A Continuum of Extinction Rates for the Fast Diffusion Equation

TL;DR

This paper identifies a continuous spectrum of extinction rates for solutions to the fast diffusion equation, showing how these rates depend on initial data decay and providing explicit formulas.

Contribution

It introduces a continuum of extinction rates for the fast diffusion equation, explicitly linking them to initial data decay rates, which was not previously known.

Findings

Extinction rates form a continuum depending on initial decay.

Explicit formulas for extinction rates are derived.

Rates are valid for a subrange of the exponent m in (0,1).

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.