Positive solutions $u$ to the fractional fast diffusion equation $\partial_t u + (-\Delta)^s (u^\frac{N-2s}{N+2s}) = 0$ on $(0, T^*) \times \mathbb R^N$ are known to extinguish in finite time $T^* < \infty$ provided the initial datum is regular enough. Moreover, as $t \to T^*$, the extinction is governed by a certain profile $U_{T_*, z, \lambda}$.

In this talk, we strengthen the above assertions by giving optimal quantitative rates of convergence. More precisely, we prove the bound $\frac{u(t,\cdot)}{U_{T_*, z, \lambda}(t,\cdot)} - 1 = \mathcal O( (T_*-t)^\frac{N+2s}{N-2s+2})$, in a natural weighted energy norm. The main point here is that the exponent $\frac{N+2s}{N-2s+2}$ is sharp. This is the analogue of recent results by Bonforte and Figalli (CPAM, 2021) and Akagi (ARMA, 2023) valid for $s = 1$ and bounded domains $\Omega \subset \mathbb R^N$. The additional difficulty, besides including the fractional setting, is the degeneracy of the linearized operator stemming from the symmetries of the limit equation on $\mathbb R^N$, which is not present in the domain case. As a consequence, our result is new also in the local case $s = 1$.

We obtain similar results for the fractional fast diffusion equation $\partial_t u + (-\Delta)^s (u^m) = 0$ on bounded domains $\Omega \subset \mathbb R^N$, with subcritical exponent $m \in (\frac{N-2s}{N+2s}, 1)$.

This is joint work with Meng Yu (Goethe-Universität Frankfurt).