  We study existence and non-existence of global solutions to the semilinear
heat equation with a drift term and a power-like source term, on
Cartan-Hadamard manifolds. Under suitable assumptions on Ricci and sectional
curvatures, we show that global solutions cannot exists if the initial datum is
large enough. Furthermore, under appropriate conditions on the drift term,
global existence is obtained, if the initial datum is sufficiently small. We
also deal with Riemannian manifolds whose Ricci curvature tends to zero at
infinity sufficiently fast.
