Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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AN INTERPOLATING HARNACK INEQUALITY FOR NONLINEAR HEAT EQUATION ON A SURFACE

  • Received : 2020.07.20
  • Accepted : 2020.12.09
  • Published : 2021.07.31
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Abstract

In this short note we prove new differential Harnack inequalities interpolating those for the static surface and for the Ricci flow. In particular, for 0 ≤ 𝜀 ≤ 1, α ≥ 0, 𝛽 ≥ 0, 𝛾 ≤ 1 and u being a positive solution to $${\frac{{\partial}u}{{\partial}t}}={\Delta}u-{\alpha}u\;{\log}\;u+{\varepsilon}Ru+{\beta}u^{\gamma}$$ on closed surfaces under the flow ${\frac{\partial}{{\partial}t}}g_{ij}=-{\varepsilon}Rg_{ij}$ with R > 0, we prove that $${\frac{\partial}{{\partial}t}}{\log}\;u-{\mid}{\nabla}\;{\log}\;u{\mid}^2+{\alpha}\;{\log}\;u-{\beta}u^{{\gamma}-1}+\frac{1}{t}={\Delta}\;{\log}\;u+{\varepsilon}R+{\frac{1}{t}{}\geq}0$$.

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Acknowledgement

Research supported by Zhejiang Provincial Natural Science Foundation of China (Grant Number LY18A010022) and NSFC (Grant Number 11971355). The authors are grateful to the anonymous reviewer for very valuable suggestions on the original manuscript which we have followed to improve the paper significantly.

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