Abstract
We will discuss some recent developments in the area of nonlinear
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 C. Kenig and F. Merle. The ideas used are natural extensions of
the ones used earlier, by many authors, to study critical nonlinear
elliptic problems, for instance in the context of the Yamabe problem
and in the study of harmonic maps. They also build up 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, C. Kenig and F. Merle. The method will be illustrated
with some concrete examples.