Nonlinear Diffusion Equations on Graphs: Global Well-Posedness, Blow-Up Analysis and Applications
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910911772491776 |
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| author | Shao, Mengqiu Yang, Yunyan Zhao, Liang |
| author_facet | Shao, Mengqiu Yang, Yunyan Zhao, Liang |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_10844 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nonlinear Diffusion Equations on Graphs: Global Well-Posedness, Blow-Up Analysis and Applications Shao, Mengqiu Yang, Yunyan Zhao, Liang Analysis of PDEs 35R02, 35B44, 35K55, 34D06 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. |
| title | Nonlinear Diffusion Equations on Graphs: Global Well-Posedness, Blow-Up Analysis and Applications |
| topic | Analysis of PDEs 35R02, 35B44, 35K55, 34D06 |
| url | https://arxiv.org/abs/2504.10844 |