A Continuum of Extinction Rates for the Fast Diffusion Equation

Abstract

We find a continuum of extinction rates for solutions $u(y,\tau)\ge 0$ of the fast diffusion equation $u_\tau=\Delta u^m$ in a subrange of exponents $m\in (0,1)$. The equation is posed in $\ren$ for times up to the extinction time $T>0$. The rates take the form $\|u(\cdot,\tau)\|_\infty\sim (T-\tau)^\theta$ \ for a whole interval of $\theta>0$. These extinction rates depend explicitly on the spatial decay rates of initial data.