On ancient solutions of the heat equation

TL;DR

This paper provides explicit formulas for positive ancient solutions of the heat equation in Euclidean and Riemannian settings, showing their structure as Laplace transforms and characterizing polynomial growth solutions.

Contribution

It introduces explicit representation formulas for ancient solutions and characterizes polynomial growth solutions as polynomials in time, extending previous understanding.

Findings

Explicit formulas for positive ancient solutions in Euclidean space.

Representation of solutions as Laplace transforms of elliptic solutions.

Finite-dimensional space of polynomial growth solutions, which are polynomials in time.

Abstract

An explicit representation formula for all positive ancient solutions of the heat equation in the Euclidean case is found. In the Riemannian case with nonnegative Ricci curvature, a similar but less explicit formula is also found. Here it is proven that any positive ancient solution is the standard Laplace transform of positive solutions of the family of elliptic operator $\Delta - s$ with $s>0$. Further relaxation of the curvature assumption is also possible. It is also shown that the linear space of ancient solutions of polynomial growth has finite dimension and these solutions are polynomials in time.