Boundedness and Finite-Time Blow-up in a Chemotaxis System with Flux Limitation and Logistic Source

The chemotaxis system ∗ $\{\begin{array}{c}{u}_t=\mathrm{\Delta }u-\chi \mathrm{\nabla }\cdot (u{|\mathrm{\nabla }v|}^{p-2}\mathrm{\nabla }v)+\lambda u-\mu u^{\kappa },\hfill \\ 0=\mathrm{\Delta }v+u-h(u,v)\hfill \end{array}$ \(\begin{aligned} \textstyle\begin{cases} u_{t}=\Delta u - \chi \nabla \cdot (u|\nabla v|^{p-2}\nabla v) + \lambda u - \mu u^{\kappa }, \\ 0=\Delta v + u - h(u,v) \end{cases}\displaystyle \end{aligned}\) is considered in a smoothly bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$ $\Omega \subset \mathbb{R}^{n}$ ( $n\in \mathbb{N}$ $n \in \mathbb{N}$ ), where $\chi >0$ $\chi > 0$ , $p>1$ $p > 1$ , $\lambda \ge 0$ $\lambda \ge 0$ , $\mu >0$ $\mu > 0$ , $\kappa >1$ $\kappa > 1$ , and $h=v$ $h = v$ or $h=\frac{1}{|\mathrm{\Omega }|}{\int }_{\mathrm{\Omega }}u$ $h = \frac{1}{|\Omega |} \int _{\Omega } u$ . It is firstly proved that if $n=1$ $n = 1$ and $p>1$ $p > 1$ is arbitrary, or $n\ge 2$ $n \ge 2$ and $p\in (1,\frac{n}{n-1})$ $p \in (1, \frac{n}{n-1})$ , then for all continuous initial data a corresponding no-flux type initial-boundary value problem for $(\ast )$ $(\ast )$ admits a globally defined and bounded weak solution. Secondly, it is shown that if $n\ge 2$ $n \ge 2$ , $\mathrm{\Omega }=B_R(0)\subset {\mathbb{R}}^n$ $\Omega = B_{R}(0) \subset \mathbb{R}^{n}$ is a ball with some $R>0$ $R > 0$ , $p>\frac{n}{n-1}$ $p > \frac{n}{n-1}$ and $\kappa >1$ $\kappa > 1$ is small enough, then one can find a nonnegative radially symmetric function $u_0$ $u_{0}$ and a weak solution of $(\ast )$ $(\ast )$ with initial datum $u_0$ $u_{0}$ which blows up in finite time.