Abstract
Abstract In this paper, we show global existence and non-existence results for the heat equation with sum of the squares of smooth vector fields on $$\mathbb {R}^{n}$$ satisfying Hörmander’s rank condition with a non-linearity of the form f ( u ), where f is a suitable function and u is the solution. In particular, when $$f(u)=u^p$$ , we calculate the critical Fujita exponent. We also give necessary conditions for blow-up or, alternatively, a sufficient condition for the existence of positive global solutions for time-dependent nonlinearities of the type $$\varphi (t)f(u)$$ .