Blow-up and global existence for semilinear parabolic equations on infinite graphs

TL;DR

This paper studies the conditions under which solutions to the semilinear heat equation on infinite graphs either exist globally or blow up, depending on the spectrum of the Laplacian and the behavior of the source term.

Contribution

It provides new criteria linking the spectrum of the Laplacian and the nonlinearity for global existence or blow-up of solutions on infinite graphs.

Findings

Existence of global solutions depends on the behavior of $f$ near zero and the spectral gap $ ext{lambda}_1(G)$.

Blow-up occurs for solutions with certain initial conditions when the spectrum and $f$ interact.

The spectrum's positivity plays a crucial role in the solution behavior.

Abstract

We investigate existence of global in time solutions and blow-up of solutions to the semilinear heat equation posed on infinite graphs. The source term is a general function $f(u)$. We always assume that the infimum of the spectrum of the Laplace operator $\lambda_1(G)$ on the graph is positive. According to an interaction between the behavior of $f$ close to $0$ and the value $\lambda_1(G)$, we get the existence of a global in time solution or blow-up of any nonnegative solution, provided that the initial datum is nontrivial.