Abstract
ABSTRACT This paper investigates the Fujita critical exponent for a heat equation with nonlinear memory posed on the Heisenberg groups. A sharp threshold is identified such that, for exponent values less than or equal to this critical value, no global solution exists, regardless of the choice of nonnegative initial data. Conversely, for exponent values above the threshold, global positive solutions exist under small initial data conditions. These findings extend the work of Cazenave, Dickstein, and Weissler, which addressed similar problems in the Euclidean setting. Furthermore, the paper provides lifespan estimates for local solutions under various initial data conditions. The analysis relies on the test function method and the Banach fixed‐point theorem to establish the main results.