Harnack Differential Inequalities for the Parabolic Equation ut=LF(u) on Riemannian Manifolds and Applications |
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Authors: | Wen WANG |
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Affiliation: | 1. School of Mathematics and Statistics, Hefei Normal University, Hefei 230601, P. R. China;2. School of Mathematical Science, University of Science and Technology of China, Hefei 230026, P. R. China |
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Abstract: | In this paper, let (Mn, g) be an n-dimensional complete Riemannian manifold with the mdimensional Bakry-Émery Ricci curvature bounded below. By using the maximum principle, we first prove a Li-Yau type Harnack differential inequality for positive solutions to the parabolic equation #br#ut = LF(u) = ΔF(u)-∇f·∇F(u), #br#on compact Riemannian manifolds Mn, where F ∈ C2(0,∞), F' > 0 and f is a C2-smooth function defined on Mn. As application, the Harnack differential inequalities for fast diffusion type equation and porous media type equation are derived. On the other hand, we derive a local Hamilton type gradient estimate for positive solutions of the degenerate parabolic equation on complete Riemannian manifolds. As application, related local Hamilton type gradient estimate and Harnack inequality for fast dfiffusion type equation are established. Our results generalize some known results. |
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Keywords: | Parabolic equation Li-Yau type Harnack differential inequality local Hamilton type gradient estimate fast diffusion equation Porous media equation |
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