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arXiv · 2406.15069

Blow-up and global existence for semilinear parabolic equations on infinite graphs

Gabriele Grillo·Giulia Meglioli·Fabio Punzo
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Abstract

We investigate existence of global in time solutions and blow-up of solutions to the semilinear heat equation posed on infinite graphs. The source term is a general function $f(u)$. We always assume that the infimum of the spectrum of the Laplace operator $\lambda_1(G)$ on the graph is positive. According to an interaction between the behavior of $f$ close to $0$ and the value $\lambda_1(G)$, we get the existence of a global in time solution or blow-up of any nonnegative solution, provided that the initial datum is nontrivial.

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Gabriele Grillo, Giulia Meglioli, Fabio Punzo. 2024-06-21. Blow-up and global existence for semilinear parabolic equations on infinite graphs. https://arxiv.org/abs/2406.15069

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