A Semi-linear Shifted Wave Equation on the Hyperbolic Spaces with Application on a Quintic Wave Equation on ${\mathbb{R}}^2$

In this paper we consider a semi-linear, defocusing, shifted wave equation on the hyperbolic space $${\mathrm{\partial }}_t^{2}u-({\mathrm{\Delta }}_{{\mathbb{H}}^n}+{\rho }^2)u=-\mathrm{\mid }u{\mathrm{\mid }}^{p-1}u,(x,t)\in {\mathbb{H}}^n\times \mathbb{R};$$ and introduce a Morawetz-type inequality $${\int }_{-T_{-}}^{T_{+}}{\int }_{{\mathbb{H}}^n}\mathrm{\mid }u{\mathrm{\mid }}^{p+1}d\mu dt<CE,$$ where $E$ is the energy. Combining this inequality with a well-posedness theory, we can establish a scattering result for solutions with initial data in $H^{1\mathrm{/}2,1\mathrm{/}2}\times H^{1\mathrm{/}2,-1\mathrm{/}2}({\mathbb{H}}^n)$ if $2\le n\le 6$ and $1<p<p_c=1+4\mathrm{/}(n-2)$. As another application we show that a solution to the quintic wave equation ${\mathrm{\partial }}_t^{2}u-\mathrm{\Delta }u=-\mathrm{\mid }u{\mathrm{\mid }}^{4}u$ on ${\mathbb R}^2$ scatters if its initial data are radial and satisfy the conditions $$\mathrm{\mid }\mathrm{\nabla }{u}_0(x)\mathrm{\mid },\mathrm{\mid }{u}_1(x)\mathrm{\mid }\le A(\mathrm{\mid }x\mathrm{\mid }+1{)}^{-3\mathrm{/}2-\epsilon };\mathrm{\mid }{u}_0(x)\mathrm{\mid }\le A(\mathrm{\mid }x\mathrm{\mid }{)}^{-1\mathrm{/}2-\epsilon };\epsilon >0.$$