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Chapter 1. Basic Notions 1 Chapter 2. Finite Speed of Propagation of Signals 5 References 14 Chapter 3 Hyperbolic Equations with Constant Coefficients 15 3.1 The Domain of Influence 15 3.2 Spacelike Hypersurfaces 19 3.3- The Initial Value Problem on Spacelike Hypersurfaces 23 3.4. Characteristic Surfaces 25 315. Solution of the Initial Value Problem by the Radon Transform 29 3,6. Conservation of Energy 33 References 34 Chapter 4. Hyperbolic Equations with Variable Coefficients 37 4.1. Equations with a Single Space Variable 37 4.2. Characteristic Surfaces 39 4.3. Energy Inequalities for Symmetric Hyperbolic Systems 41 4.4. Energy inequalities for Solutions of Second-Order Hyperbolic Equations 45 4.5. Energy Inequalities for Higher-Order Hyperbolic Equations 46 References 53 Chapter 5. Pseudodifferential Operators and Energy Inequalities 55 References 60 Chapter 6. Existence of Solutions 61 6.1 Equivalence of the Initial Value Problem and the Periodic Problem 61 6.2. Negative Norms 63 6.3. Solution of the Periodic Problem 65 6.4. A Local Uniqueness Theorem 66 References 67 Chapter 7. Waves and Rays 69 Introduction 69 7.1. The Initial Value Problem for Distributions 71 7.2. Progressing Waves 74 7.3. Integrals of Compound Distributions 77 74 An A pproximate Riemann Function and the Generalized Huygens Principle 79 References 82 Chapter 8 Finite Difference Approximation to Hyperbolic Equations 83 8, . Consistency 83 8.2. Domain of Dependence 84 8.3 Stability and Convergence 85 8.4. Higher-Order Schemes and Their Stability 88 8.5. The Gibbs Phenomenon 96 8.6. The Computation of Discontinuous Solutions of Linear Hyperbolic Equations 98 87. Schemes in More Than One Space Variable 103 .8 The Stability of Difference Schemes 108 References 119 Chapter 9 Scattering Theory 121 9.. Asymptotic Behavior of Solutions of the Wave Equation 121 9.2 The Lax-Phillips Scatering Theory 125 9.3 The Associated Semigroup 129 9.4. Back to the Wave Equation in the Exteior of an Obstacle 132 95. The Semigroup Associated with Scattering by an Obstacle 139 9.6. Analytic Form of the Scattering Matrix 144 9.7. Scattering of Automorphic Waves 154 References 163 Chaper 10. Hyperbolic Systems of Conservation Laws 165 10. Scalar Equations; Basics 165 10.2. The Initial Value Problem for Admissible Solutions 169 10.3 Hyperbolic Systems of Conversation Laws 178 10.4. The Viscosity Method and Entropy 184 105. Finite Difference Methods 189 10 6. The Flow of Compressible luids 193 Reterences 197

Library of Congress subject headings for this publication: Differential equations, Hyperbolic