Cauchy Problems

Abstract

In this chapter we study systematically well-posedness of the Cauchy problem. Given a closed operator A on a Banach space X we will see in Section 3.1 that the abstract Cauchy problem

$$\left\{\begin{array}{lcr} u^\prime(t) = Au(t) \,\,(t\geq0), \\ u(0) = x, \end{array}\right.$$

is mildly well-posed (i.e., for each

$$x \,\epsilon \,X$$

there exists a unique mild solution) if and only if the resolvent of A is a Laplace transform; and this in turn is the same as saying that A generates a C₀-semigroup. Well-posedness in a weaker sense will lead to generators of integrated semigroups (Section 3.2).