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

TL;DR

This paper establishes the global existence of solutions for semilinear damped wave equations with critical nonlinearity, considering initial data in negative order Sobolev spaces, in Euclidean and Heisenberg group settings.

Contribution

It extends the theory of damped wave equations by handling initial data in negative Sobolev spaces and includes the critical case in both Euclidean and Heisenberg spaces.

Findings

Global existence for $n\, ext{up to}\,6$ in Euclidean space.

Results also apply to the Heisenberg group for $n=1,2$.

Initial data in negative Sobolev spaces suffices for well-posedness.

Abstract

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 $n=1,2$.