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.
翻译:在这项工作中,我们分析一个PDE-ODE问题模型,以模拟Glioblastoma的演变,其中包括一个古代非线性非线性扩散术语,其扩散速度在血管方面呈上升趋势。首先,我们证明,通过在ODE系统中人工扩散和固定点参数,采用正规化技术,全球在时间上存在薄弱的解决办法。此外,关键点的稳定性结果在参数上受到一些限制。最后,我们为模型设计了一个完全离散的有限要素方案,以保持对持续问题的精确和能源估计。