  In this work we study the decidability of a class of global modal logics
arising from Kripke frames evaluated over certain residuated lattices, known in
the literature as modal many-valued logics. We exhibit a large family of these
modal logics which are undecidable, in contrast with classical modal logic and
propositional logics defined over the same classes of algebras. This family
includes the global modal logics arising from Kripke frames evaluated over the
standard Lukasiewicz and Product algebras. We later refine the previous result,
and prove that global modal Lukasiewicz and Product logics are not even
recursively axiomatizable. We conclude by solving negatively the open question
of whether each global modal logic coincides with its local modal logic closed
under the unrestricted necessitation rule.
