In this paper we consider a semi-linear, energy-critical, shifted wave equation on the hyperbolic space ${ \mathbb H}^n$ with $3 \leq n \leq 5$: \[ \partial_t^2 u - (\Delta_{ { \mathbb H}^n} + \rho^2) u = \zeta |u|^{ 4/(n-2)} u, \quad (x,t)\in { \mathbb H}^n \times { \mathbb R}. \] Here $\zeta = \pm 1$ and $\rho = (n-1)/2$ 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 \[ \int_{ -T_-}^{ T_+} \int_{ { \mathbb H}^n} \frac{ \rho (\cosh |x|) |u(x,t)|^{ 2n/(n-2)}}{ \sinh |x|} d\mu(x) dt \leq n { \mathcal E}, \] 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.
Categories: math.AP