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arXiv · 2601.18549

Semilinear Diffusion Equations on Infinite Graphs: The Dissipative and Lipschitz Cases

Elvise Berchio·Davide Bianchi·Alberto G. Setti·Maria Vallarino
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Abstract

We study a class of semilinear diffusion equations on infinite, connected, weighted graphs, focusing on two types of nonlinearities: monotone decreasing and Lipschitz continuous. Under minimal structural assumptions on the graph, we establish existence, uniqueness, and regularity of mild solutions for initial data in $\ell^p$ spaces, with $1\leq p<\infty$. Our approach relies on time discretization via an implicit Euler scheme and an exhaustion technique using Dirichlet subgraphs. As a by-product, we obtain existence and uniqueness results for a related time-independent equation. Finite-time extinction and positivity for solutions under a specific forcing term are also proved.

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Elvise Berchio, Davide Bianchi, Alberto G. Setti, Maria Vallarino. 2026-01-26. Semilinear Diffusion Equations on Infinite Graphs: The Dissipative and Lipschitz Cases. https://arxiv.org/abs/2601.18549

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