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arXiv · 2412.15145

Ancient caloric functions and parabolic frequency on graphs

Tang-Kai Lee·Archana Mohandas
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Abstract

We study ancient solutions to discrete heat equations on some weighted graphs. On a graph of the form of a product with $\bb Z,$ we show that there are no non-trivial ancient solutions with polynomial growth. This result is parallel to the case of finite graphs, which is also discussed. Along the way, we prove a backward uniqueness result for solutions with appropriate decaying rate based on a monotonicity formula of parabolic frequency.

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Tang-Kai Lee, Archana Mohandas. 2024-12-19. Ancient caloric functions and parabolic frequency on graphs. https://arxiv.org/abs/2412.15145

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