In this paper, we study the initial-boundary value problem of a repulsion Keller–Segel system with a logarithmic sensitivity modelling the reinforced random walk. By establishing an energy–dissipation identity, we prove the existence of classical solutions in two dimensions as well as existence of weak solutions in the three-dimensional setting. Moreover, it is shown that the weak solutions enjoy an eventual regularity property, i.e., it becomes regular after certain time T > 0. An exponential convergence rate towards the spatially homogeneous steady states is obtained as well. We adopt a new approach developed recently by the author to study the eventual regularity. The argument is based on observation of the exponential stability of constant solutions in scaling-invariant spaces together with certain dissipative property of the global solutions in the same spaces.

中文翻译：

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关于具有对数灵敏度的排斥 Keller-Segel 系统

在本文中，我们研究了斥力 Keller-Segel 系统的初始边值问题，该系统具有对增强随机游走建模的对数灵敏度。通过建立能量耗散恒等式，我们证明了经典解在二维中的存在以及在三维设置中弱解的存在。此外，表明弱解具有最终规律性，即在一定时间后变为规律性吨> 0. 也获得了向空间均匀稳态的指数收敛速度。我们采用作者最近开发的一种新方法来研究最终规律。该论点基于对尺度不变空间中常数解的指数稳定性的观察，以及相同空间中全局解的某些耗散特性。