scholarly journals Gradient estimates for a weighted nonlinear parabolic equation and applications

2020 ◽  
Vol 18 (1) ◽  
pp. 1150-1163
Author(s):  
Abimbola Abolarinwa ◽  
Nathaniel K. Oladejo ◽  
Sulyman O. Salawu
Keyword(s):  
Parabolic Equation ◽  
Riemannian Manifold ◽  
Nonlinear Parabolic Equation ◽  
Type Theorem ◽  
Gradient Estimates ◽  
Liouville Type ◽  
Liouville Type Theorem ◽  
Harnack Inequalities ◽  
Nonlinear Parabolic ◽  
Lower Boundedness

Abstract This paper derives elliptic gradient estimates for positive solutions to a nonlinear parabolic equation defined on a complete weighted Riemannian manifold. Applications of these estimates yield Liouville-type theorem, parabolic Harnack inequalities and bounds on weighted heat kernel on the lower boundedness assumption for Bakry-Émery curvature tensor.

scholarly journals Gradient Estimates for a Weighted Γ-nonlinear Parabolic Equation Coupled with a Super Perelman-Ricci Flow and Implications

2021 ◽  
Author(s):  
Ali Taheri
Keyword(s):  
Parabolic Equation ◽  
Ricci Flow ◽  
Nonlinear Parabolic Equation ◽  
Type Theorem ◽  
Gradient Estimates ◽  
Liouville Type ◽  
Liouville Type Theorem ◽  
Harnack Inequalities ◽  
Nonlinear Parabolic ◽  
Special Cases

AbstractThis article studies a nonlinear parabolic equation on a complete weighted manifold where the metric and potential evolve under a super Perelman-Ricci flow. It derives elliptic gradient estimates of local and global types for the positive solutions and exploits some of their implications notably to a general Liouville type theorem, parabolic Harnack inequalities and classes of Hamilton type dimension-free gradient estimates. Some examples and special cases are discussed for illustration.

scholarly journals Liouville theorems and elliptic gradient estimates for a nonlinear parabolic equation involving the Witten Laplacian

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Ali Taheri
Keyword(s):  
Parabolic Equation ◽  
Nonlinear Parabolic Equation ◽  
Gradient Estimates ◽  
Elliptic Type ◽  
Liouville Type ◽  
Witten Laplacian ◽  
Smooth Metric Measure Space ◽  
Nonlinear Parabolic ◽  
Global Estimates ◽  
Liouville Type Results

Abstract In this paper, we establish local and global elliptic type gradient estimates for a nonlinear parabolic equation on a smooth metric measure space whose underlying metric and potential satisfy a ( k , m ) {(k,m)} -super Perelman–Ricci flow inequality. We discuss a number of applications and implications including curvature free global estimates and some constancy and Liouville type results.

A remark on the mixed scalar curvature of a manifold with two orthogonal totally umbilical distributions

2019 ◽  
Vol 19 (3) ◽  
pp. 291-296 ◽  
Author(s):  
Sergey Stepanov ◽  
Irina Tsyganok
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Riemannian Manifolds ◽  
Type Theorem ◽  
Complete Riemannian Manifold ◽  
Warped Products ◽  
Liouville Type ◽  
Liouville Type Theorem ◽  
Totally Umbilical

Abstract We prove a Liouville-type theorem for two orthogonal complementary totally umbilical distributions on a complete Riemannian manifold with non-positive mixed scalar curvature. This is applied to some special types of complete doubly twisted and warped products of Riemannian manifolds.

A gradient estimate on a manifold with convex boundary

1997 ◽  
Vol 127 (1) ◽  
pp. 171-179 ◽  
Author(s):  
Zhongmin Qian
Keyword(s):  
Parabolic Equation ◽  
Riemannian Manifold ◽  
Elliptic Equation ◽  
Nonlinear Parabolic Equation ◽  
Compact Riemannian Manifold ◽  
Gradient Estimate ◽  
Markov Semigroup ◽  
Complete Manifold ◽  
Convex Boundary ◽  
Nonlinear Parabolic

We present a simple probability approach for establishing a gradient estimate for a solution of an elliptic equation on a compact Riemannian manifold with convex boundary, or on a noncompact complete manifold. Our method can also be applied to derive a similar gradient estimate for a nonlinear parabolic equation, and an abstract gradient estimate for a Markov semigroup.

scholarly journals Global gradient estimates for nonlinear parabolic operators

2021 ◽  
Vol 27 ◽  
pp. 21
Author(s):  
Serena Dipierro ◽  
Zu Gao ◽  
Enrico Valdinoci
Keyword(s):  
Parabolic Equation ◽  
Riemannian Manifold ◽  
Diffusion Coefficients ◽  
Boundary Data ◽  
Complete Riemannian Manifold ◽  
Ambient Space ◽  
Gradient Estimates ◽  
Interior Estimate ◽  
Nonlinear Parabolic ◽  
Euclidean Setting

We consider a parabolic equation driven by a nonlinear diffusive operator and we obtain a gradient estimate in the domain where the equation takes place. This estimate depends on the structural constants of the equation, on the geometry of the ambient space and on the initial and boundary data. As a byproduct, one easily obtains a universal interior estimate, not depending on the parabolic data. The setting taken into account includes sourcing terms and general diffusion coefficients. The results are new, to the best of our knowledge, even in the Euclidean setting, though we treat here also the case of a complete Riemannian manifold.

Sign in / Sign up

Export Citation Format

Share Document