Behaviour near extinction for the fast diffusion equation on bounded domains

We consider the Fast Diffusion Equation posed in a bounded smooth domain Ω ⊂ R^d with homogeneous Dirichlet conditions. It is known that in the exponent range m_s = (d − 2)_+/(d + 2) < m < 1 all bounded positive solutions u(t, x) of such problem extinguish in a finite time T = T (u), and also that such solutions approach a separate variable solution u(t, x) ∼ (T −t)1/(1−m)S(x), as t →T^−. Here, we are interested in describing the behaviour of the solutions near the extinction time in that range of exponents. We first show that the convergence v(x, t) = 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, i.e., v →S. For m close to 1 we get such rates by means of entropy methods and weighted Poincaré inequalities. The analysis of the latter point makes an essential use of fine properties of a associated stationary elliptic problem in the limit m→1, and such a study has an independent interest.