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Fast diffusion flow on manifolds of nonpositive curvature

Bonforte, Matteo; Grillo, Gabriele; Vazquez, Juan-Luis (2008), Fast diffusion flow on manifolds of nonpositive curvature, Journal of Evolution Equations, 8, 1, p. 99-128. http://dx.doi.org/10.1007/s00028-007-0345-4

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Type
Article accepté pour publication ou publié
Date
2008
Journal name
Journal of Evolution Equations
Volume
8
Number
1
Publisher
Springer
Pages
99-128
Publication identifier
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Author(s)
Bonforte, Matteo

Grillo, Gabriele

Vazquez, Juan-Luis
Abstract (EN)
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 p −L q smoothing effects of the type ∥u(t)∥ q ≤ Ct −α ∥u 0∥γ 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.
Subjects / Keywords
Nonlinear evolutions; singular parabolic equations; fast diffusion; Riemannian manifolds; asymptotics