  In this work we analyse a PDE-ODE problem modelling the evolution of a
Glioblastoma, which includes an anisotropic nonlinear diffusion term with a
diffusion velocity increasing with respect to vasculature. First, we prove the
existence of global in time weak-strong solutions using a regularization
technique via an artificial diffusion in the ODE-system and a fixed point
argument. In addition, stability results of the critical points are given under
some constraints on parameters. Finally, we design a fully discrete finite
element scheme for the model which preserves the pointwise and energy estimates
of the continuous problem.
