New from International Press, July 2008

Lectures on Non-Linear Wave Equations

Second Edition

Christopher D. Sogge (Johns Hopkins University)

This much-anticipated revised second edition of Christopher Sogge’s 1995 work provides a self-contained account of the basic facts concerning the linear wave equation and the methods from harmonic analysis that are necessary when studying nonlinear hyperbolic differential equations.

Publication details

Hardcover. 203 pages.
ISBN-13: 978-1-57146-173-5
ISBN-10: 1-57146-173-6
2000 MSC: Primary 35L70, Secondary 42B25.
Newly published: July 2008
Publisher: International Press of Boston
List price: $69.00. Discounts may apply.

Full description

This much-anticipated revised second edition of Christopher Sogge’s 1995 work provides a self-contained account of the basic facts concerning the linear wave equation and the methods from harmonic analysis that are necessary when studying nonlinear hyperbolic differential equations. Sogge examines quasilinear equations with small data where the Klainerman-Sobolev inequalities and weighted space-time estimates are introduced to prove global existence results. New simplified arguments are given in the current edition that allow one to handle quasilinear systems with multiple wave speeds. The next topic concerns semilinear equations with small initial data. John’s existence theorem for R¹⁺³ is discussed with blow-up problems and some results for the spherically symmetric case. After this, general Strichartz estimates are treated. A proof of the endpoint Strichartz estimates of Keel and Tao and the Christ-Kiselev lemma are given, the material being new in this edition. Using the Strichartz estimates, the critical wave equation in R¹⁺³ is studied.

Table of Contents

• Chapter 1: Background and groundwork
  • Linear wave equation: a review
  • Energy inequality: a first version
  • Existence and uniqueness for linear equations
  • Local existence for quasilinear equations
  • Local existence for semilinear equations in (1 + 3)-dimensions
  • Notes
• Chapter 2: Quasilinear equations with small data
  • Klainerman-Sobolev inequalities
  • Global existence in higher dimensions
  • A weighted energy estimate
  • Almost global existence for symmetric systems
  • Null condition and global existence when n=3
  • The restriction theorem and local existence revisited
  • Notes
• Chapter 3: Semilinear equations with small data
  • Strichartz’s estimate for the wave equation
  • John’s existence theorem for R¹⁺³
  • Blow-up for small powers
  • Notes
• Chapter 4: General Strichartz estimates
  • The endpoint Strichartz estimates of Keel and Tao
  • The Christ-Kiselev lemma and inhomogeneous estimates
  • An application: Existence theorems for rough data
  • Improved results under spherical symmetry
  • Notes
• Chapter 5: Global existence for semilinear equations with large data
  • Main results
  • Energy estimates and the subcritical case
  • A decay lemma and the critical case
  • Notes
• Appendix: Some tools from classical analysis

About the author

Christopher Sogge received his PhD from E.M. Stein at Princeton University in 1985. He has been an NSF Postdoctoral Fellow (1985-1988), a Sloan Research Fellow (1988-1989), a Guggenheim Fellow (2005-2006), and a recipient of a Presidential Young Investigator Award (1988-1993). He has held positions at the University of Chicago (1985-1989) and UCLA (1989-1996), and he currently is a Professor at the Johns Hopkins University. Professor Sogge’s research interests include Fourier analysis, partial differential equations, and geometry.

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