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

Abstract

In this paper we consider a semi-linear, defocusing, shifted wave equation on the hyperbolic space \[ \partial_t^2 u - (\Delta_{{\mathbb H}^n} + \rho^2) u = - |u|^{p-1} u, \quad (x,t)\in {\mathbb H}^n \times {\mathbb R}; \] and introduce a Morawetz-type inequality \[ \int_{-T_-}^{T_+} \int_{{\mathbb H}^n} |u|^{p+1} d\mu dt < C E, \] 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 $H^{1/2,1/2} \times H^{1/2,-1/2}({\mathbb H}^n)$ if $2 \leq n \leq 6$ and $1<p<p_c = 1+ 4/(n-2)$. As another application we show that a solution to the quintic wave equation $\partial_t^2 u - \Delta u = - |u|^4 u$ on ${\mathbb R}^2$ scatters if its initial data are radial and satisfy the conditions \[ |\nabla u_0 (x)|, |u_1 (x)| \leq A(|x|+1)^{-3/2-\varepsilon};\quad |u_0 (x)| \leq A(|x|)^{-1/2-\varepsilon};\quad \varepsilon >0. \]