The nonlinear porous medium equation for the f-Laplacian: Hamilton-Souplet-Zhang type gradient estimates and implications
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918214914539520 |
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| author | Taheri, Ali Vahidifar, Vahideh |
| author_facet | Taheri, Ali Vahidifar, Vahideh |
| contents | This article presents new gradient estimates for positive solutions to the nonlinear porous medium equation (NPME) in the context of smooth metric measure spaces. The diffusion operator here is the f-Laplacian and the gradient estimates of interest are mainly of Hamilton-Souplet-Zhang types. These estimates are established using a variety of methods and techniques and several implications, most notably, to parabolic Liouville-type results and characterisation of ancient solutions are given. The problem is posed in the general framework where the metric and potential evolve with time and the proofs make use of natural lower bounds on the time derivative of the metric and the Bakry-Émery m-Ricci curvature tensors. Our results extend and improve various existing ones in the literature. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_17997 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The nonlinear porous medium equation for the f-Laplacian: Hamilton-Souplet-Zhang type gradient estimates and implications Taheri, Ali Vahidifar, Vahideh Analysis of PDEs 53C44, 58J60, 58J35, 60J60 This article presents new gradient estimates for positive solutions to the nonlinear porous medium equation (NPME) in the context of smooth metric measure spaces. The diffusion operator here is the f-Laplacian and the gradient estimates of interest are mainly of Hamilton-Souplet-Zhang types. These estimates are established using a variety of methods and techniques and several implications, most notably, to parabolic Liouville-type results and characterisation of ancient solutions are given. The problem is posed in the general framework where the metric and potential evolve with time and the proofs make use of natural lower bounds on the time derivative of the metric and the Bakry-Émery m-Ricci curvature tensors. Our results extend and improve various existing ones in the literature. |
| title | The nonlinear porous medium equation for the f-Laplacian: Hamilton-Souplet-Zhang type gradient estimates and implications |
| topic | Analysis of PDEs 53C44, 58J60, 58J35, 60J60 |
| url | https://arxiv.org/abs/2511.17997 |