  This paper deals with a general system of equations and conditions arising
from a mathematical model of prostate cancer growth with chemotherapy and
antiangiogenic therapy that has been recently introduced and analyzed (see [P.
Colli et al., Mathematical analysis and simulation study of a phase-field model
of prostate cancer growth with chemotherapy and antiangiogenic therapy effects,
Math. Models Methods Appl. Sci. 30 (2020), 1253-1295], preprint in
arXiv:1907.11618 [math.AP]). The related system includes two evolutionary
operator equations involving fractional powers of selfadjoint, nonnegative,
unbounded linear operators having compact resolvents. Both equations contain
nonlinearities and in particular the equation describing the dynamics of the
tumor phase variable has the structure of a Allen-Cahn equation with
double-well potential and additional nonlinearity depending also on the other
variable, which represents the nutrient concentration. The equation for the
nutrient concentration is nonlinear as well, with a term coupling both
variables. For this system we design an existence, uniqueness and continuous
dependence theory by setting up a careful analysis which allows the
consideration of nonsmooth potentials and the treatment of continuous
nonlinearities with general growth properties.
