Solutions to partial differential equations or systems often, over specific time periods, exhibit smooth behaviour. Given sufficient time, however, they almost invariably undergo a brutal change in behaviour, and this phenomenon has become known as "blowup". In this book, the author provides an overview of what is known about this situation and discusses many of the open problems concerning it. The book deals with classical solutions of global problems for hyperbolic equations or systems. The approach is based on the display and study of two local blowup mechanisms, which the author calls the "ordinary differential equation mechanism" and the "geometric blowup mechanism". It introduces, via energy methods, the concept of lifespan, related to the nonlinear propagation of regularity (from the past to the future). It addresses specifically the question of whether or not there will be blowup in a solution, and it classifies those methods used to give positive answers to the question. The material corresponds to a one semester course for students or researchers with a basic elementary knowledge of partial differential equations, especially of hyperbolic type including such topics as the Cauchy problem, wave operators, energy inequalities, finite speed of propagation, and symmetric systems. It contains a complete bibliography reflecting the high degree of activity among mathematicians interested in the problem.

作品目录

CHAPTER I. The Two Basic Blowup Mechanisms

Introduction

A. The ODE mechanism

B. The geometric blowup mechanism

C. Combinations of the two mechanisms

Notes

CHAPTER II. First Concepts on Global Cauchy Problems

Introduction

1. Short time existence

2. Lifespan and blowup criterion

3. Blowup or not? Functional methods

4. Blowup or not? Comparison and averaging methods

Notes

CHAPTER III. Semilinear Wave Equations

Introduction

1. Semilinear blowup criteria

2. Maximal influence domain

3. Maximal influence domains for weak solutions

4. Blowup rates at the boundary of the maximal influence domain

5. An example of a sharp estimate of the lifespan

Notes

CHAPTER IV. Quasilinear Systems in One Space Dimension

Introduction

1. The scalar case

2. Riemann invariants, simple waves, and L1-boundedness

3. The case of 2 x 2 systems

4. General systems with small data

5. Rotationally invariant wave equations

Notes

CHAPTER V. Nonlinear Geometrical Optics and Applications

Introduction

1. Quasilinear systems in one space dimension

2. Quasilinear wave equations

3. Further results on the wave equation

BIBLIOGRAPHY

INDEX

Notes