Concerning the Wave equation on Asymptotically Euclidean Manifolds

Abstract

We obtain KSS, Strichartz and certain weighted Strichartz estimate for the wave equation on $(\R^d, \mathfrak{g})$, $d \geq 3$, when metric $\mathfrak{g}$ is non-trapping and approaches the Euclidean metric like $ x ^{- ρ}$ with $ρ>0$. Using the KSS estimate, we prove almost global existence for quadratically semilinear wave equations with small initial data for $ρ> 1$ and $d=3$. Also, we establish the Strauss conjecture when the metric is radial with $ρ>0$ for $d= 3$.

Final version. To appear in Journal d'Analyse Mathematique