On a semilinear heat equation on infinite graphs I: blow-up for large initial data
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914420585660416 |
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| author | Punzo, Fabio Zucchero, Federico |
| author_facet | Punzo, Fabio Zucchero, Federico |
| contents | We investigate finite-time blow-up of solutions to the Cauchy problem for a semilinear heat equation posed on infinite graphs. Assuming that the initial datum is sufficiently large, we establish a general blow-up criterion valid on arbitrary infinite graphs.
We then apply this result to specific classes of graphs, including trees and the integer lattice. The approach developed in the paper can be regarded as a discrete counterpart of Kaplan's method, suitably adapted to the graph setting. In a companion paper, which is the second part of this work, we also complement the blow-up analysis by addressing arbitrary initial data and proving global existence for sufficiently small data. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_24110 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On a semilinear heat equation on infinite graphs I: blow-up for large initial data Punzo, Fabio Zucchero, Federico Analysis of PDEs We investigate finite-time blow-up of solutions to the Cauchy problem for a semilinear heat equation posed on infinite graphs. Assuming that the initial datum is sufficiently large, we establish a general blow-up criterion valid on arbitrary infinite graphs. We then apply this result to specific classes of graphs, including trees and the integer lattice. The approach developed in the paper can be regarded as a discrete counterpart of Kaplan's method, suitably adapted to the graph setting. In a companion paper, which is the second part of this work, we also complement the blow-up analysis by addressing arbitrary initial data and proving global existence for sufficiently small data. |
| title | On a semilinear heat equation on infinite graphs I: blow-up for large initial data |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2603.24110 |