Flow Decomposition and Sharp Integral Fujita Criteria on Weighted Graphs

Abstract

We study the Fujita phenomenon for semilinear heat inequalities generated by variable-speed Laplacians on infinite weighted graphs. Assuming that the graph carries a proper adapted path metric, we establish an integral volume-growth criterion forcing every nonnegative global classical supersolution on the open cylinder $(0,\infty)\times V$ to vanish, without assuming an initial value or trace. We prove that a nontrivial supersolution of this kind exists if and only if the equation has a positive global Cauchy solution for some nonzero point-source datum. A complementary heat-kernel construction gives global Cauchy solutions for all sufficiently small point-source data when the same volume integral converges and a matching anchored heat-kernel upper bound is available. This proves sharpness on integer lattices and on a family of logarithmically perturbed weighted half-lines; in the latter examples, even the exponent of an iterated logarithm can determine the existence--nonexistence alternative. A finer nonexistence criterion couples intrinsic volume growth with the capacity of intrinsic annuli. Its proof combines parabolic testing, a Laplace--resolvent reduction, and a pathwise decomposition of resolvent currents. The nonexistence results require no volume-doubling property, Poincar\'e inequality, heat-kernel bound, or stochastic completeness.