  In this work, we analytically investigate a degenerating PDE system for phase
separation and complete damage processes considered on a nonsmooth
time-dependent domain with mixed boundary conditions. The evolution of the
system is described by a \textit{degenerating} Cahn-Hilliard equation for the
concentration, a doubly nonlinear differential inclusion for the damage
variable and a \textit{degenerating} quasi-static balance equation for the
displacement field. All these equations are highly nonlinearly coupled. Because
of the doubly degenerating character of the system, the doubly nonlinear
differential inclusion and the nonsmooth domain, the structure of the model is
very complex from an analytical point of view.
  A novel approach is introduced for proving existence of weak solutions for
such degenerating coupled system. To this end, we first establish a suitable
notion of weak solutions, which consists of weak formulations of the diffusion
and the momentum balance equation, a variational inequality for the damage
process and a total energy inequality. To show existence of weak solutions,
several new ideas come into play. Various results on shrinking sets and its
corresponding local Sobolev spaces are used. It turns out that, for instance,
on open sets which shrink in time a quite satisfying analysis in Sobolev spaces
is possible. The presented analysis can handle highly nonsmooth regions where
complete damage takes place. To mention only one difficulty, infinitely many
completely damaged regions which are not connected with the Dirichlet boundary
may occur in arbitrary small time intervals.
