Elliptic gradient estimates for a nonlinear heat equation and applications

Wu, Jia-Yong

doi:10.1016/j.na.2016.11.014

Cited by 39 publications

(18 citation statements)

References 33 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Therefore, our estimate is better than those in [Jia16,Wu17] for the case a ≤ 0. Moreover, the Liouville type results we obainned in the case a ≤ 0 for the equation (1.2) are also better than those in [DKN18, HM15, Li15, Jia16, Wu17].…”

Section: Introductionmentioning

confidence: 53%

Sharp gradient estimates for a heat equation in Riemannian manifolds

Dung

Dũng

2019

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

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.

show abstract

Section: Introductionmentioning

confidence: 53%

Sharp gradient estimates for a heat equation in Riemannian manifolds

Dung

Dũng

2019

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…He obtained a local gradient estimate for the positive smooth solutions of (1.3). Then, Wu improved his results under the assumption that the Bakry-Émery curvature is bounded from below (see [19], [20]). Later, in [10], [11], and [7], Huang and Ma, Khanh and the first author studied gradient estimates of Hamilton type and Souplet-Zhang type for equation (1.3).…”

Section: Introductionmentioning

confidence: 96%

“…In fact, these methods are standard and well-known. They are used in many works (see also [6]- [7], [10]- [11], [15], [20] and the references therein). More precisely, we first estimate the lower bound of the evolution operator acting on a suitable function in terms of the evolution solution.…”

Section: Introductionmentioning

confidence: 99%

Gradient estimates for some evolution equations on complete smooth metric measure spaces

Dũng¹,

Linh²,

Thu³

2020

Publ. Math. Debrecen

View full text Add to dashboard Cite

In this paper, we consider the following general evolution equation ut = ∆ f u + au log α u + bu on a smooth metric measure space (M n , g, e −f dv). We give a local gradient estimate of Souplet-Zhang type for positive smooth solutions of this equation provided that the Bakry-Émery curvature is bounded from below. When f is constant, we investigate the general evolution equation on compact Riemannian manifolds with nonconvex boundary satisfying an interior rolling R-ball condition. We show a gradient estimate of Hamilton type on such manifolds.

show abstract

“…This paper is an extention of [25]. In that paper, the author proved elliptic gradient estimates and Liouville type theorem for positive solutions to a nonlinear parabolic equation…”

mentioning

confidence: 95%

Elliptic gradient estimates and Liouville theorems for a weighted nonlinear parabolic equation

Abolarinwa

2019

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

Let (M N , g, e −f dv) be a complete smooth metric measure space with ∞-Bakry-Émery Ricci tensor bounded from below. We derive elliptic gradient estimates for positive solutions of a weighted nonlinear parabolic equationwhere (x, t) ∈ M N × (−∞, ∞) and α is an arbitrary constant. As Applications we prove a Liouville-type theorem for positive ancient solutions and Harnacktype inequalities for positive bounded solutions. 1.2. Smooth metric measure spaces. A smooth metric measure space is denoted by the triple (M N , g, e −f dv), where (M N , g) is an N-dimensional complete manifold with the Riemannian metric tensor g, volume element dv and f is a C ∞ real-valued function on M. Smooth metric measure spaces are naturally endowed with analogue of Laplace-Beltrami operator, called weighted Laplacian and analogue of Ricci tensor, called Bakry-Émery tensor. The weighted Laplacian defined by ∆ f := ∆ − ∇f, ∇· , where ∆ is the Laplace-Beltrami operator, is symmetric and self-adjoint with respect to the weighted measure e −f dv. The m-Bakry-Émery tensor is defined by Ric m f := Ric + ∇ 2 f − 1 m df ⊗ df for some constant m > 0, where Ric is the Ricci tensor of the manifold and ∇ 2 is the Hessian with respect to the metric g. When m is infinite we have the ∞-Bakry-Émery tensor Ric f = lim m→∞ Ric m f := Ric + ∇ 2 f. This tensor is related to the gradient Ricci solitonwhere λ is a real constant. A Ricci soliton is said to be shrinking, steady or expanding depending on whether λ is positive, zero or negative respectively. Ricci

show abstract

Elliptic gradient estimates for a nonlinear heat equation and applications

Cited by 39 publications

References 33 publications

Sharp gradient estimates for a heat equation in Riemannian manifolds

Sharp gradient estimates for a heat equation in Riemannian manifolds

Gradient estimates for some evolution equations on complete smooth metric measure spaces

Elliptic gradient estimates and Liouville theorems for a weighted nonlinear parabolic equation

Contact Info

Product

Resources

About