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arXiv · 2405.18173

Life span of solutions to a semilinear parabolic equation on locally finite graphs

Yuanyang Hu·Mingxin Wang
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Abstract

Let $G=(V,E)$ be a locally finite connected graph. We develop the first eigenvalue method on $G$ introduced in 1963 by Kaplan \cite{Kaplan} on Euclidean space, the discrete Phragmén-Lindelöf principle of parabolic equations and upper and lower solutions method on $G$. Using these methods, we establish the estimates and asymptotic behaviour of the life span of solutions to a semilinear heat equation with initial data $λψ(x)$ for different scales of $λ$ on $G$ under some different conditions. Our results are different from the continuous case, which is related to the structure of the graph $G$.

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Yuanyang Hu, Mingxin Wang. 2024-05-28. Life span of solutions to a semilinear parabolic equation on locally finite graphs. https://arxiv.org/abs/2405.18173

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