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Glen Wheeler:Ancient solutions of the heat equation with polynomial growth——2
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https://www.youtube.com/watch?v=mOM50kazqUk Australian Geometric PDE Seminar Homepage:https://oz-geom-pde.github.io/season01/ Season 01-Ancient Solutions of Geometric Flows We study a variety of ancient solutions to geometric flows including the Heat Equation, the Curve Shortening Flow, the Ricci flow, the Mean Curvature Flow and Semi-Linear Paraoblic Equations. Ancient solutions of the heat equation of polynomial growth 02 Speaker: Glen Wheeler (University of Wollongong) Date: 28 October 2020 Abstract:Glen Wheeler continues the ten-step approach to non-negative ancient solutions to the heat equation on Euclidean space. He works through the representation theorem of Lin and Zhang for such solutions, which expresses the solution pointwise as integrals of Borel measures in a specific form. This is used later in part 2 to bound the dimension of the space of ancient solutions with a specific (polynomial) growth rate and obtain a beautiful decomposition into a polynomial in time with coefficients given by functions polyharmonic in space. Fanghua Lin, Q. S. Zhang——“On Ancient Solutions of the Heat Equation”:https://doi.org/10.1002/cpa.21820 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.
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