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

Abstract

For a smooth bounded domain $\Omega \subseteq {\mathbb{R}}^n$, $n\ge 3$, 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 $$\Vert w(\cdot ,\tau ){\Vert }_{L^{\infty }(\Omega )}={\gamma }_0(T-\tau {)}^{\frac{n+2}{4}}{\left|\ln (T-\tau )\right|}^{\frac{n+2}{2(n-2)}}(1+o(1))$$ 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 $\Omega =B_1(0):=\{x\in {\mathbb{R}}^n||x|<1\}\subset {\mathbb{R}}^n$.

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