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 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 $L^{\infty }$ decay estimates of solutions of nonlinear parabolic equation

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Hui Wang ◽  
Caisheng Chen
Keyword(s):  
Parabolic Equation ◽  
Weak Solutions ◽  
Nonlinear Parabolic Equation ◽  
Divergence Form ◽  
Decay Estimates ◽  
Laplacian Equation ◽  
Estimates Of Solutions ◽  
Nonlinear Parabolic ◽  
Doubly Nonlinear ◽  
Doubly Nonlinear Parabolic Equation

AbstractIn this paper, we are interested in $L^{\infty }$ L ∞ decay estimates of weak solutions for the doubly nonlinear parabolic equation and the degenerate evolution m-Laplacian equation not in the divergence form. By a modified Moser’s technique we obtain $L^{\infty }$ L ∞ decay estimates of weak solutiona.

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