Behaviour near extinction for the Fast Diffusion Equation on bounded domains

We consider the Fast Diffusion Equation $u_t=\Delta u^m$ posed in a bounded smooth domain $\Omega\subset \RR^d$ with homogeneous Dirichlet conditions; the exponent range is $m_s=(d-2)_+/(d+2)<m<1$. It is known that bounded positive solutions $u(t,x)$ of such problem extinguish in a finite time $T$, and also that 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 the behaviour of the solutions near the extinction time. We first show that the convergence $u(t,x)\,(T-t)^{-1/(1-m)}$ to $S(x)$ takes place uniformly in the relative error norm. Then, we study the question of rates of convergence of the rescaled flow. For $m$ close to 1 we get such rates by means of entropy methods and weighted Poincar\'e inequalities. The analysis of the latter point makes an essential 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 an independent interest.