Federico Zucchero | Searcharxiv

arXiv2026

On a semilinear heat equation on infinite graphs I: blow-up for large initial data

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.

math.AP↗

arXiv2026

On a semilinear heat equation on infinite graphs II: blow-up for arbitrary initial data and global existence

This paper is the second part of the study initiated in a companion work and is devoted to finite-time blow-up and global existence for a semilinear heat equation on infinite weighted graphs. We first establish basic results on mild and classical solutions (which, to the best of our knowledge, were not previously available in the setting of graphs) proving their equivalence under suitable assumptions and showing the existence of a solution between a given sub- and supersolution. We then analyze blow-up and global existence on $\mathbb Z^N$, providing proofs based on methods different from those used on $\mathbb Z^N$ in the existing literature. Moreover, for graphs with positive spectral gap, we prove global existence for small initial data. In contrast with previous functional analytic approaches yielding mild solutions, our method relies on the construction of global-in-time supersolutions and leads to the existence of classical solutions.

math.AP↗