Introduction

Abstract

There has been a tremendous amount of recent activity dealing with the subject of solutions to partial differential equations which blow-up in a finite time. The mathematical theory for this is extensive and reviews may be found in Levine (1990) and Samarskii et al. (1994). However, finite time blow-up and other very rapid instabilities occur in situations in mechanics and other areas of applied mathematics, and studies of these phenomena have very recently been gaining momentum. The object of this book is to present an account of various instances in applied mathematics where the solution, or one of its derivatives, to a partial differential equation or to a coupled system of partial differential equations, blows up in a finite time leading to a catastrophic instability. We also deal with some situations where the solution grows very rapidly but need not cease to exist in a finite time. The problems for which we include an exposition are of importance in real life and hence justify our inclusion of them. The main emphasis here is to include recent developments in blow-up or rapid growth of solutions to practical problems which occur in some field of mechanics.