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 $\Vert u(\cdot ,\tau ){\Vert }_{\infty }\sim (T-\tau {)}^{\theta }$ \ for a whole interval of $\theta >0$. These extinction rates depend explicitly on the spatial decay rates of initial data.