Vanishing time behavior of solutions to the fast diffusion equation

Abstract

Let $n\ge 3$, $0<m<\frac{n-2}{n}$ and $T>0$. 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 \cite{DKS}, we study the second-order asymptotics of the self-similar solutions associated with $\beta$ 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 $\beta$, 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 $n\ge 3$ and $m=\frac{n-2}{n+2}$ which corresponds to the Yamabe flow on ${\mathbb{R}}^n$ with metric $g=u^{\frac{4}{n+2}}d{x}^2$.