2017/12/12 by Fanghua Lin, Qi S. Zhang, Lin, Fanghua +1 · 3 citations
Mathematics · Physics and Astronomy · #35K05 #58J35 #Advanced Differential Geometry Research #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.1712.04091
openalex publication_date 2017/12/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 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.