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 ${\mathbb{R}}^n$, $n\le 6$, 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 $\gamma \in (0,n\mathrm{/}2)$. A similar result also applies to the damped wave equation in the Heisenberg group ${\mathbb{H}}^n$, with $n=1,2$.