# A note on time analyticity for ancient solutions of the heat equation

## TL;DR

This paper proves that ancient solutions of the heat equation with exponential growth are time-analytic on certain manifolds, establishing a precise condition for backward heat equation solvability in this class.

## Contribution

It demonstrates time analyticity for ancient solutions with exponential growth and characterizes solvability conditions for the backward heat equation.

## Key findings

- Ancient solutions with exponential growth are time-analytic on ^n or manifolds with Ricci curvature bounded below.
- A necessary and sufficient condition for backward heat equation solvability in exponential growth class.
- Time analyticity does not hold for generic solutions without exponential growth.

## Abstract

It is well known that generic solutions of the heat equation are not analytic in time in general. Here it is proven that ancient solutions with exponential growth are analytic in time in ${\M} \times (-\infty, 0]$. Here $\M=\R^n$ or is a manifold with Ricci curvature bounded from below. Consequently a necessary and sufficient condition is found on the solvability of backward heat equation in the class of functions with exponential growth.