Perturbations of non-autonomous second-order abstract Cauchy problems
In this talk we present time-dependent perturbations of second-order non-autonomous abstract Cauchy problems associated to a family of operators with constant domain. We make use of the equivalence to a first-order non-autonomous abstract Cauchy problem in a product space, which we elaborate in full detail. As an application we provide a perturbed non-autonomous wave equation.
Autonomous second-order abstract Cauchy problems which often occur in the context of wave equations, have been studied intensively by several authors in the past. In contrast to the first-order problem, where (classical) solutions are given by C0-semigroups, one needs another solution concept for the second-order case, the so-called cosine and sine families. Like the Hille–Yosida generation theorem for strongly continuous semigroups, one can also characterize generators of cosine families.
Non-autonomous second-order abstract Cauchy problems have been studied first by Kozak and later by Bochenek, Winiarska and Lan, just to mention a few. The classical idea helps to reduce the non-autonomous second-order abstract Cauchy problem again to a first-order problem.
The goal is to establish a bounded perturbation result for non-autonomous second-order abstract Cauchy problems. As mentioned above, we also discuss the non-autonomous wave equation as an example.
This is joint work with C. Seifert (Technical University Hamburg, Germany).