Parabolic p-Hardy thresholds on infinite graphs: Global flows, universal blow-up, and time-dependent potentials

We study a parabolic equation driven by the graph p-Laplacian and a nonnegative potential whose time-dependent strength has the same amplitude homogeneity as diffusion. On connected infinite weighted graphs with bounded weighted degree and measure bounded away from zero, we identify the sharp autonomous threshold through the associated Hardy quotient. At and below this threshold, every square-summable solution is global and has nonincreasing norm. Above it, every maximal solution issued from nontrivial bounded nonnegative data blows up in finite time, independently of its initial amplitude. Using graph Agmon–Allegretto–Piepenbrink theory, we relate the dynamics to nonnegativity and criticality of the spatial Schrödinger form, classify square-summable equilibria in the critical case, and prove asymptotic stationarity. For time-dependent potentials, we obtain an improved energy estimate, localized blow-up time bounds, persistent-supercritical blow-up criteria, and ground-state envelopes. Examples on the integer lattice and regular trees demonstrate stationary solutions of arbitrary size and the necessity of localized criteria on infinite graphs.