This paper deals with the homogeneous Neumann boundary value problem for chemotaxis system $$\begin{aligned} \textstyle\begin{cases} u_{t} = \Delta u - \nabla \cdot (u\nabla v)+\kappa u-\mu u^{\alpha }, & x\in \Omega, t>0, \\ v_{t} = \Delta v - uv, & x\in \Omega, t>0, \end{cases}\displaystyle \end{aligned}$$ in a smooth bounded domain $\Omega \subset \mathbb{R}^{N}(N\geq 2)$ , where $\alpha >1$ and $\kappa \in \mathbb{R},\mu >0$ for suitably regular positive initial data. When $\alpha \ge 2$ , it has been proved in the existing literature that, for any $\mu >0$ , there exists a weak solution to this system. We shall concentrate on the weaker degradation case: $\alpha <2$ . It will be shown that when $N<6$ , any sublinear degradation is enough to guarantee the global existence of weak solutions. In the case of $N\geq 6$ , global solvability can be proved whenever $\alpha >\frac{4}{3}$ . It is interesting to see that once the space dimension $N\ge 6$ , the qualified value of α no longer changes with the increase of N.

中文翻译：

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逻辑阻尼在多大程度上影响高维趋化系统的整体溶解度？

本文讨论了趋化系统的齐次Neumann边值问题$$ \ begin {aligned} \ textstyle \ begin {cases} u_ {t} = \ Delta u-\ nabla \ cdot（u \ nabla v）+ \ kappa u -\ mu u ^ {\ alpha}，＆x \ in \ Omega，t> 0，\\ v_ {t} = \ Delta v-uv，＆x \ in \ Omega，t> 0，\ end {cases} \ displaystyle \ end {aligned} $$在一个光滑有界域$ \ Omega \ subset \ mathbb {R} ^ {N}（N \ geq 2）$中，其中$ \ alpha> 1 $和$ \ kappa \ in \ mathbb {R}，\ mu> 0 $可得到适当正则的正初始数据。当$ \ alpha \ ge 2 $时，现有文献证明，对于$ \ mu> 0 $，该系统存在一个弱解。我们将专注于较弱的降级情况：$ \ alpha <2 $。将显示出，当$ N <6 $时，任何亚线性降级足以保证弱解的整体存在。在$ N \ geq 6 $的情况下，只要$ \ alpha> \ frac {4} {3} $，就可以证明全局可解性。有趣的是，一旦空间维$ N \ ge 6 $，α的合格值就不会随着N的增加而变化。