Global well-posedness and scattering of energy-critical defocusing nonlinear Schrödinger equation on high-dimensional hyperbolic spaces

Abstract

We prove global well-posedness and scattering for the energy-critical defocusing nonlinear Schrödinger equation \begin{equation*} \begin{cases} i\partial_t u + Δu = |u|^{\frac{4}{d-2}} u, u(0) = u_0, \end{cases} \end{equation*} on hyperbolic space $\mathbb{H}^d(d \geq 4)$. We mainly utilize the concentration-compactness/rigidity method. We establish a linear profile decomposition on high-dimensional hyperbolic spaces and prove that the nonlinear profile on hyperbolic spaces can be approximated by the solutions of the energy-critical defocusing nonlinear Schrödinger equation on Euclidean spaces, then obtaining the global well-posedness and scattering of the energy-critical defocusing nonlinear Schrödinger equation on the high-dimensional hyperbolic space.