Recent Developments on the Global Behavior to Critical Nonlinear Dispersive Equations

We will discuss some recent developments in the area of non-linear dispersive and wave equations, concentrating on the long time behavior of solutions to critical problems. The issues that arise are global well-posedness, scattering and finite time blow-up. In this direction we will discuss a method to study such problems (which we call the "concentration compactness/rigidity theorem" method) developed by the author and Frank Merle. The ideas used here are natural extensions of the ones used earlier, by many authors, to study critical non-linear elliptic problems, for instance in the context of the Yamabe problem and in the study of harmonic maps. They also build on earlier works on energy critical defocusing problems. Elements of this program have also proved fundamental in the determination of "universal profiles" at the blow-up time. This has been carried out in recent works of Duyckaerts, the author and Merle. The method will be illustrated with concrete examples, from works of several authors.