On Ancient Solutions of the Heat Equation

Fanghua Lin, Q. S. Zhang

Research output: Contribution to journalArticlepeer-review


An explicit representation formula with Martin boundary 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 operators Δ – 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.

Original languageEnglish (US)
Pages (from-to)2006-2028
Number of pages23
JournalCommunications on Pure and Applied Mathematics
Issue number9
StatePublished - Sep 2019

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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