Extinction behaviour for the fast diffusion equations with critical exponent and Dirichlet boundary conditions

For a smooth bounded domain $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$, $n\ge 3$, we consider the fast diffusion equation with critical sobolev exponent $$\frac{\mathrm{\partial }w}{\mathrm{\partial }\tau }=\mathrm{\Delta }{w}^{\frac{n-2}{n+2}}$$ under Dirichlet boundary condition $w(\cdot ,\tau )=0$ on $\mathrm{\partial }\mathrm{\Omega }$. Using the parabolic gluing method, we prove existence of an initial data $w_0$ such that the corresponding solution has extinction rate of the form $$\mathrm{\parallel }w(\cdot ,\tau ){\mathrm{\parallel }}_{L^{\mathrm{\infty }}(\mathrm{\Omega })}={\gamma }_0(T-\tau {)}^{\frac{n+2}{4}}{\mid \ln (T-\tau )\mid }^{\frac{n+2}{2(n-2)}}(1+o(1))$$ as $t\to T^{-}$, here $T>0$ is the finite extinction time of $w(x,\tau )$. This generalizes and provides rigorous proof of a result of Galaktionov and King \cite{galaktionov2001fast} for the radially symmetric case $\Omega =B_1(0) : = {x\in $\mathbb{R}$^n||x| < 1}\subset$\mathbb{R}$^n$.