# Super fast vanishing solutions of the fast diffusion equation

## TL;DR

This paper constructs explicit subsolutions and supersolutions for the fast diffusion equation, demonstrating the existence of solutions that vanish at a specific rate near the extinction time, extending recent results in the field.

## Contribution

It extends recent work by constructing solutions with prescribed decay rates for the fast diffusion equation, providing new insights into their vanishing behavior.

## Key findings

- Existence of solutions with decay rate $(T-t)^{(1+eta)/(1-m)}$ near extinction
- Construction of subsolutions and supersolutions for the fast diffusion equation
- Extension of previous results to broader parameter ranges

## Abstract

We will extend a recent result of B.Choi, P.Daskalopoulos and J.King. For any $n\ge 3$, $0<m<\frac{n-2}{n+2}$ and $\gamma>0$, we will construct subsolutions and supersolutions of the fast diffusion equation $u_t=\frac{n-1}{m}\Delta u^m$ in $\mathbb{R}^n\times (t_0,T)$, $t_0<T$, which decay at the rate $(T-t)^{\frac{1+\gamma}{1-m}}$ as $t\nearrow T$. As a consequence we obtain the existence of unique solution of the Cauchy problem $u_t=\frac{n-1}{m}\Delta u^m$ in $\mathbb{R}^n\times (t_0,T)$, $u(x,t_0)=u_0(x)$ in $\mathbb{R}^n$, which decay at the rate $(T-t)^{\frac{1+\gamma}{1-m}}$ as $t\nearrow T$ when $u_0$ satisfies appropriate decay condition.