Panagiota Daskalopoulos

On the extinction profile of solutions to fast diffusion

Abstract We study the extinction behaviour of solutions to the fast diffusion equation ut = Δum on ℝ N × (0, T), in the range of exponents . We show that if the initial value 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 behaviour the bound from above by a Barenblatt solution B (vanishing at T) is crucial: we construct a class of solutions u with initial value u 0 = B(1 + o(1)), near |x| » 1, which live longer than B and change behaviour at T. The behaviour 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 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.

Eternal solutions to the Ricci flow on ℝ2

We provide the classification of eternal (or ancient) solutions of the two-dimensional Ricci flow, which is equivalent to the fast diffusion equation

All Time Smooth Solutions of the One-Phase Stefan Problem and the Hele-Shaw Flow

\frac{\partial u}{\partial t} = \Delta \log u

Hölder regularity of solutions of degenerate elliptic and parabolic equations

on

C^{1,1} regularity for degenerate elliptic obstacle problems

\R^2 \times \R.

Free-boundary regularity on the focusing problem for the Gauss Curvature Flow with flat sides

We show that, under the necessary assumption that for every

Borderline Estimates for Fully Nonlinear Elliptic Equations

t \in \R

The evolution of complete non-compact graphs by powers of Gauss curvature

, the solution

Nonradial solvability structure of super-diffusive nonlinear parabolic equations

u(\cdot, t)

On the Evolution of Convex Hypersurfaces by the Q k Flow

defines a complete metric of bounded curvature and bounded width,