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.

中文翻译：

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依赖于脉管系统扩散的混合肿瘤模型的理论和数值分析

在这项工作中，我们分析了胶质母细胞瘤进化的PDE-ODE问题，该问题包括各向异性的非线性扩散项，其扩散速度相对于脉管系统增加。首先，我们通过ODE系统中的人工扩散和不动点参数，使用正则化技术证明了时间上弱强全局解的存在性。另外，在一些参数约束下给出了临界点的稳定性结果。最后，我们为模型设计了一个完全离散的有限元方案，该方案保留了连续问题的逐点估计和能量估计。