Radial Fast Diffusion on the Hyperbolic Space

We consider radial solutions to the fast diffusion equation $u_t=\mathrm{\Delta }{u}^m$ on the hyperbolic space ${\mathbb{H}}^N$ for $N\ge 2$, $m\in (m_s,1)$, $m_s=\frac{N-2}{N+2}$. By radial we mean solutions depending only on the geodesic distance $r$ from a given point $o\in {\mathbb{H}}^N$. We investigate their fine asymptotics near the extinction time $T$ in terms of a separable solution of the form $\mathcal{V}(r,t)=(1-t\mathrm{/}T{)}^{1\mathrm{/}(1-m)}{V}^{1\mathrm{/}m}(r)$, where $V$ is the unique positive energy solution, radial w.r.t. $o$, to $-\Delta V=c,V^{1/m}$forasuitable$c>0$, a semilinear elliptic problem thoroughly studied in \cite{MS08}, \cite{BGGV}. We show that $u$ converges to ${\mathcal V}$inrelativeerror,inthesensethat$|{u^m(\cdot,t)}/{{\mathcal V}^m(\cdot,t)}-1|_\infty\to0$as$t\to T^-$. In particular the solution is bounded above and below, near the extinction time $T$, by multiples of $(1-t\mathrm{/}T{)}^{1\mathrm{/}(1-m)}{e}^{-(N-1)r\mathrm{/}m}$.