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;

D O I：

暂无

中图分类号：

学科分类号：

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.

引用

页码：99 / 128

页数：29