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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| Accès en ligne: | https://arxiv.org/abs/2402.00006 |
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| _version_ | 1866929454847098880 |
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| author | Shen, Bin Zhu, Yuhan |
| author_facet | Shen, Bin Zhu, Yuhan |
| contents | In this paper, we explore the positive solutions to the Finslerian nonlinear equation $$\frac{\partial u}{\partial t} = Δ^{\nabla u} u + au\log u + bu,$$ which is related to Ricci solitons and serves as the Euler-Lagrange equation to the Finslerian log-energy functional. We then obtain the global gradient estimate of its positive solution on a compact Finsler metric measure space with the weighted Ricci curvature bounded below. Furthermore, using a new comparison theorem developed by the first author, we also establish a local gradient estimate on a non-compact forward complete Finsler metric measure spaces with the mixed weighted Ricci curvature bounded below, as well as finite bounds of misalignment and some non-Riemannian curvatures. Lastly, we prove the Harnack inequalities and a Liouville-type theorem of such solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_00006 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Li-Yau Estimates for a Nonlinear Parabolic Equation on Finsler Manifolds Shen, Bin Zhu, Yuhan Differential Geometry In this paper, we explore the positive solutions to the Finslerian nonlinear equation $$\frac{\partial u}{\partial t} = Δ^{\nabla u} u + au\log u + bu,$$ which is related to Ricci solitons and serves as the Euler-Lagrange equation to the Finslerian log-energy functional. We then obtain the global gradient estimate of its positive solution on a compact Finsler metric measure space with the weighted Ricci curvature bounded below. Furthermore, using a new comparison theorem developed by the first author, we also establish a local gradient estimate on a non-compact forward complete Finsler metric measure spaces with the mixed weighted Ricci curvature bounded below, as well as finite bounds of misalignment and some non-Riemannian curvatures. Lastly, we prove the Harnack inequalities and a Liouville-type theorem of such solutions. |
| title | Li-Yau Estimates for a Nonlinear Parabolic Equation on Finsler Manifolds |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2402.00006 |