Local Aronson-Bénilan estimates and entropy formulae for porous medium and fast diffusion equations on manifolds

Ricci流下多孔介质方程的Aronson-Bénilan估计

Aronson-Bénilan estimates for the porous medium equation under the Ricci flow

本文研究有界非负曲率完备流形上与Ricci流耦合的带线性强迫项的多孔介质方程，推导了其正解的Aronson-Bénilan型和Li-Yau-Hamilton型微分Harnack估计。

Aronson-Bénilan gradient estimates for porous medium equations under lower bounds of $N$-weighted Ricci curvature with $N < 0$

The Aronson-Bénilan gradient estimate for the porous medium equation has been studied as a counterpart to the Li-Yau gradient estimate for the heat equation. In this paper, we give the Aronson-Bénilan gradient estimates for the porous medium equation on weighted Riemannian manifolds under lower bounds of $N$-weighted Ricci curvature with $\varepsilon$-range for some $N < 0$. This is a generalization of those estimates under constant lower $N$-weighted Ricci curvature bounds with $N\in [n,\infty)$.

Local Aronson-Benolan type gradient estimates for the porous medium type equation under Ricci

In this paper, we investigate some new local Aronson-Bénilan type gradient estimates for positive solutions of the porous medium equation $$ u_{t}=Δu^{m}, $$ under Ricci flow. As application, the related Harnack inequalities are derived. Our results generalize known results. These results in the paper can be regard as generalizing the gradient estimates of Lu-Ni-Vázquez-Villani and Huang-Huang-Li to the Ricci flow.

Gradient estimates for $u_t=ΔF(u)$ on manifolds and some Liouville-type theorems

In this paper, we first prove a localized Hamilton-type gradient estimate for the positive solutions of Porous Media type equations: $$u_t=ΔF(u),$$ with $F'(u) > 0$, on a complete Riemannian manifold with Ricci curvature bounded from below. In the second part, we study Fast Diffusion Equation (FDE) and Porous Media Equation (PME): $$u_t=Δ(u^p),\qquad p>0,$$ and obtain localized Hamilton-type gradient estimates for FDE and PME in a larger range of $p$ than that for Aronson-Bénilan estimate, Harnack inequalities and Cauchy problems in the literature. Applying the localized gradient estimates for…

Bochner formula and Bernstein type estimates on locally finite graphs

In this paper, we consider three typical problems on a locally finite connected graph. The first one is to study the Bochner formula for the Laplacian operator on a locally finite connected graph. We use the Bochner formula to derive the Bernstein type estimate of the heat equation. The second is to derive the Reilly type formula of the Laplacian operator. The last one is to obtain global positive solution to porous-media equation via the use of Aronson-Benilan argument. There is not much work in the direction of the study of nonlinear heat equations on locally finite connected graphs.