Harnack Estimation for Nonlinear, Weighted, Heat-Type Equation along Geometric Flow and Applications
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 wei...
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MDPI AG
2023-05-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/11/2516 |
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| author | Yanlin Li Sujit Bhattacharyya Shahroud Azami Apurba Saha Shyamal Kumar Hui |
| author_facet | Yanlin Li Sujit Bhattacharyya Shahroud Azami Apurba Saha Shyamal Kumar Hui |
| author_sort | Yanlin Li |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-11T03:02:47Z |
| format | Article |
| id | doaj.art-951a477a0b294407b9db53206757d1f6 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T03:02:47Z |
| publishDate | 2023-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-951a477a0b294407b9db53206757d1f62023-11-18T08:13:05ZengMDPI AGMathematics2227-73902023-05-011111251610.3390/math11112516Harnack Estimation for Nonlinear, Weighted, Heat-Type Equation along Geometric Flow and ApplicationsYanlin Li0Sujit Bhattacharyya1Shahroud Azami2Apurba Saha3Shyamal Kumar Hui4School of Mathematics, Key Laboratory of Cryptography of Zhejiang Province, Hangzhou Normal University, Hangzhou 311121, ChinaDepartment of Mathematics, The University of Burdwan, Golapbag, Burdwan 713104, West Bengal, IndiaDepartment of Mathematics, Faculty of Sciences, Imam Khomeini International University, Qazvin 34148-96818, IranDepartment of Mathematics, The University of Burdwan, Golapbag, Burdwan 713104, West Bengal, IndiaDepartment of Mathematics, The University of Burdwan, Golapbag, Burdwan 713104, West Bengal, IndiaThe 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.https://www.mdpi.com/2227-7390/11/11/2516Harnack inequalityHarnack estimateweighted Laplacianparabolic equationgradient estimate |
| spellingShingle | Yanlin Li Sujit Bhattacharyya Shahroud Azami Apurba Saha Shyamal Kumar Hui Harnack Estimation for Nonlinear, Weighted, Heat-Type Equation along Geometric Flow and Applications Mathematics Harnack inequality Harnack estimate weighted Laplacian parabolic equation gradient estimate |
| title | Harnack Estimation for Nonlinear, Weighted, Heat-Type Equation along Geometric Flow and Applications |
| title_full | Harnack Estimation for Nonlinear, Weighted, Heat-Type Equation along Geometric Flow and Applications |
| title_fullStr | Harnack Estimation for Nonlinear, Weighted, Heat-Type Equation along Geometric Flow and Applications |
| title_full_unstemmed | Harnack Estimation for Nonlinear, Weighted, Heat-Type Equation along Geometric Flow and Applications |
| title_short | Harnack Estimation for Nonlinear, Weighted, Heat-Type Equation along Geometric Flow and Applications |
| title_sort | harnack estimation for nonlinear weighted heat type equation along geometric flow and applications |
| topic | Harnack inequality Harnack estimate weighted Laplacian parabolic equation gradient estimate |
| url | https://www.mdpi.com/2227-7390/11/11/2516 |
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