Local Aronson-Bénilan estimates for the porous medium and fast diffusion equations on collapsing manifolds
Abstract
We establish local Aronson-Bénilan estimates for positive solutions to the porous medium and fast diffusion equations on complete noncompact Riemannian manifolds with a small scale-invariant integral bound on the negative part of the Ricci curvature in Lp, where p>n/2. No uniform lower bound on the volume of unit balls is assumed, and hence the results apply to manifolds that may collapse. For the fast diffusion equation, we obtain the estimate in the range 1−2/n<m<1. The auxiliary function method, which was designed for the heat equation under integral curvature conditions, is not directly applicable to those two nonlinear equations. The pressure operator in both cases contains a solution-dependent coefficient, which produces a residual term involving ∂t J in the evolution inequality. Its treatment requires a positive lower bound for J, and moreover, control of −∂tlog J. We obtain these bounds by linearizing the auxiliary equation, applying Duhamel's formula, and combining local Dirichlet heat kernel estimates with a singular Volterra inequality. Additionally, in the fast diffusion case, the negativity of v changes the sign structure of the evolution inequality, so a separate coercive estimate is required to close the argument. Local Aronson-Bénilan estimates follow for both diffusion regimes and integrating them along space-time curves gives the corresponding Harnack inequalities.
Related
- Hamilton-Type Gradient Estimates for Porous Medium and Fast Diffusion Equations on Riemannian Manifolds for all exponents
- Differential Harnack Estimates and Frequency Monotonicity for Filtration Equations on Riemannian Manifolds
- Stability and gradient estimates for fully nonlinear elliptic equations on hermitian manifolds
- Schauder estimates for the linearized very fast diffusion equations in bounded domains
- Uniform estimates for complex Monge-Ampère equations II: Hermitian manifolds
BibTeX
@misc{miao2026local,
title = {{Local Aronson-Bénilan estimates for the porous medium and fast diffusion equations on collapsing manifolds}},
author = {Jinpeng Miao and Bin Shen},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.09962},
eprint = {2611.09962},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.09962},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Local Aronson-Bénilan estimates for the porous medium and fast diffusion equations on collapsing manifolds
AU - Jinpeng Miao
AU - Bin Shen
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.09962
UR - https://arxiv.org/abs/2611.09962
AB - We establish local Aronson-Bénilan estimates for positive solutions to the porous medium and fast diffusion equations on complete noncompact Riemannian manifolds with a small scale-invariant integral bound on the negative part of the Ricci curvature in L^p, where p>n/2. No uniform lower bound on the volume of unit balls is assumed, and hence the results apply to manifolds that may collapse. For the fast diffusion equation, we obtain the estimate in the range 1−2/n<m<1. The auxiliary function method, which was designed for the heat equation under integral curvature conditions, is not directly applicable to those two nonlinear equations. The pressure operator in both cases contains a solution-dependent coefficient, which produces a residual term involving ∂_t J in the evolution inequality. Its treatment requires a positive lower bound for J, and moreover, control of −∂_tlog J. We obtain these bounds by linearizing the auxiliary equation, applying Duhamel's formula, and combining local Dirichlet heat kernel estimates with a singular Volterra inequality. Additionally, in the fast diffusion case, the negativity of v changes the sign structure of the evolution inequality, so a separate coercive estimate is required to close the argument. Local Aronson-Bénilan estimates follow for both diffusion regimes and integrating them along space-time curves gives the corresponding Harnack inequalities.
ER -