Concerning the Wave equation on Asymptotically Euclidean Manifolds

TL;DR

This paper derives key estimates for the wave equation on asymptotically Euclidean manifolds and uses them to prove almost global existence and the Strauss conjecture in specific geometric settings.

Contribution

It establishes new Strichartz and KSS estimates for wave equations on non-trapping asymptotically Euclidean manifolds, leading to results on existence and conjectures.

Findings

Proved KSS, Strichartz, and weighted Strichartz estimates for wave equations.

Established almost global existence for small data in certain geometric conditions.

Confirmed the Strauss conjecture for radial metrics with specific decay rates.

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 ^{- \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 $d= 3$.