SIAM Journal on Mathematical Analysis, Volume 53, Issue 1, Page 973-1003, January 2021.
In recent years, a lot of attention has been drawn to the question of whether logistic kinetics is sufficient to enforce the global existence of classical solutions or to prevent finite-time blow-up in various chemotaxis models. However, for several important chemotaxis models, only in the space two dimensional setting has it been shown that logistic kinetics is sufficient to enforce the global existence of classical solutions (see [K. Fujie, M. Winkler, and T. Yokota, Nonlinear Anal., 109 (2014), pp. 56--71], [J. I. Tello and W. Winkler, Comm. Partial Differential Equations, 32 (2007), pp. 849--877]). The current paper gives a confirmed answer to the above question for the following parabolic-elliptic chemotaxis system with singular sensitivity and logistic source in any space dimensional setting: $u_t=\Delta u-\chi\nabla\cdot (\frac{u}{v} \nabla v)+u(a(x,t)-b(x,t) u)$ and $0=\Delta v-\mu v+\nu u$ under the homogeneous Neumann boundary conditions in a bounded domain $\Omega \subset \mathbb{R}^n$, where $\chi$ is the singular chemotaxis sensitivity coefficient, $a(x,t)$ and $b(x,t)$ are positive smooth functions, and $\mu,\nu$ are positive constants. We prove that, for every given nonnegative initial data $0\not\equiv u_0\in C^0(\bar \Omega)$, the system has a unique globally defined classical solution $(u(t,x;u_0),v(t,x;u_0))$ with $u(0,x;u_0)=u_0(x)$, which shows that, in any space dimensional setting, logistic kinetics is sufficient to enforce the global existence of classical solutions and hence prevents the occurrence of finite-time blow-up even for arbitrarily large $\chi$. In addition, the solutions are shown to be uniformly bounded under the conditions $a_{\inf}>\frac{\mu \chi^2}{4}$ when $\chi \leq 2$ and $a_{\inf}>\mu (\chi-1)$ when $\chi>2$.

中文翻译：

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在任何尺寸设置下具有奇异灵敏度的抛物线-椭圆趋化模型中逻辑源的有限时间爆破预防

SIAM数学分析杂志，第53卷，第1期，第973-1003页，2021年1月。
近年来，人们对逻辑动力学是否足以增强经典解的整体存在性或防止各种趋化性模型中的有限时间爆炸问题已引起了很多关注。但是，对于一些重要的趋化模型，只有在空间二维环境中，才证明逻辑动力学足以强制经典解的整体存在（请参阅[K. Fujie，M。Winkler和T. Yokota，非线性肛门，第109页（2014），第56--71页，[JI Tello和W. Winkler，Comm。偏微分方程，32（2007），第849--877页]）。对于以下具有抛物线椭圆趋化系统的奇异灵敏度和逻辑源，在任何空间尺寸设置下，当前论文都给出了上述问题的肯定答案：$ u_t = \ Delta u- \ chi \ nabla \ cdot（\ frac {u} {v} \ nabla v）+ u（a（x，t）-b（x，t）u）$和$ 0 = \ Delta v- \ mu v + \ nu u $在有界域$ \ Omega \ subset \ mathbb {R} ^ n $中的齐次Neumann边界条件下，其中$ \ chi $是奇异的趋化敏感性系数$ a（x， t）$和$ b（x，t）$是正光滑函数，而$ \ mu，\ nu $是正常数。我们证明，对于C ^ 0（\ bar \ Omega）$中的每个给定非负初始数据$ 0 \ not \ equiv u_0 \，系统具有唯一的全局定义的经典解$（u（t，x; u_0），v （t，x; u_0））$具有$ u（0，x; u_0）= u_0（x）$，这表明在任何空间维设置中，逻辑动力学都足以强制经典解的全局存在，因此即使对于任意大的\\ chi $，也可以防止发生限时爆炸。此外，