Super fast vanishing solutions of the fast diffusion equation

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.

37 pages, typos corrected, reference updated