We recount and discuss some of the most important methods and blow-up criteria for analyzing solutions of Keller-Segel chemotaxis models. First, we discuss the results concerning the global existence, boundedness and blow-up of solutions to parabolic-elliptic type models. Thereafter we describe the global existence, boundedness and blow-up of solutions to parabolic-parabolic models. The numerical analysis of these models is still at a rather early stage only. We recollect quite a few of the known results on numerical methods also and direct the attention to a number of open problems in this domain.

中文翻译：

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Keller-Segel趋化模型：综述

我们叙述并讨论了一些最重要的方法和爆破标准，用于分析Keller-Segel趋化性模型的解决方案。首先，我们讨论有关抛物线型椭圆模型的整体存在性，有界性和爆炸性的结果。此后，我们描述了抛物线-抛物线模型解的整体存在性，有界性和爆炸性。这些模型的数值分析仍处于早期阶段。我们还回顾了许多关于数值方法的已知结果，并将注意力转移到该领域中许多未解决的问题上。