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

For a smooth bounded domain Ω⊆Rn$\Omega \subseteq \mathbb {R}^n$, n⩾3$n\geqslant 3$, we consider the fast diffusion equation with critical sobolev exponent ∂w∂τ=Δwn−2n+2\begin{equation*} \hspace*{7pc}\frac{\partial w}{\partial \tau } =\Delta w^{\frac{n-2}{n+2}}\hspace*{-7pc} \end{equation*}under Dirichlet boundary condition w(·,τ)=0$w(\cdot , \tau ) = 0$ on ∂Ω$\partial \Omega$. Using the parabolic gluing method, we prove existence of an initial data w0$w_0$ such that the corresponding solution has extinction rate of the form ∥w(·,τ)∥L∞(Ω)=γ0(T−τ)n+24ln(T−τ)n+22(n−2)(1+o(1))\begin{equation*} \hspace*{13pc}\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)) \end{equation*}as t→T−$t\rightarrow T^-$, here T>0$T > 0$ is the finite extinction time of w(x,τ)$w(x, \tau )$. This generalizes a result of Galaktionov and King [30] for the radially symmetric case Ω=B1(0):={x∈Rn||x|