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


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

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

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

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


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

Chapter 2 Cauchy
Problem for Nonlinear Hyperbolic Systems in Diagonal Form
...........................................................25
2.1 The
Single Nonlinear Hyperbolic Equation ...............................25

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

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

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

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

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

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

3.4 m(m>
2)-Step Completely Reducible Systems with Constant Eigenvalues
..............................................................67
3.5 Non-completely
Reducible Systems ..........................................74

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


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

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

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

4.3 Mixed
Initial Boundary Value Problem for Hyperbolic Geometric Flow
......................................................................99
4.4 Dissipative
Hyperbolic Geometric Flow .................................. 107

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

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

Chapter 5 Life-Span
of Classical Solutions to Hyperbolic Geometric Flow in Two Space Variables with
Slow Decay Initial Data .............................................. 127
5.1 Intention
and Signi.cances .................................................... 127

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

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

Chapter 6 Nonlinear
Hyperbolic Systems with Relaxation ...... 153
6.1 Introduction
......................................................................... 153

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

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

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

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

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

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

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

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

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

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