Elliptic gradient estimates for a weighted heat equation and applications

Wu, Jia-Yong

doi:10.1007/s00209-015-1432-9

Cited by 51 publications

(63 citation statements)

References 24 publications

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“…Finally, by using the same argument in the proof of Theorem 1.1 and a Laplacian comparison theorem given by Wei and Wylie [WW09] (see also [Bri13]), it is worth to notice that our gradient estimates are also valid for a positive solution to [Wu15], our gradient estimate is sharp. The paper has two sections.…”

Section: Introductionmentioning

confidence: 77%

“…From this remark, we observe that if Ric f ≥ 0, let u be a positive soltion to (2.15), where a ≤ 0 and u = e o(d(x)+|t|) then u must be constant, namely u ≡ 1. The Example 1.2 in [Wu15] implies that the estimate (2.16) is sharp. Indeed, let us recall the example.…”

Section: Sharp Gradient Estimatesmentioning

confidence: 95%

“…This example confirms that our gradient estimate is sharp in both spatial and time directions. In [Wu15], to obtain a Liouville property, the author had to require that the potential function f is bounded and u = e o(d(x)+ √ |t|) .…”

Section: Sharp Gradient Estimatesmentioning

confidence: 99%

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Sharp gradient estimates for a heat equation in Riemannian manifolds

Dung

Dũng

2019

Proc. Amer. Math. Soc.

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In this paper, we prove sharp gradient estimates for a positive solution to the heat equation u t = ∆u + au log u in complete noncompact Riemannian manifolds. As its application, we show that if u is a positive solution of the equation u t = ∆u and log u is of sublinear growth in both spatial and time directions then u must be constant. This gradient estimate is sharp since it is well-known that u(x, t) = e x+t satisfying u t = ∆u. We also emphasize that our results are better than those given by Jiang ([Jia16]), Souplet-Zhang ([SZ06]), Wu ([Wu15, Wu17]), and others.

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Section: Introductionmentioning

confidence: 77%

Section: Sharp Gradient Estimatesmentioning

confidence: 95%

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Sharp gradient estimates for a heat equation in Riemannian manifolds

Dung

Dũng

2019

Proc. Amer. Math. Soc.

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“…Zhang [29] obtained an elliptic type gradient estimate for a bounded positive solution of the heat equation after inserting a necessary logarithmic correction term. Li-Yau type and Hamilton-Souplet-Zhang type gradient estimates have been obtained for other nonlinear equations on manifolds, see for example [4,5,9,17,19,23,24,27,[31][32][33] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Gradient estimates and Harnack inequalities of a parabolic equation under geometric flow

Zhao

2020

Journal of Mathematical Analysis and Applications

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In this paper, we consider a manifold evolving by a general geometric flow and study parabolic equationWe establish space-time gradient estimates for positive solutions and elliptic type gradient estimates for bounded positive solutions of this equation. By integrating the gradient estimates, we derive the corresponding Harnack inequalities. Finally, as applications, we give gradient estimates of some specific parabolic equations.

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“…On a smooth metric measure space, in [13], Wu gave a gradient estimate of Souplet-Zhang type for the equation…”

Section: Introductionmentioning

confidence: 99%