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

The chemotaxis system

$$\begin{aligned} \textstyle\begin{cases} u_{t}=\Delta u - \chi \nabla \cdot (u|\nabla v|^{p-2}\nabla v) + \lambda u - \mu u^{\kappa }, \\ 0=\Delta v + u - h(u,v) \end{cases}\displaystyle \end{aligned}$$

(∗)

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