The semilinear wave equation on asymptotically euclidean manifolds

TL;DR

This paper studies the semilinear wave equation on asymptotically Euclidean manifolds, establishing long and global existence results for small initial data using advanced spectral and dispersive estimates.

Contribution

It introduces new dispersive estimates for the linear wave equation on non-trapping asymptotically Euclidean manifolds, leading to existence results for the nonlinear case.

Findings

Long time existence for d=3

Global existence for d>3

Established dispersive inequalities for the linear operator

Abstract

We consider the quadratically semilinear wave equation on R^d, d>=3, equipped with a Riemannian metric. This metric is non-trapping and approaches the Euclidean metric polynomially at infinity. Using Mourre estimates and the Kato theory of smoothness, we obtain a Keel-Smith-Sogge type inequality for the linear equation. Thanks to this estimate, we prove long time existence (d=3) and global existence (d>3) for the nonlinear problem with small initial data.