ABSTRACT
Chapter 2
Abstract Distribution
Methods
2.1 The Cauchy problem
In the previous chapter we showed that under dierent conditions on the
resolvent of an operator A : D(A) X ! X , the Cauchy problem
u
(t) = Au(t); t 2 [0; T ); T 1; u(0) = x; (CP)
has a unique solution for x from dierent correctness classes stable with
respect to x in dierent norms. In this section, we consider the Cauchy
problem (CP) for any x 2 X in the space of distributions. Our main aim
is to obtain necessary and sucient conditions for well-posedness in the
space of distributions in terms of the resolvent of A. We show that they
are the same as in the case when A is the generator of an integrated semi-
group. That is, only the Cauchy problem with a generator of an integrated
semigroup can be well-posed in the space of distributions.
