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 $m_s=(N-2)_+/(N+2) 0$, 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 Poincare 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 Poincare inequalities and sharp convergence rates for linear flows.

中文翻译：

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通用有界域上快速扩散方程的急剧消光率

我们研究了快速扩散方程 $u_t=\Delta u^m$ 的齐次狄利克雷问题，提出在平滑有界域 $\Omega\subset \mathbb{R}^N$ 中，在指数范围 $m_s=(N -2)_+/(N+2) 0$，并且他们接近一个单独的变量解 $u(t,x)\sim (Tt)^{1/(1-m)}S(x)$ , 作为 $t\to T^-$。最近已经表明 $v(x,t)=u(t,x)\,(Tt)^{-1/(1-m)}$ 趋向于 $S(x)$ 作为 $t\to T^-$，统一在相对误差范数中。从这个结果开始，我们研究了精细的渐近行为并证明了相对误差的收敛速度。该证明基于依赖于（改进的）加权庞加莱不等式的熵方法，我们证明在通用有界域上是正确的。该方法的另一个重要方面是“几乎正交性”的新概念，