A semi-linear shifted wave equation on the hyperbolic spaces with application on a quintic wave equation on R²

In this paper we consider a semi-linear, defocusing, shifted wave equation on the hyperbolic space ∂[subscript t][superscript 2]u- (∆ℍ[superscript n] +ρ[superscript 2] )u = -|u| p[superscript -1] u, (x,t) ∈ ℍ n × ℝ, and we introduce a Morawetz-type inequality (Formula presented) where ε is the energy. Combining this inequality with a well-posedness theory, we can establish a scattering result for solutions with initial data in H[superscript 1/2,1/2] × H[superscript 1/2,−1/2](ℍ[superscript n) if 2 ≤ n ≤ 6 and 1 0