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 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 on \({{\mathbb{R}}}^{n}\) or on groups, 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 local loss of regularity phenomenon depending on the step of the group (for stratified groups) and on the order of the considered operator.

在本文中，我们考虑具有时间依赖性Hölder（或更常规）非负传播速度的梯度Lie群上次椭圆均质左不变算子的波动方程。示例包括Heisenberg组或一般分层Lie组上次Laplacian的时变波方程，或\（{{\ mathbb {R}}}上高阶算子的\（p \）-演化方程。^ {n} \）或分组显示，在所有这些情况下，我们的结果都是新的。我们秉承Colombini，De Giorgi和Spagnolo的经典成果的精神，确立了清晰合理的结果。特别是，我们根据组的步骤（对于分层的组）和所考虑的算子的顺序来描述有趣的局部不规则现象。