University Lecture Series 1990; 64 pp; softcover Volume: 2 Reprint/Revision History: third printing 1997 ISBN10: 0821870017 ISBN13: 9780821870013 List Price: US$20 Member Price: US$16 Order Code: ULECT/2
 This is the second volume in the University Lecture Series, designed to make more widely available some of the outstanding lectures presented in various institutions around the country. Each year at Lehigh University, a distinguished mathematical scientist presents the Pitcher Lectures in the Mathematical Sciences. This volume contains the Pitcher lectures presented by Fritz John in April 1989. The lectures deal with existence in the large of solutions of initial value problems for nonlinear hyperbolic partial differential equations. As is typical with nonlinear problems, there are many results and few general conclusions in this extensive subject, so the author restricts himself to a small portion of the field, in which it is possible to discern some general patterns. Presenting an exposition of recent research in this area, the author examines the way in which solutions can, even with small and very smooth initial data, "blow up" after a finite time. For various types of quasilinear equations, this time depends strongly on the number of dimensions and the "size" of the data. Of particular interest is the formation of singularities for nonlinear wave equations in three space dimensions. Table of Contents  Equations in one space variable
 Blowup in higher dimensions
 Longtime existence for solutions of nonlinear wave equations with small initial data
 Appendix I. Uniqueness for nonlinear wave equations
 Appendix II. Klainerman's inequality
