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

TL;DR

This paper studies finite-time blow-up of solutions to a semilinear heat equation on infinite graphs, establishing a general blow-up criterion for large initial data and applying it to specific graph classes.

Contribution

It introduces a novel blow-up criterion for semilinear heat equations on infinite graphs, extending Kaplan's method to the discrete graph setting.

Findings

Blow-up occurs for sufficiently large initial data on infinite graphs.

The blow-up criterion applies to various graph structures, including trees and lattices.

The approach provides a foundation for analyzing global existence in a companion paper.

Abstract

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.