Sharp Extinction Rates for Fast Diffusion Equations on Generic Bounded Domains

We investigate the homogeneous Dirichlet problem for the fast diffusion equation u(t) = Delta u(m), posed in a smooth bounded domain omega subset of Double-struck capital R-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) similar to (T - t)S1/(1 - m)(x) as t -> 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 -> T-, uniformly in the relative error norm. Starting from this result, we investigate the fine asymptotic behavior and prove sharp rates of convergence for the relative error. The proof is based on an entropy method relying on an (improved) weighted Poincare inequality, which 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 of as a nonlinear analogue of the classical orthogonality condition needed to obtain improved Poincare inequalities and sharp convergence rates for linear flows. (c) 2019 Wiley Periodicals, Inc.