The cubic fourth-order Schrodinger equation

Abstract

We investigate the cubic defocusing fourth order Schrödinger equation $i{u}_t+{\Delta }^{2}u+|u{|}^{2}u=0$ 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 $5\le n\le 8$.

38 pages, references added