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

Huai-Dong Cao, Meng Zhu

Abstract

In this paper we study the porous medium equation (PME) coupled with the Ricci flow on complete manifolds with bounded nonnegative curvature operator. In particular, we derive Aronson-Bénilan and Li-Yau-Hamilton type differential Harnack estimates for positive solutions to the PME, with a linear forcing term, under the Ricci flow.

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

Abstract

In this paper we study the porous medium equation (PME) coupled with the Ricci flow on complete manifolds with bounded nonnegative curvature operator. In particular, we derive Aronson-Bénilan and Li-Yau-Hamilton type differential Harnack estimates for positive solutions to the PME, with a linear forcing term, under the Ricci flow.

Paper Structure

This paper contains 5 sections, 18 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. The Main Quantity for PME and FDE
  3. Aronson-Bénilan and Li-Yau-Hamilton Estimates for PME
  4. Case 1: $b\geq2$
  5. Case 2: $1\leq b\leq 2$

Key Result

Theorem 1.1

Let $(M^n, g_{ij}(t))$, $t\in[0, T)$, be a complete solution to the Ricci flow with bounded and nonnegative curvature operator at each time $t$. If $u$ is a bounded smooth positive solution to eq2 with $p>1$ and $v=\frac{p}{p-1}u^{p-1}$, then we have on $M\times(0,T)$, where $d=\max\{2\alpha, 1\}$ and $\alpha=\frac{n(p-1)}{1+n(p-1)}$.

Theorems & Definitions (29)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • ...and 19 more