We consider a model describing the evolution of a tumor inside a host tissue
in terms of the parameters φp, φd (proliferating and dead
cells, respectively), u (cell velocity) and n (nutrient concentration). The
variables φp, φd satisfy a Cahn-Hilliard type system with
nonzero forcing term (implying that their spatial means are not conserved in
time), whereas u obeys a form of the Darcy law and n satisfies a
quasistatic diffusion equation. The main novelty of the present work stands in
the fact that we are able to consider a configuration potential of singular
type implying that the concentration vector (φp,φd) is
constrained to remain in the range of physically admissible values. On the
other hand, in view of the presence of nonzero forcing terms, this choice gives
rise to a number of mathematical difficulties, especially related to the
control of the mean values of φp and φd. For the resulting
mathematical problem, by imposing suitable initial-boundary conditions, our
main result concerns the existence of weak solutions in a proper regularity
class.Comment: 41 page