We consider the fast diffusion equation on a nonparabolic Riemannian manifold M. Existence of weak solutions holds. Then we show that the validity of Euclidean–type Sobolev inequalities implies that smoothing effects into L^\infty hold true. The converse holds if m is sufficiently close to one. We then consider the case in which the manifold has the addition gap property min σ(−\Delta) > 0, \Delta being the Laplace-Beltrami operator. In that case solutions vanish in finite time, and we estimate from below and from above the extinction time.

Fast diffusion flow on manifolds of nonpositive curvature

GRILLO, GABRIELE;
2008

Abstract

We consider the fast diffusion equation on a nonparabolic Riemannian manifold M. Existence of weak solutions holds. Then we show that the validity of Euclidean–type Sobolev inequalities implies that smoothing effects into L^\infty hold true. The converse holds if m is sufficiently close to one. We then consider the case in which the manifold has the addition gap property min σ(−\Delta) > 0, \Delta being the Laplace-Beltrami operator. In that case solutions vanish in finite time, and we estimate from below and from above the extinction time.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11311/563090
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