Hamilton's gradient estimates and Liouville theorems for u_t=Δu^m+aulog u+bu on Riemannian manifolds

Abstract

In this paper, we apply Nash-Moser iteration and Saloff-Coste's Sobolev inequalities to derive gradient estimates for positive solutions to a class of nonlinear parabolic equations of the form u_t=Δu^m+a(x, t)ulog u+b(x, t)u, 1−2/n<m<1 or 1<m<1+1/(1+√(2n)), on a complete Riemannian manifold (M,g) of dimension n, where $a(x, t)$, $b(x, t)$ are C¹ functions. For a=b=0, this equation becomes the porous medium equation (PME) or the fast diffusion equation (FDE) when m>1 or m<1, respectively. As consequences, we obtain Liouville type theorems for the corresponding constant coefficient equations when 1−1/(√(n−1))<m<1 and 1<m<1+1/(√(n−1)).

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BibTeX RIS

BibTeX

@misc{sun2026hamiltons,
  title = {{Hamilton's gradient estimates and Liouville theorems for $u_t=Δu^m+au\log u+bu$ on Riemannian manifolds}},
  author = {Jun Sun and Jiaming Yang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.31393},
  eprint = {2609.31393},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.31393},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Hamilton's gradient estimates and Liouville theorems for u_t=Δu^m+aulog u+bu on Riemannian manifolds
AU  - Jun Sun
AU  - Jiaming Yang
PY  - 2026
DA  - 2026/09/25
PB  - arXiv
DO  - 10.48550/arxiv.2609.31393
UR  - https://arxiv.org/abs/2609.31393
AB  - In this paper, we apply Nash-Moser iteration and Saloff-Coste's Sobolev inequalities to derive gradient estimates for positive solutions to a class of nonlinear parabolic equations of the form u_t=Δu^m+a(x, t)ulog u+b(x, t)u, 1−2/n<m<1 or 1<m<1+1/(1+√(2n)), on a complete Riemannian manifold (M,g) of dimension n, where $a(x, t)$, $b(x, t)$ are C^1 functions. For a=b=0, this equation becomes the porous medium equation (PME) or the fast diffusion equation (FDE) when m>1 or m<1, respectively. As consequences, we obtain Liouville type theorems for the corresponding constant coefficient equations when 1−1/(√(n−1))<m<1 and 1<m<1+1/(√(n−1)).
ER  -

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