Seminar 2. The Abstract Cauchy Problem

This session investigates the problem of finding solutions to the abstract Cauchy problem. It will be shown that even for a densely defined operator, a classical solution does not exist for all initial data. This leads to the concept of a mild (or generalised) solution, defined by the action of the semigroup. The differentiability of these mild solutions and their relation to the domain of the semigroup generator are examined. The properties of a $C_0$-semigroup and its connection to the original differential equation are studied in detail.