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Singular evolution on maniforlds, their smoothing properties, and soboleve inequalities

Pages: 130 - 137, Issue Special, September 2007

 Abstract        Full Text (201.6K)              

Matteo Bonforte - CEREMADE, Université Paris Dauphine, Place de Lattre de Tassigny, F-75775 Paris Cédex 16, France (email)
Gabriele Grillo - Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy (email)

Abstract: The evolution equation $u_ = \Delta_pu$, posed on a Riemannian manifold, is studied in the singular range $p \in 2$ (1; 2). It is shown that if the manifold supports a suitable Sobolev inequality, the smoothing effect $||u(t)||\infty\leq C ||u(0)||_q^\gamma$/$t^\alpha$ holds true for suitable for $\alpha, \gamma$and that the converse holds if $p$ is sufficiently close to 2, or in the degenerate range $p$ > 2. In such ranges, the Sobolev inequality and the smoothing efect are then equivalent

Keywords:  Singular evolutions, Riemannian manifolds, Sobolev inequalities, smoothing effects.
Mathematics Subject Classification:  Primary: 47J35; Secondary: 35B45, 58J35, 35B65, 35K55.

Received: September 2006;      Revised: January 2007;      Published: September 2007.