论文标题
在排斥凯勒 - 具有对数灵敏度的识别系统
On a Repulsion Keller--Segel System with a Logarithmic Sensitivity
论文作者
论文摘要
在本文中，我们研究了排斥凯勒（Keller）的初始有限价值问题 - segel系统具有对数灵敏度，对加强随机行走进行了建模。通过建立能量驱动身份，我们证明了在三维环境中二维以及弱解的存在。此外，已经表明，弱解决方案享有最终的规律性属性，即，在一定时间后$ t> 0 $。也获得了指数的收敛速率，也获得了空间均匀的稳态。我们采用了作者\ cite {J19}最近开发的一种新方法来研究最终的规律性。该论点是基于对缩放不变空间中常数溶液的指数稳定性以及在同一空间中全局溶液的某些耗散特性的指数稳定性。
In this paper, we study the initial-boundary value problem of a repulsion Keller--Segel system with a logarithmic sensitivity modeling 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 enjoys an eventual regularity property, i.e., it becomes regular after certain time $T>0$. An exponential convergence rate toward the spatially homogeneous steady states is obtained as well. We adopt a new approach developed recently by the author \cite{J19} 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.
