  We investigate the dynamics of a nonlinear system modeling tumor growth with
drug application. The tumor is viewed as a mixture consisting of proliferating,
quiescent and dead cells as well as a nutrient in the presence of a drug. The
system is given by a multi-phase flow model: the densities of the different
cells are governed by a set of transport equations, the density of the nutrient
and the density of the drug are governed by rather general diffusion equations,
while the velocity of the tumor is given by Brinkman's equation. The domain
occupied by the tumor in this setting is a growing continuum $\Omega$ with
boundary $\partial \Omega$ both of which evolve in time. Global-in-time weak
solutions are obtained using an approach based on penalization of the boundary
behavior, diffusion and viscosity in the weak formulation. Both the solutions
and the domain are rather general, no symmetry assumption is required and the
result holds for large initial data. This article is part of a research program
whose aim is the investigation of the effect of drug application in tumor
growth.
