The content of this book corresponds to a one-semester course taught at the University of Paris-Sud (Orsay) in the spring 1994. It is accessible to students or researchers with a basic elementary knowledge of Partial Dif- ferential Equations, especially of hyperbolic PDE (Cauchy problem, wave operator, energy inequality, finite speed of propagation, symmetric systems, etc.). This course is not some final encyclopedic reference gathering all avail- able results. We tried instead to provide a short synthetic view of what we believe are the main results obtained so far, with self-contained proofs. In fact, many of the most important questions in the field are still completely open, and we hope that this monograph will give young mathe- maticians the desire to perform further research. The bibliography, restricted to papers where blowup is explicitly dis- cussed, is the only part we tried to make as complete as possible (despite the new preprints circulating everyday) j the references are generally not mentioned in the text, but in the Notes at the end of each chapter.Basic references corresponding best to the content of these Notes are the books by Courant and Friedrichs [CFr], Hormander [HoI] and [Ho2], Majda [Ma] and Smoller [Sm], and the survey papers by John [J06], Strauss [St] and Zuily [Zu].
