In this paper, we introduce a log entropy for the $p$-heat equation and prove its monotonicity on compact Riemannian manifold with nonnegative Ricci curvature. Moreover, we prove the concavity of $p$-entropy power for positive solutions to the weighted $p$-heat equation on the closed Riemannian manifold with curvature dimension $CD(-K,m)$ condition, where $K\ge 0$ and $m\ge n$. Finally, we obtain an NIW(LIW) formula for $p$-heat equation, which establishes a connection between $p$-entropy power($p$-log entropy), $p$-Fisher information and $p$-$W$-entropy.

中文翻译：

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的对数熵公式和熵幂 $p$黎曼流形上的热方程

在本文中，我们引入了对数熵 $p$-热方程，并证明其在具有非负Ricci曲率的紧致黎曼流形上的单调性。此外，我们证明了$p$熵权用于加权解的正解 $p$曲率尺寸的封闭黎曼流形上的热方程 $Cd（-ķ，米）$ 条件，在哪里 $ķ\ge 0$ 和 $米\ge ñ$。最后，我们获得一个NIW（LIW）公式，用于$p$-heat方程，它之间建立了联系 $p$熵功率（$p$-log熵）， $p$-渔夫信息和 $p$--$\mathrm{w\; ^}$-熵。