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

Bonforte, Matteo; Grillo, Gabriele (2007), Singular evolution on maniforlds, their smoothing properties, and Sobolev inequalities, Discrete and Continuous Dynamical Systems. Series A, p. 130-137. http://dx.doi.org/10.3934/proc.2007.2007.130

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Type
Article accepté pour publication ou publié
Date
2007
Journal name
Discrete and Continuous Dynamical Systems. Series A
Publisher
AIMS
Pages
130-137
Publication identifier
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Author(s)
Bonforte, Matteo
Grillo, Gabriele
Abstract (EN)
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 .
Subjects / Keywords
Singular evolutions; Riemannian manifolds; Sobolev inequalities; smoothing effect