Time analyticity for the heat equation under Bakry-\'Emery Ricci curvature condition

TL;DR

This paper extends the analyticity in time of heat equation solutions with quadratic growth to complete noncompact Riemannian manifolds with Bakry-Émery Ricci curvature bounds, including gradient Ricci solitons, and explores solvability conditions and $L^p$ space solutions.

Contribution

It generalizes time analyticity results for heat equations to broader geometric settings with quadratic growth conditions, including gradient Ricci solitons, and characterizes solvability and $L^p$ analyticity.

Findings

Time analyticity extends to all gradient Ricci solitons.

Provides necessary and sufficient conditions for backward heat equation solvability.

Proves $L^p$ space solutions are analytic in time.

Abstract

Inspired by Hongjie Dong and Qi S. Zhang's article \cite{ZQ2}, 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-\'Emery 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 $p\in[2,+\infty)$ and prove its analyticity with respect to time.