*Accepted Paper*

**Inserted:** 3 jun 2019

**Last Updated:** 3 jun 2019

**Journal:** Comm. Pure Appl. Math.

**Year:** 2019

**Abstract:**

We investigate the homogeneous Dirichlet problem for the Fast Diffusion Equation $u_t=\Delta u^m$, posed in a smooth bounded domain $\Omega\subset \RR^N$, in the exponent range $m_s=(N-2)_+/(N+2)<m<1$. It is known that bounded positive solutions extinguish in a finite time $T>0$, and also that they approach a separate variable solution $u(t,x)\sim (T-t)^{1/(1-m)}S(x)$, as $t\to T^-$, where $S$ belongs to the set of solutions to a suitable elliptic problem and depends on the initial datum $u_0$. It has been shown recently that $v(x,t)=u(t,x)\,(T-t)^{-1/(1-m)}$ tends to $S(x)$ as $t\to T^-$, uniformly in the relative error norm. Starting from this result, we investigate the fine asymptotic behaviour and prove sharp rates of convergence for the relative error. The proof is based on an entropy method relying on a (improved) weighted Poincar\'e inequality, that we show to be true on generic bounded domains. Another essential aspect of the method is the new concept of ``almost orthogonality'', which can be thought as a nonlinear analogous of the classical orthogonality condition needed to obtain improved Poincar\'e inequalities and sharp convergence rates for linear flows.

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