This book deals with blow-up, or at least very rapid growth, of a solution to a system of partial differential equations that arise in practical physics situations. It begins with a relatively simple account of blow-up in systems of interaction-diffusion equations.
Then the book concentrates on mechanics applications. In particular it deals with the Euler equations, Navier--Stokes equations, models for glacier physics, Korteweg--de-Vries equations, and ferro-hydrodynamics. Blow-up is treated in Volterra equations, too, stressing how these equations arise in mechanics, e.g. in combustion theory. The novel topic of chemotaxis in mathematical biology is also presented. There is a chapter on change of type, from hyperbolic to elliptic, addressing three new and important applications: instability in soils, instability in sea ice dynamics, and also instability in pressure-dependent viscosity flow. Finally, the book includes an exposition of exciting work, very recent and on-going, dealingwith rapid energy growth in parallel shear flows.
The book addresses graduate students as well as researchers in mechanics and applied mathematics.

1. Introduction.- 1.1 Blow-Up in Partial Differential Equations in Applied Mathematics.- 1.2 Methods of Establishing Non-existence and Growth Solutions.- 1.3 Finite Time Blow-Up Systems with Convection.- 2. Analysis of a First-Order System.- 2.1 Conditional Decay of Solutions.- 2.2 Boundedness of Solutions.- 2.3 Unconditional Decay of Solutions.- 2.4 Global Non-existence of Solutions.- 2.5 Numerical Results by Finite Elements.- 3. Singularities for Classical Fluid Equations.- 3.1 Breakdown for First-Order Systems.- 3.2 Blow-Up of Solutions to the Euler Equations.- 3.3 Blow-Up of Solutions to the Navier-Stokes Equations.- 4. Catastrophic Behaviour in Other Non-linear Fluid Theories.- 4.1 Non-existence on Unbounded Domains.- 4.2 A Model for a Second Grade Fluid in Glacier Physics.- 4.3 Blow-Up for Generalised KdeV Equations.- 4.4 Very Rapid Growth in Ferrohydrodynamics.- 4.5 Temperature Blow-Up in an Ice Sheet.- 5. Blow-Up in Volterra Equations.- 5.1 Blow-Up for a Solution to a VolterraEquation.- 5.2 Blow-Up for a Solution to a System of Volterra Equations.- 6. Chemotaxis.- 6.1 Mathematical Theories of Chemotaxis.- 6.2 Blow-Up in Chemotaxis When There Are Two Diffusion Terms.- 6.3 Blow-Up in Chemotaxis with a Single Diffusion Term.- 7. Change of Type.- 7.1 Instability in a Hypoplastic Material.- 7.2 Instability in a Viscous Plastic Model for Sea Ice Dynamics.- 7.3 Pressure Dependent Viscosity Flow.- 8. Rapid Energy Growth in Parallel Flows.- 8.1 Rapid Growth in Incompressible Viscous Flows.- 8.2 Transient Growth in Compressible Flows.- 8.3 Shear Flow in Granular Materials.- 8.4 Energy Growth in Parallel Flows of Superimposed Viscous Fluids.