Parabolic frequency monotonicity for two nonlinear equations under Ricci flow

TL;DR

The paper investigates parabolic frequency monotonicity for positive solutions of two nonlinear parabolic equations under the Ricci flow on closed manifolds with bounded Ricci curvature. It constructs a weighted framework using the conjugate heat kernel and defines a time-dependent parabolic frequency $U(t)$ for each equation, employing Li–Yau–type gradient estimates and Hamilton-type inequalities to derive differential inequalities that enforce monotonicity when the weight $h(t)$ is negative (increasing $U$) or positive (decreasing $U$). The main contributions are explicit monotonicity results for a nonlinear reaction-diffusion equation $(\partial_t-\Delta)u = a\,u + |\nabla u|^2$ and for a nonlinear heat-type equation $(\partial_t-\Delta)u = \lambda u^p$, including the construction of $U(t)$ with appropriate exponential factors and error terms, and the deduction of integral type Harnack inequalities from these monotonicity properties. This work extends prior results on parabolic frequency under Ricci flow (notably LLX-2023) to nonlinear settings, providing tools for unique continuation and quantitative control of positive solutions in evolving geometries with bounded curvature.

Abstract

In this paper, we consider the parabolic frequency for positive solutions of two nonlinear parabolic equations under the Ricci flow on closed manifolds. We obtain the monotonicity of parabolic frequency for the solution of two nonlinear parabolic equations with bounded Ricci curvature, then we apply the parabolic frequency monotonicity to get some integral type Harnack inequalities and we use -K1 instead of the lower bound 0 of Ricci curvature from Theorem 4.3 in 16, where K1 is any positive constant.

Theorems & Definitions (27)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Corollary 1.4
• Corollary 1.5
• Remark 1.6
• Lemma 3.1
• proof
• Lemma 3.2
• proof