We consider time-dependent inverse problems in a mathematical setting using
Lebesgue-Bochner spaces. Such problems arise when one aims to recover
parameters from given observations where the parameters or the data depend on
time. There are various important applications being subject of current
research that belong to this class of problems. Typically inverse problems are
ill-posed in the sense that already small noise in the data causes tremendous
errors in the solution. In this article we present two different concepts of
ill-posedness: temporally (pointwise) ill-posedness and uniform ill-posedness
with respect to the Lebesgue-Bochner setting. We investigate the two concepts
by means of a typical setting consisting of a time-depending observation
operator composed by a compact operator. Furthermore we develop regularization
methods that are adapted to the respective class of ill-posedness.Comment: 21 pages, no figure