Harnack Estimation for Nonlinear, Weighted, Heat-Type Equation along Geometric Flow and Applications

Yanlin Li (), Sujit Bhattacharyya, Shahroud Azami, Apurba Saha and Shyamal Kumar Hui
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Yanlin Li: School of Mathematics, Key Laboratory of Cryptography of Zhejiang Province, Hangzhou Normal University, Hangzhou 311121, China
Sujit Bhattacharyya: Department of Mathematics, The University of Burdwan, Golapbag, Burdwan 713104, West Bengal, India
Shahroud Azami: Department of Mathematics, Faculty of Sciences, Imam Khomeini International University, Qazvin 34148-96818, Iran
Apurba Saha: Department of Mathematics, The University of Burdwan, Golapbag, Burdwan 713104, West Bengal, India
Shyamal Kumar Hui: Department of Mathematics, The University of Burdwan, Golapbag, Burdwan 713104, West Bengal, India

Mathematics, 2023, vol. 11, issue 11, 1-14

Abstract: The method of gradient estimation for the heat-type equation using the Harnack quantity is a classical approach used for understanding the nature of the solution of these heat-type equations. Most of the studies in this field involve the Laplace–Beltrami operator, but in our case, we studied the weighted heat equation that involves weighted Laplacian. This produces a number of terms involving the weight function. Thus, in this article, we derive the Harnack estimate for a positive solution of a weighted nonlinear parabolic heat equation on a weighted Riemannian manifold evolving under a geometric flow. Applying this estimation, we derive the Li–Yau-type gradient estimation and Harnack-type inequality for the positive solution. A monotonicity formula for the entropy functional regarding the estimation is derived. We specify our results for various different flows. Our results generalize some works.

Keywords: Harnack inequality; Harnack estimate; weighted Laplacian; parabolic equation; gradient estimate (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (2)

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