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We consider the Fast Diffusion Equation $u_t=\Delta u^m$ posed in a boundedsmooth domain $\Omega\subset \RR^d$ with homogeneous Dirichlet conditions; theexponent range is $m_s=(d-2)_+/(d+2)<m<1$. It is known that bounded positivesolutions $u(t,x)$ of such problem extinguish in a finite time $T$, and alsothat such solutions approach a separate variable solution $u(t,x)\sim(T-t)^{1/(1-m)}S(x)$, as $t\to T^-$. Here we are interested in describing thebehaviour of the solutions near the extinction time. We first show that theconvergence $u(t,x)\,(T-t)^{-1/(1-m)}$ to $S(x)$ takes place uniformly in therelative error norm. Then, we study the question of rates of convergence of therescaled flow. For $m$ close to 1 we get such rates by means of entropy methodsand weighted Poincar\'e inequalities. The analysis of the latter point makes anessential use of fine properties of the associated stationary elliptic problem$-\Delta S^m= {\bf c} S$ in the limit $m\to 1$, and such a study has anindependent interest.

Behaviour near extinction for the Fast Diffusion Equation on bounded domains