Interpolating between constrained Li-Yau and Chow-Hamilton Harnack inequalities for a nonlinear parabolic equation. (English) Zbl 1253.53065
The Harnack inequality for positive solutions of parabolic equation is extensively studied. The author is mainly concerned with an interpolating phenomenon and this phenomenon dates back to the study of B. Chow in [J. Partial Differ. Equations 11, No. 2, 137–140 (1998; Zbl 0943.58017)] obtained a family of Li-Yau-Hamilton inequalities on a closed surface with positive scalar curvature connecting the Li-Yau Harnack inequality for the positive solutions to the heat equation with the Chow-Hamilton linear trace Harnack estimate for the Ricci flow on a closed surface with positive scalar curvature.
In the paper under review the author proves a differential Harnack inequality for nonlinear parabolic equation associated with epsilon-Ricci flow on a closed surface (with positive scalar curvature)
\[ \partial_t=\Delta f-f\log f+\epsilon R f \]
where the scalar curvature and metric is involved by the epsilon Ricci flow \[ \partial_t g=-\epsilon R g. \]
The author obtains a differential inequality regarding two solutions of the parabolic nonlinear equation above. This result is a generalization of joint work of the author with Y. Zhen in [Arch. Math. 94, No. 6, 591–600 (2010; Zbl 1198.53078)], where the main concern is to study the linear equation
\[ \partial_t=\Delta f+\epsilon R f. \]
The proofs are similar and by rather direct but involved computation.
In the paper under review the author proves a differential Harnack inequality for nonlinear parabolic equation associated with epsilon-Ricci flow on a closed surface (with positive scalar curvature)
\[ \partial_t=\Delta f-f\log f+\epsilon R f \]
where the scalar curvature and metric is involved by the epsilon Ricci flow \[ \partial_t g=-\epsilon R g. \]
The author obtains a differential inequality regarding two solutions of the parabolic nonlinear equation above. This result is a generalization of joint work of the author with Y. Zhen in [Arch. Math. 94, No. 6, 591–600 (2010; Zbl 1198.53078)], where the main concern is to study the linear equation
\[ \partial_t=\Delta f+\epsilon R f. \]
The proofs are similar and by rather direct but involved computation.
Reviewer: Weiyong He (Eugene)
MSC:
| 53C44 | Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) |
| 46B70 | Interpolation between normed linear spaces |
| 47A57 | Linear operator methods in interpolation, moment and extension problems |
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