This paper is concerned with a parabolic–parabolic–elliptic system arising as a simplified model for the initial phase of tumor-related angiogenesis. As essential characteristics, this system contains a cascade-like coupling of two chemotaxis processes involving signal production as in classical Keller–Segel systems, linked to a further repulsive cross-diffusive mechanism. It is shown that in $n$-dimensional bounded convex domains with $n\le 3$ and for any given suitably regular initial data, a suitable assumption on largeness of the parameter corresponding to the latter repulsion term ensures global existence of a bounded classical solution, and hence rules out the possibility of finite-time explosions which in the absence of any such chemorepulsion are known to occur at least in a certain borderline case.

本文关注的是抛物线-抛物线-椭圆形系统，它是肿瘤相关血管生成初始阶段的简化模型。作为基本特征，该系统包含两个趋化过程的级联耦合，涉及经典的Keller-Segel系统中涉及信号产生的过程，并与进一步的排斥性交叉扩散机制相联系。结果表明$ñ$维有界凸域 $ñ\le 3$ 对于任何给定的适当正则初始数据，对与后一个推斥项相对应的参数的大小的适当假设可确保有界经典解的整体存在，因此排除了在不存在任何此类限制的情况下进行有限时间爆炸的可能性已知至少在某些临界情况下会发生化学脉冲。