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<p>In this article, we employ new approaches to study gradient estimates for the positive solutions to the porous medium equations (PME) and fast diffusion equations (FDE) on a complete noncompact Riemannian manifold. Firstly for the PME case, we use the method from [<CitationRef CitationID="CR1">1</CitationRef>, <CitationRef CitationID="CR2">2</CitationRef>] to obtain a Li–Yau type gradient estimate, which improves the one of Huang, Huang and Li in [<CitationRef CitationID="CR3">3</CitationRef>]. Moreover, for the FDE case, we modify the approach by Davies [<CitationRef CitationID="CR4">4</CitationRef>] and derive a Li–Yau type gradient estimate, which polishes up the one of Lu, Ni, Vazquez and Villani in [<CitationRef CitationID="CR5">5</CitationRef>].</p>

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New Approaches to Study Gradient Estimates for Porous Medium and Fast Diffusion Equations

  • 期刊论文

  • 发表日期：

    2026年3月10日

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  • Shansong Huang,
  • Bin Shen

摘要

In this article, we employ new approaches to study gradient estimates for the positive solutions to the porous medium equations (PME) and fast diffusion equations (FDE) on a complete noncompact Riemannian manifold. Firstly for the PME case, we use the method from [1, 2] to obtain a Li–Yau type gradient estimate, which improves the one of Huang, Huang and Li in [3]. Moreover, for the FDE case, we modify the approach by Davies [4] and derive a Li–Yau type gradient estimate, which polishes up the one of Lu, Ni, Vazquez and Villani in [5].