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

Abstract

For a smooth bounded domain $\Omega\subseteq\mathbb{R}^n$, $n\geq 3$, we consider the fast diffusion equation with critical sobolev exponent $$\frac{\partial w}{\partial\tau} =\Delta w^{\frac{n-2}{n+2}}$$ under Dirichlet boundary condition $w(\cdot, \tau) = 0$ on $\partial\Omega$. 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 $$\|w(\cdot, \tau)\|_{L^\infty(\Omega)} = \gamma_0(T-\tau)^{\frac{n+2}{4}}\left|\ln(T-\tau)\right|^{\frac{n+2}{2(n-2)}}(1+o(1))$$ 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 $\Omega =B_1(0) : = \{x\in \mathbb{R}^n||x| < 1\}\subset\mathbb{R}^n$.