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arXiv 2006.01308math.AP
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具有临界指数和Dirichlet边界条件的快扩散方程的灭绝性

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

Yannick Sire, Juncheng Wei, Youquan Zheng

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摘要：

对于光滑有界区域$Ω\subseteq\mathbb{R}^n$，$n\geq 3$，我们考虑$\partialΩ$上具有临界Sobolev指数$$\frac{\partial w}{\partialτ} =Δw^{\frac{n-2}{n+2}}$$的快扩散方程在Dirichlet边界条件$w(\cdot, τ) = 0$下的解.利用抛物型胶合方法，证明了初始数据$w_0$的存在性，使得相应解的消光率为$$\|w(\cdot, τ)\|_{L^\infty(Ω)} = γ_0(T-τ)^{\frac{n+2}{4}}\left|\ln(T-τ)\right|^{\frac{n+2}{2(n-2)}}(1+o(1))$$ = $t\to T^-$，其中$T > 0$是$w(x, τ)$的有限消光时间.这推广并严格证明了Galaktionov和King \cite{galaktionov2001fast}关于径向对称情形$Ω=B_1(0) : = \{x\in \mathbb{R}^n||x| < 1\}\subset\mathbb{R}^n$的一个结果.

英文摘要：

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 $Ω=B_1(0) : = \{x\in \mathbb{R}^n||x| < 1\}\subset\mathbb{R}^n$.

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