Perturbations of non-autonomous second-order abstract Cauchy problems

TL;DR

This paper investigates how time-dependent perturbations affect second-order non-autonomous abstract Cauchy problems, providing a detailed equivalence to first-order problems and applying results to a perturbed wave equation.

Contribution

It introduces a method to analyze perturbations of second-order non-autonomous problems via an equivalence to first-order systems, with detailed theoretical development and application.

Findings

Established equivalence between second-order and first-order non-autonomous problems.

Derived conditions for perturbations to preserve well-posedness.

Applied theory to a perturbed non-autonomous wave equation.

Abstract

In this paper 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.