Partial functional differential equation arise from many biological, chemical and physical systems which are characterized by both spatial and temporal variables and exhibit various spatiotemporal patterns. The systematic study of such equations from the dynamical systems and semi-groups point of view began in the 70s, and considerable advances have been achieved since then. A partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives. PDEs are used to formulate problems involving functions of several variables, and are either solved by hand, or used to create a relevant computer model. PDEs can be used to describe a wide variety of phenomena such as sound, heat, electrostatics, electrodynamics, fluid flow, or elasticity. These seemingly distinct physical phenomena can be formalised identically in terms of PDEs, which shows that they are governed by the same underlying dynamic. This book provides an introduction to the qualitative theory and applications of partial functional differential equations from the viewpoint of dynamical systems. The main emphasis of the book is on reaction-diffusion equations with delayed nonlinear reaction terms and on the joint effect of the time delay and spatial diffusion on the spatial-temporal patterns of the considered systems.
