Basic Theory of Evolutionary Equations

Abstract

In the last chapter, we presented a theory describing solutions of a linear 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))

and then the nonlinear evolutionary equation

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

((40.2))

.