  In this paper, we study the global properties of a class of evolution-like
differential operator with a 0-order perturbation defined on the product of
$r+1$ tori and $s$ spheres $\mathbb{T}^{r+1}\times(\mathbb{S}^{3})^s$, with $r$
and $s$ non-negative integers. By varying the values of $r$ and $s$, we show
that it is possible to recover results already known in the literature and
present new results. The main tool used in this study is Fourier analysis,
taken partially with respect to each copy of the torus and sphere. We obtain
necessary and sufficient conditions related to Diophantine inequalities, change
of sign and connectivity of level sets associated the operator's coefficients.
