Bulletin of the
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ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2021; 58(6): 1539-1561

Online first article November 8, 2021      Printed November 30, 2021

https://doi.org/10.4134/BKMS.b210087

Copyright © The Korean Mathematical Society.

Perelman type entropy formulae and differential Harnack estimates for weighted doubly nonlinear diffusion equations under curvature dimension condition

Yu-Zhao Wang

Shanxi University

Abstract

We prove Perelman type $\mathcal{W}$-entropy formulae and differential Harnack estimates for positive solutions to weighed doubly nonlinear diffusion equation on weighted Riemannian manifolds with $CD(-K,m)$ condition for some $K\ge0$ and $m\ge n$, which are also new for the non-weighted case. As applications, we derive some Harnack inequalities.

Keywords: Weighted doubly nonlinear diffusion equations, Perelman type entropy formula, differential Harnack estimates, Bakry-\'Emery Ricci curvature, curvature dimension condition.

MSC numbers: Primary 58J35, 35K92, 35K55

Supported by: This work was financially supported by NSFC No. 11701347.

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