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.

Radial fast diffusion on the hyperbolic space

GRILLO, GABRIELE;MURATORI, MATTEO
2014

Abstract

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.
Fast diffusion; Nonlinear evolution equations; Asymptotics
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11311/762749
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