Radial Fast Diffusion on the Hyperbolic Space

Abstract

We consider radial solutions to the fast diffusion equation $u_t=\Delta u^m$ 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 $\mathcal{V}(r,t)=(1-t/T{)}^{1/(1-m)}{V}^{1/m}(r)$, where $V$ is the unique positive energy solution, radial w.r.t. $o$, to $-\Delta V=c{V}^{1/m}$ for a suitable $c>0$, a semilinear elliptic problem thoroughly studied in \cite{MS08}, \cite{BGGV}. We show that $u$ converges to $\mathcal{V}$ in relative error, in the sense that $\Vert u^m(\cdot ,t)/{\mathcal{V}}^m(\cdot ,t)-1{\Vert }_{\infty }\to 0$ as $t\to T^{-}$. In particular the solution is bounded above and below, near the extinction time $T$, by multiples of $(1-t/T{)}^{1/(1-m)}{e}^{-(N-1)r/m}$.

To appear in Proc. London Math. Soc

References in corpus (1)

• Asymptotics of the fast diffusion equation via entropy estimates

Cited by in corpus (3)

• Laplacian cut-offs, porous and fast diffusion on manifolds and other applications
• Unconditional existence of conformally hyperbolic Yamabe flows
• The porous medium equation with large initial data on negatively curved Riemannian manifolds