Optimal estimates for Fractional Fast diffusion equations

We obtain a priori estimates with best constants for the solutions of the fractional fast diffusion equation ut+(-δ)σ/2um=0, posed in the whole space with 0<σ<2, 0<m≤1. The estimates are expressed in terms of convenient norms of the initial data, the preferred norms being the L1-norm and the Marcinkiewicz norm. The estimates contain exact exponents and best constants. We also obtain optimal estimates for the extinction time of the solutions in the range m near 0 where solutions may vanish completely in finite time. Actually, our results apply to equations with a more general nonlinearity. Our main tools are symmetrization techniques and comparison of concentrations. Classical results for σ=2 are recovered in the limit.