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

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.