, Volume 6, Issue 2, pp 121-141
Date: 25 Dec 2011

Critical non-linear dispersive equations: global existence, scattering, blow-up and universal profiles

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We discuss recent progress in the understanding of the global behavior of solutions to critical non-linear dispersive equations. The emphasis is on global existence, scattering and finite time blow-up. For solutions that are bounded in the critical norm, but which blow-up in finite time, we also discuss the issue of universal profiles at the blow-up time.

Communicated by: Toshiyuki Kobayashi
This article is based on the 9th Takagi Lectures that the author delivered at Research Institute for Mathematical Sciences, Kyoto University on June 4, 2011.
Supported in part by NSF grants DMS-0456583 and DMS-0968472.