论文标题
具有关键指数和Dirichlet边界条件的快速扩散方程的灭绝行为
Extinction behaviour for the fast diffusion equations with critical exponent and Dirichlet boundary conditions
论文作者
论文摘要
对于平滑界域$ω\ subseteq \ mathbb {r}^n $，$ n \ geq 3 $，我们考虑具有关键Sobolev endent $ sobolev endent $ sobolev endem $ $ \ frac {\ partial w} {\ partial w} {\partialτ} { $ w（\ cdot，τ）= 0 $ oN $ \partialΩ$。使用抛物线胶合法，我们证明存在初始数据$ W_0 $，因此相应的解决方案的灭绝率$ \ | w（\ cdot，τ）\ | _ _ {l^\ infty（ω）} = γ_0（t-τ）^{\ frac {n+2} {4}} \ left | \ ln（t-τ）\ right |^{\ frac {n+2} {n+2} {2（n-2）}}}（n-2）}}（1+o（1+o（1+o（1+o（1））$ t as $ t as $ t \ to^to^to^ - t^ - $ tiention $ tiete $ tiention $ tient $ nise $ nise $ nime $ nise of intient y是pin $ nise of intien of intien of intien nime of。这将概括并提供严格的证明Galaktionov和king \ cite {galaktionov2001fast}的结果，用于径向对称情况$ω= b_1（0）：= \ {x \ in \ in \ mathbb {r}^n || x | <1 \} \ subset \ mathbb {r}^n $。
For a smooth bounded domain $Ω\subseteq\mathbb{R}^n$, $n\geq 3$, we consider the fast diffusion equation with critical sobolev exponent $$\frac{\partial w}{\partialτ} =Δw^{\frac{n-2}{n+2}}$$ under Dirichlet boundary condition $w(\cdot, τ) = 0$ on $\partialΩ$. 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 $$\|w(\cdot, τ)\|_{L^\infty(Ω)} = γ_0(T-τ)^{\frac{n+2}{4}}\left|\ln(T-τ)\right|^{\frac{n+2}{2(n-2)}}(1+o(1))$$ as $t\to T^-$, here $T > 0$ is the finite extinction time of $w(x, τ)$. This generalizes and provides rigorous proof of a result of Galaktionov and King \cite{galaktionov2001fast} for the radially symmetric case $Ω=B_1(0) : = \{x\in \mathbb{R}^n||x| < 1\}\subset\mathbb{R}^n$.
