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 \(\nu \ge 2\) on \({\mathbb {G}}\) with power-type nonlinearity \(|u|^p\) and initial data taken from negative order homogeneous Sobolev space \(\dot{H}^{-\gamma }({\mathbb {G}}), \gamma >0,\) for the critical exponent case \(p=1+\frac{2\nu }{Q+2\gamma }.\) We also explore the diffusion phenomenon of the higher-order hypoelliptic damped wave equations on graded Lie groups with initial data belonging to Sobolev spaces of negative order. 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.
