On Manifolds and Some Liouville Theorems of Porous Media Equations
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چکیده
(1.1) ut = ∆F (u) on a complete Riemannian manifold (M,g) of dimension n ≥ 1 with Ric(M) ≥ −k for some k ≥ 0. Here F ∈ C2(0,∞), F ′ > 0, and ∆ is the Laplace-Beltrami operator of the metric g. There is a lot of literature on this kind of topics. For example, related problems such as Porous Media Equations have been considered by D.G. Aronson [1], G. Auchmuty and D. Bao [2], M.A. Herrero and M. Pierre [6] and S.T. Yau [10]. It is well known that, in the study of geometric analysis as well as other elliptic or parabolic equations, the gradient estimate and the Harnack inequality play a most important role. To begin with, let us review some main results on the gradient estimate and the Harnack inequality. The first one is the Harnack-type differential inequality for the heat equation by P. Li and S.T. Yau [7], Theorem A (P. Li and S.T. Yau [7]). Let M be a complete manifold of dimension n ≥ 2 with Ricci(M) ≥ −k for some k ≥ 0. Suppose that u is any positive solution to the heat equation in B(x0, R)× [t0 − T, t0]. Then for a > 1, |∇u|2 u2 − at u ≤ cn (
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تاریخ انتشار 2008