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arXiv · 2504.10844

Nonlinear Diffusion Equations on Graphs: Global Well-Posedness, Blow-Up Analysis and Applications

Mengqiu Shao·Yunyan Yang·Liang Zhao
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Abstract

For a nonlinear diffusion equation on graphs whose nonlinearity violates the Lipschitz condition, we prove short-time solution existence and characterize global well-posedness by establishing sufficient criteria for blow-up phenomena and quantifying blow-up rates. These theoretical results are then applied to model complex dynamical networks, with supporting numerical experiments. This work mainly makes two contributions: (i) generalization of existing results for diffusion equations on graphs to cases with nontrivial potentials, producing richer analytical results; (ii) a new PDE approach to model complex dynamical networks, with preliminary numerical experiments confirming its validity.

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Mengqiu Shao, Yunyan Yang, Liang Zhao. 2025-04-15. Nonlinear Diffusion Equations on Graphs: Global Well-Posedness, Blow-Up Analysis and Applications. https://arxiv.org/abs/2504.10844

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