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

Matteo Bonforte; Gabriele Grillo

doi:10.3934/proc.2007.2007.130

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2007, Volume 2007: 130-137. Doi: 10.3934/proc.2007.2007.130 open access

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

Matteo Bonforte^1, and
Gabriele Grillo^2,

1.
CEREMADE, Université Paris Dauphine, Place de Lattre de Tassigny, F-75775 Paris Cédex 16
2.
Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino

Received: August 31, 2006

Revised: December 31, 2006

Published: August 2007

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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

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Mathematics Subject Classification: Primary: 47J35; Secondary: 35B45, 58J35, 35B65, 35K55.

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