Time analyticity for the heat equation under Bakry-Émery Ricci curvature condition

2022/11/25 by Ling Wu, Wu, Ling
Mathematics · Physics and Astronomy · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Advanced Differential Geometry Research

paper · pdf · doi:10.48550/arxiv.2211.14000

Abstract

Inspired by Hongjie Dong and Qi S. Zhang's article \citeZQ2, we find that the analyticity in time for a smooth solution of the heat equation with exponential quadratic growth in the space variable can be extended to any complete noncompact Riemannian manifolds with Bakry-Émery Ricci curvature bounded below and the potential function being of at most quadratic growth. Therefore, our result holds on all gradient Ricci solitons. As a corollary, we give a necessary and sufficient condition on the solvability of the backward heat equation in a class of functions with the similar growth condition. In addition, we also consider the solution in certain L^p spaces with p∈[2,+∞) and prove its analyticity with respect to time.

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