Energy-critical semi-linear shifted wave equation on the hyperbolic spaces

In this paper we consider a semi-linear, energy-critical, shifted wave equation on the hyperbolic space ${\mathbb{H}}^n$ with $3\le n\le 5$: $${\mathrm{\partial }}_t^{2}u-({\mathrm{\Delta }}_{{\mathbb{H}}^n}+{\rho }^2)u=\zeta \mathrm{\mid }u{\mathrm{\mid }}^{4\mathrm{/}(n-2)}u,(x,t)\in {\mathbb{H}}^n\times \mathbb{R}\mathrm{.}$$ Here $\zeta =\pm 1$ and $\rho =(n-1)\mathrm{/}2$ are constants. We introduce a family of Strichartz estimates compatible with initial data in the energy space $H^{0,1}\times L^2({\mathbb{H}}^n)$ and then establish a local theory with these initial data. In addition, we prove a Morawetz-type inequality $${\int }_{-T_{-}}^{T_{+}}{\int }_{{\mathbb{H}}^n}\frac{\rho (\cosh \mathrm{\mid }x\mathrm{\mid })\mathrm{\mid }u(x,t){\mathrm{\mid }}^{2n\mathrm{/}(n-2)}}{\sinh \mathrm{\mid }x\mathrm{\mid }}d\mu (x)dt\le n\mathcal{E},$$ in the defocusing case $\zeta =-1$, where $\mathcal{E}$ is the energy. Moreover, if the initial data are also radial, we can prove the scattering of the corresponding solutions by combining the Morawetz-type inequality, the local theory and a pointwise estimate on radial $H^{0,1}({\mathbb{H}}^n)$ functions.