ABSTRACT
Semigroup theory provides a unified and powerful tool for the study of differential equations on Banach spaces covering systems described by ordinary differential equations, partial differential equations, functional differential equations, and their combinations. The basic idea of the semigroup method is to rewrite an evolutionary partial differential equation as an abstract ordinary differential equation, and to deal with the evolutionary partial differential equation by imitating the initial value problem of the ordinary differential equation. Then, by studying the properties of the semigroup, the people can determine whether this mild solution satisfies the abstract ordinary differential equation. This chapter is devoted to a brief review of basic results of semigroup theory and to establishing the existence, uniqueness, extension and regularity of mild solutions.
