Fast diffusion equations on riemannian manifolds


In the present paper, we first study the nonexistence of positive solutions of the following nonlinear parabolic problem u u∂u∂t((x, x = t0)) ∆ = = g(0 u u 0m() x) + ≥ V 0 (x)um + λuq in Ω × (0, T), in Ω, on ∂Ω × (0, T). Here, Ω is a bounded domain with smooth boundary in a complete non-compact Riemannian manifold M, 0 < m < 1, V ∈ L1loc(Ω), q > 0 and λ ∈ R. Next, we prove some Hardy and Leray type inequalities with remainders on a Riemannian Manifold M. Furthermore, we obtain explicit (sometimes optimal) constants for these inequalities and present several nonexistence results with help of Hardy and Leray type inequalities on the hyperbolic space Hn.

Publication Title

Differential and Integral Equations

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