New Differential Harnack Inequalities for Nonlinear Heat Equations


Jiayong WU.New Differential Harnack Inequalities for Nonlinear Heat Equations[J].Chinese Annals of Mathematics B,2020,41(2):267~284
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Jiayong WU;


This work was supported by the National Natural Science Foundation of China (No.11671141) and the Natural Science Foundation of Shanghai (No.17ZR1412800).
Abstract: This paper deals with constrained trace, matrix and constrained matrix Harnack inequalities for the nonlinear heat equation $\omega_t=\Delta\omega+a\omega\ln \omega$ on closed manifolds. A new interpolated Harnack inequality for $\omega_t=\Delta\omega-\omega\ln\omega+\varepsilon R\omega$ on closed surfaces under $\varepsilon$-Ricci f\/low is also derived. Finally, the author proves a new differential Harnack inequality for $\omega_t=\Delta\omega-\omega\ln\omega$ under Ricci f\/low without any curvature condition. Among these Harnack inequalities, the correction terms are all time-exponential functions, which are superior to time-polynomial functions.


Harnack inequality, Nonlinear heat equation, Ricci f/low


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