Hamilton-Type Gradient Estimates for Porous Medium and Fast Diffusion Equations on Riemannian Manifolds for all exponents
Abstract
We establish local Hamilton-type gradient estimates for positive C2,1 solutions of ut=Δum on complete Riemannian manifolds whose Ricci curvature is bounded from below. For every fixed m>1 and every fixed 0<m<1, there exist β=β(m,n)>0 and C=C(m,n)>0 such that a solution 0<uleqslant A in B2R(x0)×(t0−T,t0] satisfies the following local gradient estimate supBR(x0)×(t0−T/2,t0]|∇ uβ| leqslant C Aβ(1/R+√k+(A(1−m)/2)/√T), where RicMgeqslant−k. The proof uses an intrinsic quantitative alternative to locate, around each prescribed positive point, a cylinder on which the solution has a controlled upper-to-lower ratio. A local gradient estimate on this cylinder is combined with a stopping argument. As a consequence, every uniformly bounded positive ancient solution on a connected complete manifold with nonnegative Ricci curvature is constant.
Related
- Differential Harnack Estimates and Frequency Monotonicity for Filtration Equations on Riemannian Manifolds
- Interior gradient estimates for mean curvature equations on Riemannian manifolds
- Sub-Laplacian Comparison Theorem and Gradient Estimates on Complete Noncompact Contact Riemannian Manifolds
- Yau-type gradient estimates for p-harmonic functions on Riemannian manifolds with boundary under Dirichlet boundary condition
- Hamilton's gradient estimates and Liouville theorems for ut=Δum+aulog u+bu on Riemannian manifolds
BibTeX
@misc{sun2026hamiltontype,
title = {{Hamilton-Type Gradient Estimates for Porous Medium and Fast Diffusion Equations on Riemannian Manifolds for all exponents}},
author = {Jun Sun and Jiaming Yang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31373},
eprint = {2609.31373},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31373},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Hamilton-Type Gradient Estimates for Porous Medium and Fast Diffusion Equations on Riemannian Manifolds for all exponents
AU - Jun Sun
AU - Jiaming Yang
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.31373
UR - https://arxiv.org/abs/2609.31373
AB - We establish local Hamilton-type gradient estimates for positive C^2,1 solutions of u_t=Δu^m on complete Riemannian manifolds whose Ricci curvature is bounded from below. For every fixed m>1 and every fixed 0<m<1, there exist β=β(m,n)>0 and C=C(m,n)>0 such that a solution 0<uleqslant A in B_2R(x_0)×(t_0−T,t_0] satisfies the following local gradient estimate sup_B_R(x_0)×(t_0−T/2,t_0]|∇ u^β| leqslant C A^β(1/R+√k+(A^(1−m)/2)/√T), where Ric_Mgeqslant−k. The proof uses an intrinsic quantitative alternative to locate, around each prescribed positive point, a cylinder on which the solution has a controlled upper-to-lower ratio. A local gradient estimate on this cylinder is combined with a stopping argument. As a consequence, every uniformly bounded positive ancient solution on a connected complete manifold with nonnegative Ricci curvature is constant.
ER -