Dynamics of Evolutionary Equations pp 141-266 | Cite as

# Basic Theory of Evolutionary Equations

Chapter

## Abstract

In the last chapter, we presented a theory describing solutions of a linear evolutionary equation and then the nonlinear evolutionary equation
.

*ə*_{t}*u*+*Au*= 0, where \( {\partial _t}u = \frac{d}{{dt}}u \) on a Banach space*W*, in terms of*C*_{o}-semigroups. As we have seen, this theory allows one to construct mild solutions of many linear partial differential equations with constant coefficients. Our objective in this chapter is to generalize this theory so that it applies first to the linear inhomogeneous equation$$
{\partial _t}u + Au = f(t)
$$

(40.1)

$$
{\partial _t}u + Au = F(u)
$$

(40.2)

## Keywords

Banach Space Weak Solution Evolutionary Equation Strong Solution Mild Solution
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© Springer Science+Business Media New York 2002