Time-Dependent Wave Equations on Graded Groups

Abstract In this paper we consider the wave equations for hypoelliptic homogeneous left-invariant operators on graded Lie groups with time-dependent Hölder (or more regular) non-negative propagation speeds. The examples are equation sub-Laplacian Heisenberg group or general stratified groups, $p$ p -evolution higher order ${{\mathbb{R}}}^{n}$ R n already in all these cases our results being new. We establish sharp well-posedness spirit of classical result by Colombini, De Giorgi and Spagnolo. particular, describe an interesting local loss regularity phenomenon depending step (for groups) considered operator.