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

In this note, we prove the global existence of solutions to the semilinear damped wave equation with critical nonlinearity in low space dimension, under the assumption that the initial data are small in the energy space and that the initial data belong to some Sobolev space of negative order. The latter is also called vanishing condition. A similar result also applies to the damped wave equation in the Heisenberg group.