# Sharp Extinction Rates for Fast Diffusion Equations on Generic Bounded Domains

## TL;DR

This paper establishes sharp convergence rates for solutions to the fast diffusion equation near extinction time on generic bounded domains, using an entropy method and a novel 'almost orthogonality' concept.

## Contribution

It introduces a new approach combining an improved weighted Poincaré inequality and the 'almost orthogonality' concept to achieve sharp asymptotic rates.

## Key findings

- Proves sharp convergence rates for the relative error in fast diffusion equations.
- Develops an entropy method based on a weighted Poincaré inequality.
- Introduces the 'almost orthogonality' concept for nonlinear flows.

## Abstract

We investigate the homogeneous Dirichlet problem for the Fast Diffusion Equation $u_t=\Delta u^m$, posed in a smooth bounded domain $\Omega\subset \mathbb{R}^N$, 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 $u(t,x)\sim (T-t)^{1/(1-m)}S(x)$, as $t\to T^-$. 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.