The cubic fourth-order Schrodinger equation

Details The cubic fourth-order Schrodinger equation The cubic fourth-order Schrodinger equation

2008-07-30

arxiv.org/abs/0807.4916v2

Mathematics

Analysis of PDEs

38 pages, references added

Scientific paper

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 $5 \le n \le 8$.

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