This book is concerned mainly with the initial value problem for nonlinear hyperbolic partial differential equations. Some basic methods and techniques of studying hyperbolic equations are presented. In the book, global existence , singularities , blow up and life span of classical solutions to hyperbolic equations are discussed. Some related problems for hyperbolic geometric flow on Riemann surfaces are studied. The book can be used for students and postgraduates majoring in mathematics, teachers，scientists and engineers.

Preface ...................................................................................................I

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Chapter 1 Introduction..................................................................... 1

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1.1 Intention and Signi.cances ....................................................... 1

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1.2 Basic Concepts ........................................................................ 7

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1.3 Some Examples.......................................................................14

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1.4 Preliminaries ..........................................................................18

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Chapter 2 Cauchy Problem for Nonlinear Hyperbolic Systems in Diagonal Form ...........................................................25 _x000D_

2.1 The Single Nonlinear Hyperbolic Equation ...............................25

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2.2 The Classical Solutions to Single Nonlinear Hyperbolic Equation ................................................................................32 _x000D_

2.3 Nonlinear Hyperbolic Equations in Diagonal Form....................40

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Chapter 3 Singularities Caused by the Eigenvectors ....................50

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3.1 Introduction ...........................................................................50

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3.2 Completely Reducible Systems.................................................55

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3.3 2-Step Completely Reducible Systems ......................................59

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3.4 m(m> 2)-Step Completely Reducible Systems with Constant Eigenvalues ..............................................................67 _x000D_

3.5 Non-completely Reducible Systems ..........................................74

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3.6 Examples ...............................................................................76

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Chapter 4 Hyperbolic Geometric Flow on Riemannian Surfaces...........................................................................85 _x000D_

4.1 Introduction ...........................................................................85

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4.2 Cauchy Problem for Hyperbolic Geometric Flow.......................87

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4.3 Mixed Initial Boundary Value Problem for Hyperbolic Geometric Flow ......................................................................99 _x000D_

4.4 Dissipative Hyperbolic Geometric Flow .................................. 107

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4.5 Explicit Solutions..................................................................119

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4.6 Radial Solutions to Hyperbolic Geometric Flow ...................... 124

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Chapter 5 Life-Span of Classical Solutions to Hyperbolic Geometric Flow in Two Space Variables with Slow Decay Initial Data .............................................. 127 _x000D_

5.1 Intention and Signi.cances .................................................... 127

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5.2 Some Useful Lemmas ............................................................ 130

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5.3 Lower Bound of Life-Span ..................................................... 143

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Chapter 6 Nonlinear Hyperbolic Systems with Relaxation ...... 153 _x000D_

6.1 Introduction ......................................................................... 153

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6.2 Global Classical Solutions...................................................... 155

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6.3 Applications .........................................................................162

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6.4 Convergence of Approximate Solutions...................................165

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Chapter 7 Applications.................................................................. 175

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7.1 One Dimensional Hydromagnetic Dynamics............................175

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7.2 Fluid Flow on a Pipe ............................................................ 187

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7.3 Heat Conduction with Finite of Propagation .......................... 189

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7.4 A Nonlinear Systems in Viscoelasticity...................................191

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Bibliography ...................................................................................... 202

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Index .................................................................................................. 209

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