Strichartz estimates and Strauss conjecture on non-trapping asymptotically hyperbolic manifolds

We prove global-in-time Strichartz estimates for the shifted wave equations on non-trapping asymptotically hyperbolic manifolds. The key tools are the spectral measure estimates from \cite{CH2} and arguments borrowed from \cite{HZ, Zhang}. As an application, we prove the small data global existence for any power $p\in (1,1+\frac{4}{n-1})$ for the shifted wave equation in this setting, involving nonlinearities of the form $\pm \mathrm{\mid }u{\mathrm{\mid }}^p$ or $\pm \mathrm{\mid }u{\mathrm{\mid }}^{p-1}u$, which answers partially an open question raised in \cite{SSW}.