Semilinear Diffusion Equations on Infinite Graphs: The Dissipative and Lipschitz Cases
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| Main Authors: | , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866914563928096768 |
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| author | Berchio, Elvise Bianchi, Davide Setti, Alberto G. Vallarino, Maria |
| author_facet | Berchio, Elvise Bianchi, Davide Setti, Alberto G. Vallarino, Maria |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_18549 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Semilinear Diffusion Equations on Infinite Graphs: The Dissipative and Lipschitz Cases Berchio, Elvise Bianchi, Davide Setti, Alberto G. Vallarino, Maria Analysis of PDEs 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. |
| title | Semilinear Diffusion Equations on Infinite Graphs: The Dissipative and Lipschitz Cases |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2601.18549 |