# American Institute of Mathematical Sciences

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

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

 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
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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