show that if the initial data u 0 is trapped in between two Barenblatt solutions vanishing at time T , then the vanishing behaviour of u at T is given by a Barenblatt solution. We also give an example showing that for such a behavior the bound from above by a Barenblatt solution B (vanishing at T ) is crucial: we construct a class of solutions u with initial data u 0 = B (1+o (1)), near |x| >> 1, which live longer than B and change behaviour at T . The behavior of such solutions is governed by B(·, t) up to T , while for t > T the solutions become integrable and exhibit a different vanishing profile. For the Yamabe flow (m = N−2 N+2 ) the above means that these solutions u develop a singularity at time T , when the Barenblatt solution disappears, and at t > T they immediately smoothen up and exhibit the vanishing profile of a sphere.In the appendix we show how we remove the assumption on the bound on u 0 by a Barenblatt from below.