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Optimal estimates for Fractional Fast diffusion equations

We obtain a priori estimates with best constants for the solutions of the fractional fast diffusion equation $u_t+(-Δ)^{σ/2}u^m=0$, posed in the whole space with $0<σ<2$, $0<m\le 1$. The estimates are expressed in terms of convenient norms of the initial data, the preferred norms being the $L^1$-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.