Semilinear damped wave equations with data from Sobolev spaces of negative order: the critical case in Euclidean setting and in the Heisenberg space

In this note, we prove the global existence of solutions to the semilinear damped wave equation in $\mathbb{R}^n$, $n\leq6$, with critical nonlinearity under the assumption that the initial data are small in the energy space $H^1\times L^2$ and under the vanishing condition that the initial data belong to $\dot H^{-\gamma}$ for some $\gamma\in(0,n/2)$. A similar result also applies to the damped wave equation in the Heisenberg group $\mathbb{H}^n$, with $n=1,2$.