Extinction behaviour for the fast diffusion equations with critical exponent and Dirichlet boundary conditions
Year: 2020
Type: preprint
Abstract: 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 $Ω=B_1(0) : = \{x\in \mathbb{R}^n||x| < 1\}\subset\mathbb{R}^n$.
Source: arXiv (Cornell University)
Authors: Sire, Yannick, Wei, Juncheng, Zheng, Youquan
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W3033960088