Vanishing time behavior of solutions to the fast diffusion equation

Let $n \geq 3$, $0 \lt m \lt \frac{n-2}{n}$ and $T \gt 0$. We construct positive solutions to the fast diffusion equation $u_t = \Delta u^m$ in $\mathbb{R}^n \times (0, T)$, which vanish at time $T$. By introducing a scaling parameter $\beta$ inspired by $\href{https://dx.doi.org/10.4310/CAG.2019.v27.n8.a4}{\textrm{[DKS]}}$, we study the second-order asymptotics of the self-similar solutions associated with $\delta$ at spatial infinity. We also investigate the asymptotic behavior of the solutions to the fast diffusion equation near the vanishing time $T$, provided that the initial value of the solution is close to the initial value of some self-similar solution and satisfies some proper decay condition at infinity. Depending on the range of the parameter $\delta$, we prove that the rescaled solution converges either to a self-similar profile or to zero as $t \nearrow T$. The former implies asymptotic stabilization towards a self-similar solution, and the latter is a new vanishing phenomenon even for the case $n \geq 3$ and $m = \frac{n-2}{n+2}$ which corresponds to the Yamabe flow on $\mathbb{R}^n$ with metric $g = u^\frac{4}{n+2} dx^2$.

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Received 30 January 2020

Accepted 27 August 2020

Published 6 December 2023