In this paper we study the zero-flux chemotaxis-system$$ \textstyle\begin{cases} u_{t}=\Delta u -\chi\nabla\cdot(\frac{u}{v} \nabla v) \\ v_{t}=\Delta v-f(u)v \end{cases} $$in a smooth and bounded domain \(\varOmega\) of \(\mathbb{R}^{2}\), with \(\chi >0\) and \(f\in C^{1}(\mathbb{R})\) essentially behaving like \(u^{\beta}\), \(0<\beta<1\). Precisely for \(\chi<1\) and any sufficiently regular initial data \(u(x,0)\geq0\) and \(v(x,0)>0\) on \(\bar{\varOmega}\), we show the existence of global classical solutions. Moreover, if additionally \(m:=\int _{\varOmega}u(x,0) dx\) is sufficiently small, then also their boundedness is achieved.

中文翻译：

---

具有奇异敏感性的趋化消费模型解的整体存在性和有界性

在本文中，我们研究零通量趋化系统$$ \ textstyle \ begin {cases} u_ {t} = \ Delta u-\ chi \ nabla \ cdot（\ frac {u} {v} \ nabla v）\ \ v_ {t} = \ Delta vf（u）v \ end {cases} $$在\（\ mathbb {R} ^ {2} \）的光滑有界域\（\ varOmega \）中，且\ \ chi> 0 \）和\（f在C ^ {1}（\ mathbb {R}）\）中的行为本质上类似于\（u ^ {\ beta} \），\（0 <\ beta <1 \）。正是为了\（\智<1 \）和任何足够规则的初始数据\（U（x，0）\ geq0 \）和\（V（X，0）> 0 \）上\（\栏{\ varOmega} \），说明存在全局经典解。而且，如果另外\（m：= \ int _ {\ varOmega} u（x，0）dx \）足够小，那么它们的有界性也得以实现。