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

Author

Listed:
  • 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)

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.

Suggested Citation

  • Yanlin Li & Sujit Bhattacharyya & Shahroud Azami & Apurba Saha & Shyamal Kumar Hui, 2023. "Harnack Estimation for Nonlinear, Weighted, Heat-Type Equation along Geometric Flow and Applications," Mathematics, MDPI, vol. 11(11), pages 1-14, May.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:11:p:2516-:d:1159746
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    References listed on IDEAS

    as
    1. Erol Kılıç & Mehmet Gülbahar & Ecem Kavuk, 2020. "Concurrent Vector Fields on Lightlike Hypersurfaces," Mathematics, MDPI, vol. 9(1), pages 1-16, December.
    2. Engin As & Süleyman Şenyurt, 2013. "Some Characteristic Properties of Parallel -Equidistant Ruled Surfaces," Mathematical Problems in Engineering, Hindawi, vol. 2013, pages 1-7, June.
    3. Yanlin Li & Abimbola Abolarinwa & Ali H. Alkhaldi & Akram Ali, 2022. "Some Inequalities of Hardy Type Related to Witten–Laplace Operator on Smooth Metric Measure Spaces," Mathematics, MDPI, vol. 10(23), pages 1-13, December.
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    Cited by:

    1. Esmaeil Peyghan & Davood Seifipour & Ion Mihai, 2023. "On the Geometry of Kobayashi–Nomizu Type and Yano Type Connections on the Tangent Bundle with Sasaki Metric," Mathematics, MDPI, vol. 11(18), pages 1-18, September.
    2. Yanlin Li & Mahmut Mak, 2023. "Framed Natural Mates of Framed Curves in Euclidean 3-Space," Mathematics, MDPI, vol. 11(16), pages 1-14, August.

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