tailieunhanh - Gradient estimates for the porous medium type equation on smooth metric measure space
The porous medium equation arises in different applications to model diffusive phenomena. In this paper, we obtain several gradient estimates for some porous medium type equations on smooth metric measure space with N-Bakry-Emery Ricci tensor bounded from below. In particular, we improve and generalize some current gradient estimates for the porous medium equations. | Turkish Journal of Mathematics Research Article Turk J Math (2014) 38: 985 – 993 ¨ ITAK ˙ c TUB ⃝ doi: Gradient estimates for the porous medium type equation on smooth metric measure space Deng YIHUA∗ Department of Mathematics and Computing Science Hengyang Normal University Hengyang, Hunan, . China Received: • Accepted: • Published Online: • Printed: Abstract: The porous medium equation arises in different applications to model diffusive phenomena. In this paper, we obtain several gradient estimates for some porous medium type equations on smooth metric measure space with NBakry-Emery Ricci tensor bounded from below. In particular, we improve and generalize some current gradient estimates for the porous medium equations. Key words: Gradient estimates, porous medium type equation, smooth metric measure space, positive solution 1. Introduction A smooth metric measure space is a triple, (M n , g, e−f dv), where M n is a complete n -dimensional Riemannian manifold with metric g , f is a smooth real-valued function on M n , and dv is the Riemannian volume density. Smooth metric measure spaces carry a natural analog of the Laplace-Beltrami operator △, the f -Laplacian, which is also called drifting Laplacian or Witten-Laplacian, defined for a function u by △f u = △u−g(∇f, ∇u) = △u − ⟨∇f, ∇u⟩. The N-Bakry-Emery Ricci tensor is defined by RicN f = Ric + Hessf − 1 N df ⊗ df . A natural question about smooth metric measure space is which of the results about the Ricci tensor and the LaplaceBeltrami operator can be extended to the N-Bakry-Emery Ricci tensor and the f -Laplacian. For example, in [15], Yang discussed the gradient estimates for the following parabolic equation, ∂ u = △u + au log u + bu, ∂t on Riemannian manifolds. In [8], Huang and Ma considered the gradient estimates for the following parabolic equation, ∂ u = △f u + au log u + bu, ∂t on smooth
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