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

TL;DR

This paper investigates finite-time blow-up and global existence of solutions to a semilinear heat equation on infinite graphs, introducing new methods for establishing classical solutions and analyzing behavior on specific graph structures.

Contribution

It develops novel techniques for proving classical solutions and analyzes blow-up and global existence on infinite graphs, including $ ext{Z}^N$, with new proofs and conditions.

Findings

Established equivalence of mild and classical solutions on graphs.

Proved global existence for small initial data on graphs with positive spectral gap.

Provided new methods for analyzing blow-up and global existence on infinite graphs.

Abstract

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.