  We investigate the dynamics of a nonlinear model for tumor growth within a
cellular medium. In this setting the "tumor" is viewed as a multiphase flow
consisting of cancerous cells in either proliferating phase or quiescent phase
and a collection of cells accounting for the "waste" and/or dead cells in the
presence of a nutrient. Here, the tumor is thought of as a growing continuum
$\Omega$ with boundary $\partial \Omega$ both of which evolve in time. The key
characteristic of the present model is that the total density of cancerous
cells is allowed to vary, which is often the case within cellular media. We
refer the reader to the articles \cite{Enault-2010}, \cite{LiLowengrub-2013}
where compressible type tumor growth models are investigated. Global-in-time
weak solutions are obtained using an approach based on penalization of the
boundary behavior, diffusion, viscosity and pressure in the weak formulation,
as well as convergence and compactness arguments in the spirit of Lions
\cite{Lions-1998} (see also \cite{Feireisl-book, DT-MixedModel-2013}).
