The cubic fourth-order Schrodinger equation

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 $5 \le n \le 8$.