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 [Ann. Inst. Fourier, Grenoble 68 (2018), pp. 1011-1075] and arguments borrowed from [Analysis PDE 9 (2016), pp. 151-192], [Adv. Math. 271 (2015), pp. 91-111]. As an application, we prove the small data global existence for any power p ∈(1,1 + 4 n-1) for the shifted wave equation in this setting, involving nonlinearities of the form ±|u|p or ±|u|p-1u, which answers partially an open question raised in [Discrete Contin. Dyn. Syst. 39 (2019), pp. 7081-7099].