Let \(\mathbb {G}\) be a graded Lie group with homogeneous dimension Q. In this paper, we study the Cauchy problem for a semilinear hypoelliptic damped wave equation involving a positive Rockland operator \(\mathcal {R}\) of homogeneous degree \(\nu \geqslant 2\) on \(\mathbb {G}\) with power type nonlinearity \(|u|^p\) and initial data taken from negative order homogeneous Sobolev space \(\dot{H}^{-\gamma }(\mathbb {G}), \gamma >0\) . In the framework of Sobolev spaces of negative order, we prove that \(p_{\text {Crit}}(Q, \gamma , \nu ):=1+\frac{2\nu }{Q+2\gamma }\) is the new critical exponent for \(\gamma \in (0, \frac{Q}{2})\) . More precisely, we show the global-in-time existence of small data Sobolev solutions of lower regularity for \(p>p_{\text {Crit}}(Q, \gamma , \nu ) \) in the energy evolution space \( \mathcal {C}\left( [0, T], H^{s}(\mathbb {G})\right) , s\in (0, 1]\) . Under certain conditions on the initial data, we also prove a finite-time blow-up of weak solutions for \(1<p<p_{\text {Crit}}(Q, \gamma , \nu )\) . Furthermore, to precisely characterize the blow-up time, we derive sharp upper bound and lower bound estimates for the lifespan in the subcritical cases. We emphasize that our results are also new, even in the setting of higher-order differential operators on \(\mathbb {R}^n\) , and more generally, on stratified Lie groups.
