Fujita Exponent for the Fractional Sub‐Laplace Semi‐Linear Heat Equation With Forcing Term on the Heisenberg Group

ABSTRACT We study the following semi‐linear heat equation in the Heisenberg group : where denotes the fractional sub‐Laplacian of order on . We establish that the Fujita exponent, the critical threshold that delimits different dynamical regimes of this equation, is where is the homogeneous dimension of . We prove the existence of global‐in‐time solutions for the supercritical case and the nonexistence of global‐in‐time solutions for the subcritical case For the critical case we provide a class of functions for which blows up in finite time. These results extend the classical Fujita phenomenon to a sub‐Riemannian setting with the nonlocal effects of the fractional sub‐Laplacian. Our proof methods intertwine analytic techniques with the geometric structure of the Heisenberg group.