Energy-critical semi-linear shifted wave equation on the hyperbolic spaces. (Q2629092)

The author continues the discussion on the semi-linear shifted wave equation on \(\mathbb{H}^n\) \[ u_{tt}-\Delta_{\mathbb{H}^n}u-\rho ^2u=\zeta | u| ^{p-1}u,\quad (x,t)\in \mathbb{H}^n\times \mathbb{R} \] of the previous paper [\textit{R. Shen} and \textit{G. Staffilani}, Trans. Am. Math. Soc. 368, No. 4, 2809--2864 (2016; Zbl 1339.35190)], where energy-subcritical case (\(p<p_c =1+4/(n-2),\;2\leq n\leq 6\)) was considered. Here, the critical case \(p=p_c\) is discussed, whereas \(\rho =(n-1)/2,\) \(3\leq n\leq 5\). A family of Strichartz estimates compatible with initial data in the energy space \(H^{0,1}\times L^2 (\mathbb{H}^n)\) is introduced, a local theory with these initial data is established and a Morawetz-type inequality is proved. Moreover, if the initial data are radial, 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 may be proved.