摘要：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: $\alpha \frac{4}{3}$ . 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.
