scholarly journals Gradient Estimates and Harnack Inequalities of a Nonlinear Heat Equation for the Finsler-Laplacian

2021 ◽  
Vol 17 (4) ◽  
pp. 521-548
Author(s):  
Fanqi Zeng ◽  
Keyword(s):  
Heat Equation ◽  
Nonlinear Heat Equation ◽  
Gradient Estimates ◽  
Harnack Inequalities

Gradient estimates for a nonlinear heat equation under the Finsler-Ricci flow

2019 ◽  
Vol 69 (2) ◽  
pp. 409-424
Author(s):  
Fanqi Zeng ◽  
Qun He
Keyword(s):  
Heat Equation ◽  
Positive Solutions ◽  
Ricci Flow ◽  
Finsler Manifold ◽  
Nonlinear Heat Equation ◽  
Gradient Estimates ◽  
Harnack Inequalities

Abstract This paper considers a compact Finsler manifold (Mn, F(t), m) evolving under the Finsler-Ricci flow and establishes global gradient estimates for positive solutions of the following nonlinear heat equation: $$\begin{array}{} \partial_{t}u=\Delta_{m} u, \end{array} $$ where Δm is the Finsler-Laplacian. As applications, several Harnack inequalities are obtained.

scholarly journals Gradient estimates for the heat equation under the Ricci-harmonic map flow

2015 ◽  
Vol 15 (4) ◽  
Author(s):  
Mihai Bailesteanu
Keyword(s):  
Heat Equation ◽  
Positive Solutions ◽  
Ricci Flow ◽  
Harmonic Map ◽  
Gradient Estimates ◽  
Complete Manifold ◽  
Harnack Inequalities ◽  
Harmonic Map Flow

AbstractThe paper establishes a series of gradient estimates for positive solutions to the heat equation on a manifold M evolving under the Ricci flow, coupled with the harmonic map flow between M and a second manifold N. We prove Li-Yau type Harnack inequalities and we consider the cases when M is a complete manifold without boundary and when M is compact without boundary.

scholarly journals Differential Harnack estimates for conjugate heat equation under the Ricci flow

2015 ◽  
Vol 08 (04) ◽  
pp. 1550063
Author(s):  
Abimbola Abolarinwa
Keyword(s):  
Heat Equation ◽  
Ricci Flow ◽  
Beltrami Operator ◽  
Nonlinear Heat Equation ◽  
Curvature Operator ◽  
Harnack Estimate ◽  
Diffusion Operator ◽  
Harnack Inequalities ◽  
Harnack Estimates ◽  
Conjugate Heat Equation

We prove (local and global) differential Harnack inequalities for all positive solutions to the geometric conjugate heat equation coupled to the forward in time Ricci flow. In this case, the diffusion operator is perturbed with the curvature operator, precisely, the Laplace–Beltrami operator is replaced with “[Formula: see text]”, where [Formula: see text] is the scalar curvature of the Ricci flow, which is well generalized to the case of nonlinear heat equation with potential. Our estimates improve on some well known results by weakening the curvature constraints. As a by-product, we obtain some Li–Yau-type differential Harnack estimate. The localized version of our estimate is very useful in extending the results obtained to noncompact case.

GEOMETRIC INTEGRATION BY SOLUTION INTERPOLATION

2009 ◽  
Vol 20 (02) ◽  
pp. 313-322
Author(s):  
PILWON KIM
Keyword(s):  
Differential Equation ◽  
Heat Equation ◽  
Exact Solutions ◽  
Standard Method ◽  
Interpolation Method ◽  
Nonlinear Heat Equation ◽  
Geometric Integration ◽  
Numerical Schemes ◽  
New Class ◽  
Geometric Integrators

Numerical schemes that are implemented by interpolation of exact solutions to a differential equation naturally preserve geometric properties of the differential equation. The solution interpolation method can be used for development of a new class of geometric integrators, which generally show better performances than standard method both quantitatively and qualitatively. Several examples including a linear convection equation and a nonlinear heat equation are included.

scholarly journals Null controllability of a nonlinear heat equation

2002 ◽  
Vol 7 (7) ◽  
pp. 375-383 ◽  
Author(s):  
G. Aniculăesei ◽  
S. Aniţa
Keyword(s):  
Heat Equation ◽  
Dirichlet Boundary ◽  
Null Controllability ◽  
Carleman Estimates ◽  
Nonlinear Heat Equation ◽  
Homogeneous Dirichlet Boundary Condition ◽  
Linearized System ◽  
Kakutani Fixed Point ◽  
Exact Null Controllability ◽  
Kakutani Fixed Point Theorem

We study the internal exact null controllability of a nonlinear heat equation with homogeneous Dirichlet boundary condition. The method used combines the Kakutani fixed-point theorem and the Carleman estimates for the backward adjoint linearized system. The result extends to the case of boundary control.

Sign in / Sign up

Export Citation Format

Share Document