# Time-dependent wave equations on graded groups

## TL;DR

This paper studies wave equations with time-dependent speeds on graded Lie groups, establishing well-posedness and regularity loss phenomena for hypoelliptic operators like the sub-Laplacian, extending classical results to more complex structures.

## Contribution

It introduces new well-posedness results for wave equations on graded groups with time-dependent coefficients, including examples on the Heisenberg group and higher-order operators.

## Key findings

- Sharp well-posedness results for hypoelliptic wave equations
- Description of regularity loss depending on group step and operator order
- Extension of classical wave equation results to graded Lie groups

## Abstract

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.