Fast diffusion flow on manifolds of nonpositive curvature

被引:0
作者
Matteo Bonforte
Gabriele Grillo
Juan Luis Vazquez
机构
[1] Universidad Autónoma de Madrid,Departamento de Matemáticas
[2] Politecnico di Torino,Dipartimento di Matematica
[3] Ceremade,undefined
[4] Université Paris Dauphine,undefined
来源
Journal of Evolution Equations | 2008年 / 8卷
关键词
35B45; 35B65; 35K55; 35K65; Nonlinear evolutions; singular parabolic equations; fast diffusion; Riemannian manifolds; asymptotics;
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摘要
We consider the fast diffusion equation (FDE) ut = Δum (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 Lp−Lq smoothing effects of the type ∥u(t)∥q ≤ Ct−α ∥u0∥γp, 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.
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页码:99 / 128
页数:29
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