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Gradient estimates and Harnack inequalities of a parabolic equation under geometric flow

Guangwen Zhao

Abstract

In this paper, we consider a manifold evolving by a general geometric flow and study parabolic equation \[ (Δ-q(x,t)-\partial_t)u(x,t)=A(u(x,t)),\quad (x,t)\in M\times [0,T]. \] We establish space-time gradient estimates for positive solutions and elliptic type gradient estimates for bounded positive solutions of this equation. By integrating the gradient estimates, we derive the corresponding Harnack inequalities. Finally, as applications, we give gradient estimates of some specific parabolic equations.

Gradient estimates and Harnack inequalities of a parabolic equation under geometric flow

Abstract

In this paper, we consider a manifold evolving by a general geometric flow and study parabolic equation \[ (Δ-q(x,t)-\partial_t)u(x,t)=A(u(x,t)),\quad (x,t)\in M\times [0,T]. \] We establish space-time gradient estimates for positive solutions and elliptic type gradient estimates for bounded positive solutions of this equation. By integrating the gradient estimates, we derive the corresponding Harnack inequalities. Finally, as applications, we give gradient estimates of some specific parabolic equations.

Paper Structure

This paper contains 6 sections, 15 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. space-time gradient estimates for positive solutions
  3. Elliptic type gradient estimates for bounded positive solutions
  4. Applications
  5. Applications of space-time gradient estimates
  6. Applications of elliptic type gradient estimates

Key Result

Theorem 2.1

Let $(M,g(0))$ be a complete Riemannian manifold, and let $g(t)$ evolves by eqb for $t\in [0,T]$. Given $x_0$ and $R>0$, let $u$ be a positive solution to eqa in the cube $Q_{2R,T}:=\{(x,t):d(x,x_0,t)\le 2R, 0\le t\le T\}$. Suppose that there exist constants $K_1, K_2, K_3, K_4,\gamma ,\theta \ge 0$ and on $Q_{2R,T}$. Then for any $\alpha >1$ and $0<\varepsilon <1$, we have on $Q_{R,T}$, where $

Theorems & Definitions (32)

  • Theorem 2.1
  • Remark 2.2
  • Lemma 2.3: Lemma 3 in sun2011Gradient
  • Lemma 2.4
  • proof
  • proof : The proof of Theorem \ref{['tha']}
  • Remark 2.5
  • Corollary 2.6
  • proof
  • Theorem 2.7
  • ...and 22 more