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 $3 \leq n \leq 5$: $\partial_t^2 u - (Δ_{{\mathbb H}^n} + ρ^2) u = ζ|u|^{4/(n-2)} u, \quad (x,t)\in {\mathbb H}^n \times {\mathbb R}.$ Here $ζ= \pm 1$ and $ρ= (n-1)/2$ 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 $\int_{-T_-}^{T_+} \int_{{\mathbb H}^n} \frac{ρ(\cosh |x|) |u(x,t)|^{2n/(n-2)}}{\sinh |x|} dμ(x) dt \leq n {\mathcal E},$ in the defocusing case $ζ= -1$, 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.

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