On a semilinear parabolic equation with time-dependent source term on infinite graphs

TL;DR

This paper investigates the behavior of solutions to a semilinear parabolic equation with a time-dependent source on infinite graphs, establishing conditions for global existence or finite-time blow-up based on spectral and source parameters.

Contribution

It provides new criteria for solution existence or blow-up for semilinear parabolic equations on infinite graphs with time-dependent sources.

Findings

Solutions exist globally or blow up in finite time depending on parameters.

Spectral properties of the graph influence solution behavior.

Conditions for global existence are explicitly characterized.

Abstract

We are concerned with semilinear parabolic equations, with a time-dependent source term of the form $h(t)u^q$ with $q>1$, posed on an infinite graph. We assume that the bottom of the $L^2$-spectrum of the Laplacian on the graph, denoted by $\lambda_1(G)$, is positive. In dependence of $q, h(t)$ and $\lambda_1(G)$, we show global in time existence or finite time blow-up of solutions.