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

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