  In this paper, we study a system of three evolutionary operator equations
involving fractional powers of selfadjoint, monotone, unbounded, linear
operators having compact resolvents. This system constitutes a generalization
of a phase field system of Cahn-Hilliard type modelling tumor growth that has
been proposed in Hawkins-Daarud et al. (Int. J. Numer. Math. Biomed. Eng. 28
(2012), 3-24) and investigated in recent papers co-authored by the present
authors and E. Rocca. The model consists of a Cahn-Hilliard equation for the
tumor cell fraction, coupled to a reaction-diffusion equation for a function S
representing the nutrient-rich extracellular water volume fraction. Effects due
to fluid motion are neglected. The generalization investigated in this paper is
motivated by the possibility that the diffusional regimes governing the
evolution of the different constituents of the model may be of different (e.g.,
fractional) type. Under rather general assumptions, well-posedness and
regularity results are shown. In particular, by writing the equation governing
the evolution of the chemical potential in the form of a general variational
inequality, also singular or nonsmooth contributions of logarithmic or of
double obstacle type to the energy density can be admitted.
