  We consider a family of variational regularization functionals for a generic
inverse problem, where the data fidelity and regularization term are given by
powers of a Hilbert norm and an absolutely one-homogeneous functional,
respectively, and the regularization parameter is interpreted as artificial
time. We investigate the small and large time behavior of the associated
solution paths and, in particular, prove finite extinction time for a large
class of functionals. Depending on the powers, we also show that the solution
paths are of bounded variation or even Lipschitz continuous. In addition, it
will turn out that the models are "almost" mutually equivalent in terms of the
minimizers they admit. Finally, we apply our results to define and compare two
different non-linear spectral representations of data and show that only one of
it is able to decompose a linear combination of non-linear eigenfunctions into
the individual eigenfunctions. For that purpose, we will also briefly address
piecewise affine solution paths.
