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In this paper, we study the global properties of a class of evolution-likedifferential 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 showthat it is possible to recover results already known in the literature andpresent 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 obtainnecessary and sufficient conditions related to Diophantine inequalities, changeof sign and connectivity of level sets associated the operator's coefficients.

Global solvability and hypoellipticity for evolution operators on tori and spheres