  We investigate a recently proposed cross-diffusion system modelling the
growth of gliobastoma taking into account size exclusion both in the migration
and proliferation process. In addition to degenerate nonlinear cross-diffusion
the model includes reaction terms as in the Fisher-Kolmogorov equation and
linear ones modelling transition between states of proliferation and migration.
  We discuss the mathematical structure of the system and provide a complete
existence analysis in spatial dimension one. The proof is based on exploiting
partial entropy dissipation techniques and fully implicit time discretisations.
In order to prove existence of the latter appropriate variational and
fixed-point techniques are used, together with suitable a-priori estimates.
Moreover, we review the existence of travelling waves and their relation to
potential growth or decay of glioblastoma. Finally we provide extensive
numerical studies in one and two spatial dimensions, including the effect of
anisotropic diffusions as found in neural tissues.
