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 X with metric g = dt2 -g where g is the scattering metric on X. We establish the global-intime Strichartz estimate for Klein-Gordon equationwithout 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.