The cubic fourth-order Schrodinger equation

TL;DR

This paper studies the cubic defocusing fourth-order Schrödinger equation across various dimensions, establishing global well-posedness and scattering results for dimensions up to 8, and ill-posedness for higher dimensions.

Contribution

It provides the first comprehensive analysis of well-posedness and scattering for this equation in arbitrary dimensions, extending known results to higher dimensions.

Findings

Global well-posedness for n ≤ 8

Ill-posedness for n ≥ 9

Scattering results for 5 ≤ n ≤ 8

Abstract

We investigate the cubic defocusing fourth order Schr\"odinger equation $iu_t + \Delta^2u + |u|^2u=0$ in arbitrary space dimension $\mathbb{R}^n$ for arbitrary $H^2$ initial data. We prove that the equation is globally well-posed when $n \le 8$ and ill-posed when $n \ge 9$, with the additional important information that scattering holds true when $5 \le n \le 8$.