Strichartz Estimate and Nonlinear Klein–Gordon Equation on Nontrapping Scattering Space

We study the nonlinear Klein–Gordon equation on a product space M=R×X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M=\mathbb {R}\times X$$\end{document} with metric g~=dt2-g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{g}=dt^2-g$$\end{document} where g is the scattering metric 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 Hassell and Zhang (Anal PDE 9:151–192, 2016) and a Littlewood–Paley squarefunction estimate proved in Zhang (Adv Math 271: 91–111, 2015). 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.