PARABOLIC FREQUENCY MONOTONICITY FOR TWO NONLINEAR EQUATIONS UNDER RICCI FLOW

doi:10.1007/s10473-026-0212-4

数学物理学报(英文版) ›› 2026, Vol. 46 ›› Issue (2): 752-766.doi: 10.1007/s10473-026-0212-4

PARABOLIC FREQUENCY MONOTONICITY FOR TWO NONLINEAR EQUATIONS UNDER RICCI FLOW

Chuanhuan LI¹, Yi LI^2,3,*, Kairui XU⁴, Jichun ZHU⁴

1. Shanghai Institute for Mathematics and Interdisciplinary Sciences, Shanghai 200433, China;
2. Fudan University, Shanghai 200433, China;
3. Shanghai Institute for Mathematics and Interdisciplinary Sciences, Shanghai 200433, China;
4. School of Mathematics, Southeast University, Nanjing 211189, China

收稿日期:2024-01-22 修回日期:2024-11-19 发布日期:2026-05-22

PARABOLIC FREQUENCY MONOTONICITY FOR TWO NONLINEAR EQUATIONS UNDER RICCI FLOW

Chuanhuan LI¹, Yi LI^2,3,*, Kairui XU⁴, Jichun ZHU⁴

1. Shanghai Institute for Mathematics and Interdisciplinary Sciences, Shanghai 200433, China;
2. Fudan University, Shanghai 200433, China;
3. Shanghai Institute for Mathematics and Interdisciplinary Sciences, Shanghai 200433, China;
4. School of Mathematics, Southeast University, Nanjing 211189, China

Received:2024-01-22 Revised:2024-11-19 Published:2026-05-22
Contact: ^*Yi LI, E-mail: yilicms@simis.cn
About author:Chuanhuan LI, E-mail: chli@simis.cn; Kairui XU, E-mail: xukarry@163.com; Jichun ZHU, E-mail: 220231969@seu.edu.cn
Supported by:
The second author's research was supported by Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS-ID-2024-LG).

摘要/Abstract

摘要： In this paper, we study the parabolic frequency for positive solutions of two nonlinear parabolic equations under the Ricci flow on closed manifolds. The first equation is $\partial_{t}u=\Delta_{g(t)}u+au+|\nabla_{g(t)} u|^{2}$ with a constant $a$; the other one is $\partial_{t}u=\Delta_{g(t)} u+\lambda u^{p}$ with two constants $\lambda$ and $p\geq1$. Here $g(t)$ is the Riemannian metric involved by Ricci flow. We establish the monotonicity of the parabolic frequency for the solutions of two nonlinear parabolic equations with bounded Ricci curvature. Subsequently, we apply the parabolic frequency monotonicity to derive some integral type Harnack inequalities. Additionally, we use $-K_{1}$ instead of the lower bound $0$ of Ricci curvature from Theorem 1.3 in \cite{LLX-2023}, where $K_{1}$ is any positive constant.

关键词: Ricci flow, parabolic frequency, nonlinear equation

Abstract: In this paper, we study the parabolic frequency for positive solutions of two nonlinear parabolic equations under the Ricci flow on closed manifolds. The first equation is $\partial_{t}u=\Delta_{g(t)}u+au+|\nabla_{g(t)} u|^{2}$ with a constant $a$; the other one is $\partial_{t}u=\Delta_{g(t)} u+\lambda u^{p}$ with two constants $\lambda$ and $p\geq1$. Here $g(t)$ is the Riemannian metric involved by Ricci flow. We establish the monotonicity of the parabolic frequency for the solutions of two nonlinear parabolic equations with bounded Ricci curvature. Subsequently, we apply the parabolic frequency monotonicity to derive some integral type Harnack inequalities. Additionally, we use $-K_{1}$ instead of the lower bound $0$ of Ricci curvature from Theorem 1.3 in \cite{LLX-2023}, where $K_{1}$ is any positive constant.

Key words: Ricci flow, parabolic frequency, nonlinear equation

中图分类号:

58J35

Chuanhuan LI, Yi LI, Kairui XU, Jichun ZHU. PARABOLIC FREQUENCY MONOTONICITY FOR TWO NONLINEAR EQUATIONS UNDER RICCI FLOW[J]. 数学物理学报(英文版), 2026, 46(2): 752-766.

Chuanhuan LI, Yi LI, Kairui XU, Jichun ZHU. PARABOLIC FREQUENCY MONOTONICITY FOR TWO NONLINEAR EQUATIONS UNDER RICCI FLOW[J]. Acta mathematica scientia,Series B, 2026, 46(2): 752-766.

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PARABOLIC FREQUENCY MONOTONICITY FOR TWO NONLINEAR EQUATIONS UNDER RICCI FLOW

PARABOLIC FREQUENCY MONOTONICITY FOR TWO NONLINEAR EQUATIONS UNDER RICCI FLOW

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