Radial fast diffusion on the hyperbolic space

We consider positive radial solutions to the fast diffusion equationon the hyperbolic space. By radial, we mean solutions depending only on the geodesic distance r from a given point o ∈ H^N. We investigate their fine asymptotics near the extinction time T in terms of a separable solution defined in terms of the unique positive energy solution, radial with respect to o, to a semilinear elliptic problem thoroughly studied in [G. Mancini and K. Sandeep, ‘On a semilinear elliptic equation in Hn’, Ann. Sc. Norm. Super. Pisa Cl. Sci. 7 (2008) 635–671; M. Bonforte, F. Gazzola, G. Grillo and J. L. Vazquez, ‘Classification of radial solutions to the Emden–Fowler equation on the hyperbolic space’, Calc. Var. Partial Differential Equations 46 (2013) 375–401]. We show that u converges to V in relative error. Solutions are smooth, and bounds on derivatives are given as well. In particular, sharp convergence results as t → T are shown for spatial derivatives, again in the form of convergence in relative error.