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.
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
.
DOI: 10.1007/s10473-026-0212-4
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