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

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 λ₁(G) on the graph is positive. According to an interaction between the behavior of f close to $0$ and the value λ₁(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.