Differential Harnack Estimates and Entropy Formulae for Weighted p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{p}$$\end{document}-Heat Equations

In this paper, we obtain various global differential Harnack estimates for positive solutions to weighted p-heat equation on closed smooth metric measure space with a lower m-Bakry-Émery Ricci curvature bound. Moreover, Perelman type W\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {W}}$$\end{document}-entropy formulae and Li–Yau type entropy inequalities are derived for weighted p-heat equation on compact with boundary (or no boundary) smooth metric measure space with nonnegative (or negative) m-Bakry-Émery Ricci curvature, which are new in non-weighted case and generalized the results of Kotschwar and Ni (Ann Sci éc Norm Supér 42(1):1–36, 2009) and Wang et al. (Acta Math Sci Ser B Engl Ed 33(4):963–974, 2013).