Skip to main content
Log in

Fast diffusion flow on manifolds of nonpositive curvature

  • Published:
Journal of Evolution Equations Aims and scope Submit manuscript

Abstract.

We consider the fast diffusion equation (FDE) u t = Δu m (0 < m < 1) on a nonparabolic Riemannian manifold M. Existence of weak solutions holds. Then we show that the validity of Euclidean–type Sobolev inequalities implies that certain L pL q smoothing effects of the type ∥u(t)∥ q Ct −αu 0γ p , the case q = ∞ being included. 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(−Δ) > 0. In that case solutions vanish in finite time, and we estimate from below and from above the extinction time.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Matteo Bonforte.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bonforte, M., Grillo, G. & Vazquez, J.L. Fast diffusion flow on manifolds of nonpositive curvature. J. evol. equ. 8, 99–128 (2008). https://doi.org/10.1007/s00028-007-0345-4

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00028-007-0345-4

Mathematics Subject Classifications (2000):

Keywords:

Navigation