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2007, 2007(Special): 130-137. doi: 10.3934/proc.2007.2007.130

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, France
2. 

Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

Received  September 2006 Revised  January 2007 Published  September 2007

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, 35K5.
Citation: Matteo Bonforte, Gabriele Grillo. Singular evolution on maniforlds, their smoothing properties, and soboleve inequalities. Conference Publications, 2007, 2007 (Special) : 130-137. doi: 10.3934/proc.2007.2007.130
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