Ancient caloric functions and parabolic frequency on graphs
Abstract
We study ancient solutions to discrete heat equations on some weighted graphs. On a graph of the form of a product with 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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