Fujita exponent for heat equation with H\"{o}rmander vector fields

In this paper, we show global existence and non-existence results for the heat equation with some of the squares of smooth vector fields on $\Rn$ satisfying H\"{o}rmander'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 $\varphi(t)f(u)$.