Year: 2014
Type: article
Abstract: We consider positive radial solutions to the fast diffusion equation u t = Δ ( u m ) on the hyperbolic space H N for N ⩾ 2 , m ∈ ( m s , 1 ) , m s = ( N − 2 ) / ( N + 2 ) . 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 of the form V ( r , t ) = ( 1 − t / T ) 1 / ( 1 − m ) V 1 / m ( r ) , where V is the unique positive energy solution, radial with respect to o, to − Δ V = c V 1 / m for a suitable c > 0 , a semilinear elliptic problem thoroughly studied in [G. Mancini and K. Sandeep, ‘On a semilinear elliptic equation in H n ’, Ann. Sc. Norm. Super. Pisa Cl. Sci. 7 (2008) 635–671; M. Bonforte, F. Gazzola, G. Grillo and J. L. Vázquez, ‘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, in the sense that ∥ u m ( · , t ) / V m ( · , t ) − 1 ∥ ∞ → 0 as t → 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 . 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.
DOI: 10.1112/plms/pdt071
Authors: Gabriele Grillo, Matteo Muratori
Institution: Politecnico di Milano
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W1978550595