Global and almost global existence of small solutions to a dissipative wave equation in 3D with nearly null nonlinear terms

Abstract

The existence of globalsmall $\mathcal O(\varepsilon )$ solutions to quadratically nonlinear wave equations in three space dimensions under the null condition isshown to be stable under the simultaneous addition of small $\mathcal O(\nu)$ viscous dissipationand $\mathcal O(\delta)$ non-null quadratic nonlinearities, provided that $\varepsilon \delta/\nu\ll 1$.When this condition is not met, small solutions exist ``almost globally'', and in certain parameter ranges, the addition of dissipation enhances the lifespan.