Evolution semigroups and product formulas for nonautonomous Cauchy problems

In this paper, we study nonautonomous Cauchy problems (NCP) { (u) over dot(t) = A(T)u(t) u(s) = x is an element of X for a family of linear operators (A(t))(t is an element of I) on some Banac h space X by means of evolution semigroups. In particular, we characterize "stability" in the so called "hyperbolic case" on the level of evolution se migroups and derive a product formula for the solutions of (NCP). Moreover, in Section 4 we connect the "hyperbolic" and the "parabolic" case by showi ng, that integrals integral(s)(t) A(tau)d tau always define generators. Thi s yields another product formula.