Basic Theory of Evolutionary Equations

In the last chapter, we presented a theory describing solutions of a linear evolutionary equation ə_tu + 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 (40.1)$$ {\partial _t}u + Au = f(t) $$ and then the nonlinear evolutionary equation (40.2)$$ {\partial _t}u + Au = F(u) $$.