Evolution Semigroups and Product Formulas for Nonautonomous Cauchy Problems

✍ Scribed by Gregor Nickel

• Publisher: John Wiley and Sons
• Year: 2000
• Tongue: English
• Weight: 290 KB
• Volume: 212
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/(sici)1522-2616(200004)212:1<101::aid-mana101>3.0.co;2-3

✦ Synopsis

In this paper, we study nonautonomous Cauchy problems

for a family of linear operators (A(t)) t∈I on some Banach space X by means of evolution semigroups.

In particular, we characterize "stability" in the so called "hyperbolic case" on the level of evolution semigroups and derive a product formula for the solutions of (N CP ). Moreover, in Section 4 we connect the "hyperbolic" and the "parabolic" case by showing, that integrals t s A(τ ) dτ always define generators. This yields another product formula.

1. Kato's stability condition

A necessary and sufficient condition for wellposedness of nonautonomous Cauchy problems in terms of the family (A(t)) t∈I is still lacking (see, e. g., [Go1]). However, at least in the hyperbolic case, all results are based on the classical 1970 paper of Kato [Ka2] and his stability condition. In preliminary results (see, e. g., [Ka1]) the operators A(t) were assumed to generate contraction semigroups, thus Kato's stability condition was automatically satisfied (see below). It was used later, e. g., [DaP-Gr], or [DaP-Si] in combination with more or less complicated regularity conditions to obtain wellposedness of (N CP ).

In our paper, we characterize Kato's stability in the perspective of evolution semigroups and derive two different approximation formulas, see Sections 3 and 4. We start with Kato's basic definition of stability for a family of generators (A(t)) t∈I (cf.

[Pa], p. 131). Since, for the moment, we do not use evolution semigroups, we consider compact intervals I := [0, T ].

Definition 1.1. (Kato -stability.) A family (A(t), D(A(t))) t∈I of generators of C 0semigroups on a Banach space X is called Kato -stable, if there exist constants M ≥ 1