Global wellposedness and scattering for the defocusing energy-critical nonlinear Schrodinger equations of fourth order in dimensions $d\geq9$

TL;DR

This paper proves that all finite energy solutions to the defocusing energy-critical fourth-order nonlinear Schrödinger equation in dimensions nine and higher are globally well-posed and scatter in both time directions.

Contribution

It establishes global well-posedness and scattering for the energy-critical fourth-order nonlinear Schrödinger equation in high dimensions, extending previous results to $d\,\geq 9$.

Findings

All finite energy solutions are global in time.

Solutions scatter both forward and backward in time.

The result applies to dimensions $d\geq 9$.

Abstract

We consider the defocusing energy-critical nonlinear Schr\"odinger equation of fourth order $iu_t+\Delta^2 u=-|u|^\frac{8}{d-4}u$. We prove that any finite energy solution is global and scatters both forward and backward in time in dimensions $d\geq9$.