Time-dependent wave equations on graded groups

In this paper we consider the wave equations for hypoelliptic homogeneous left-invariant operators on graded Lie groups with time-dependent H\"older propagation speeds. The examples are the time-dependent wave equation for the sub-Laplacian on the Heisenberg group or on general stratified Lie groups, or $p$-evolution equations for higher order operators, already in all these cases our results being new. We establish sharp well-posedness results in the spirit of the classical result by Colombini, de Giorgi and Spagnolo. In particular, we describe an interesting loss of regularity phenomenon depending on the step of the group and on the order of the considered operator.