In this paper we study the Cauchy problem for the wave equations for hypoelliptic homogeneous left-invariant operators on graded Lie groups when the time-dependent non-negative propagation speed is regular, H\"older, and distributional. For H\"older coefficients we derive the well-posedness in the spaces of ultradistributions associated to Rockland operators on graded groups. In the case when the propagation speed is a distribution, we employ the notion of "very weak solutions" to the Cauchy problem, that was already successfully used in similar contexts in [GR15] and [RT17b]. We show that the Cauchy problem for the wave equation with the distributional coefficient has a unique "very weak solution" in an appropriate sense, which coincides with classical or distributional solutions when the latter exist. Examples include 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, the results already being new in all these cases.

中文翻译：

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亚椭圆波动方程的极弱解

在本文中，我们研究了当时间相关的非负传播速度是规则的、H\"older 和分布的时，分级 Lie 群上亚椭圆齐次左不变算子波动方程的柯西问题。对于 H\"older 系数我们推导出与分级组上的 Rockland 算子相关的超分布空间中的适定性。在传播速度是分布的情况下，我们对柯西问题采用“非常弱解”的概念，该概念已在 [GR15] 和 [RT17b] 中的类似上下文中成功使用。我们证明了具有分布系数的波动方程的柯西问题在适当意义上具有唯一的“非常弱解”，当后者存在时，这与经典或分布解相吻合。