Concerning the Wave equation on Asymptotically Euclidean Manifolds

Details Concerning the Wave equation on Asymptotically Euclidean Manifolds Concerning the Wave equation on Asymptotically Euclidean Manifolds

2008-12-30

arxiv.org/abs/0901.0022v4

Journal d'Analyse Mathematique, 112(2010), No. 1, 1-32

Mathematics

Analysis of PDEs

Final version. To appear in Journal d'Analyse Mathematique

Scientific paper

10.1007/s11854-010-0023-2

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 ^{- \rho}$ with $\rho>0$. Using the KSS estimate, we prove almost global existence for quadratically semilinear wave equations with small initial data for $\rho> 1$ and $d=3$. Also, we establish the Strauss conjecture when the metric is radial with $\rho>0$ for $d= 3$.

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