Semigroup Criteria for Blow-up and Global Existence of Semilinear Heat Equations on Metric Measure Spaces

Abstract: We study finite-time blow-up and global existence for semilinear heat equations with time-dependent reaction coefficients on metric measure spaces. For convex nonlinearities, we establish a blow-up criterion that retains the dependence on the initial datum through its linear evolution and does not require stochastic completeness. Under additional conservativity and heat kernel assumptions, we also prove blow-up at the critical Fujita exponent associated with polynomial volume growth. Our main global existence result is based on a heat-kernel-weighted functional space in which the mild solution map becomes a contraction. This yields a unified construction covering both polynomial and exponential heat kernel decay, general locally Lipschitz nonlinearities, and time-dependent reaction coefficients, without convexity or monotonicity assumptions. In particular, the global existence criterion only uses the behaviour of the nonlinearity on the range explored by the solution. Applications include Riemannian manifolds, metric graphs, and, within a separate discrete framework, weighted graphs. The arguments, especially the functional construction underlying global existence, are new and lead to new results even for the classical underlying spaces considered above.