Global wellposedness and scattering for the focusing energy-critical nonlinear Schrodinger equations of fourth order in the radial case

TL;DR

This paper proves that radial solutions to the focusing energy-critical fourth-order nonlinear Schrödinger equation are global and scatter if their energy and kinetic energy are below those of the ground state, extending understanding of solution behavior.

Contribution

It establishes global well-posedness and scattering for radial solutions below the ground state energy for the fourth-order focusing NLS.

Findings

Solutions with energy below the ground state are global and scatter.

Radial symmetry is crucial for the proof.

The ground state W acts as a threshold for solution behavior.

Abstract

We consider the focusing energy-critical nonlinear Schr\"odinger equation of fourth order $iu_t+\Delta^2 u=|u|^\frac{8}{d-4}u$. We prove that if a maximal-lifespan radial solution $u: I\times\Bbb R^d\to\mathbb{C}$ obeys $\displaystyle\sup_{t\in I}\|\Delta u(t)\|_{2}<\|\Delta W\|_{2}$, then it is global and scatters both forward and backward in time. Here $W$ denotes the ground state, which is a stationary solution of the equation. In particular, if a solution has both energy and kinetic energy less than those of the ground state $W$ at some point in time, then the solution is global and scatters.