  We prove the local well-posedness for the nonlinear fourth-order
Schr\"odinger equation (NL4S) in Sobolev spaces. We also studied the regularity
of solutions in the sub-critical case. A direct consequence of this regularity
is the global well-posedness above mass and energy spaces under some
assumptions. Finally, we show the ill-posedness for (NL4S) in some cases of the
super-critical range.

Analysis of PDEs