On Existence and Scattering with Minimal Regularity for Semilinear Wave Equations

We prove existence and scattering results for semilinear wave equations with low regularity data. We also determine the minimal regularity that is needed to ensure local existence and well-posedness, and we give counterexamples to well-posedness. More specifically, we show that equations of the type □ u= |u| p, with initial data (u, ut) in Ḣγ(Rn) × Ḣγ − 1(Rn), have a local solution if γ ≥ γ(p, n), and we construct counterexamples if γ < γ(p, n). The existence results rely on mixed-norm space-time estimates of Strichartz-type.