Strichartz estimate and nonlinear Klein-Gordon on non-trapping scattering space

TL;DR

This paper proves global Strichartz estimates and scattering for nonlinear Klein-Gordon equations on a non-trapping scattering space, extending analysis techniques to this geometric setting.

Contribution

It establishes the first global-in-time Strichartz estimates without derivative loss for Klein-Gordon on scattering manifolds and demonstrates global existence and scattering for small initial data.

Findings

Global Strichartz estimates without derivative loss

Global existence and scattering for small data

Extension of microlocal analysis techniques to scattering spaces

Abstract

We study the nonlinear Klein-Gordon equation on a product space $M=\R\times X$ with metric $\tilde{g}=dt^2-g$ where $g$ is the scattering metic on $X$. We establish the global-in-time Strichartz estimate for Klein-Gordon equation without loss of derivative by using the microlocalized spectral measure of Laplacian on scattering manifold showed in \cite{HZ} and a Littlewood-Paley squarefunction estimate proved in \cite{Zhang}. We prove the global existence and scattering for a family of nonlinear Klein-Gordon equations for small initial data with minimum regularity on this setting.