We study a modified version of an initial-boundary value problem describing the formation of colony patterns of bacteria Escherichia Coli. The original system of three parabolic equations was studied numerically and analytically and gave insights into the underlying mechanisms of chemotaxis. We focus here on the parabolic-elliptic-parabolic approximation and the hyperbolic-elliptic-parabolic limiting system which describes the case of pure chemotactic movement without random diffusion. We first construct local-in-time solutions for the parabolic-elliptic-parabolic system. Then we prove uniform a priori estimates and we use them along with a compactness argument in order to construct local-in-time solutions for the hyperbolic-elliptic-parabolic limiting system. Finally, we prove that some initial conditions give rise to solutions which blow-up in finite time in the $ L^\infty $ norm in all space dimensions. This last result is true even in space dimension 1, which is not the case for the full parabolic or parabolic-elliptic Keller-Segel systems.

中文翻译：

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描述趋化大肠杆菌菌落的双曲线-椭圆-抛物线 PDE 模型

我们研究了描述细菌大肠杆菌菌落模式形成的初始边界值问题的修改版本。对三个抛物线方程的原始系统进行了数值和分析研究，并深入了解了趋化性的潜在机制。我们在这里关注抛物线-椭圆-抛物线近似和双曲线-椭圆-抛物线限制系统，它们描述了没有随机扩散的纯趋化运动的情况。我们首先为抛物线-椭圆-抛物线系统构建局部时间解。然后我们先验地证明一致估计，我们将它们与紧凑性参数一起使用，以便为双曲椭圆抛物线限制系统构建局部时间解。最后，我们证明了一些初始条件会产生在有限时间内在所有空间维度的 $L^\infty $ 范数中爆炸的解。即使在空间维度 1 中，最后一个结果也是正确的，而对于完全抛物线或抛物线-椭圆 Keller-Segel 系统而言，情况并非如此。