A degenerate attraction-repulsion chemotaxis system with logistic-type superlinear degradation

In this paper, we consider radially symmetric solutions of the degenerate cross-diffusion system with flux limitation diffusion and logistic source,$ \begin{equation*} \begin{cases} u_t = \nabla\cdot\Big( \frac { u \nabla u }{\sqrt{u^2+ |\nabla u|^2}} \Big) - \chi \nabla \cdot (u\nabla v) + \xi \nabla \cdot (u\nabla w) + f(u), & \\ 0 = \Delta v +\alpha u - m_{1}(t), & \\ 0 = \Delta w +\gamma u - m_{2}(t), & \end{cases} \end{equation*} $in $ \Omega \times (0,\infty) $, with $ \Omega $ a ball in $ \mathbb{R}^N $, $ N\geq 1 $, and subjected to no-flux boundary conditions. The logistic dampening satisfies $ f(u) = \lambda u - \mu u^{k} $ with $ \lambda,\, \mu $ positive constants and $ k \geq 1 $. If $ N\geq 3 $ and $ \chi\alpha-\xi\gamma>0 $, under certain smallness conditions on logistic degradation, we demonstrate that the solution $ u(x,t) $ exhibits blow-up behavior in $ L^{\infty} $-norm at a finite time. Moreover, for some $ p>N $, we prove that the solution also blows up in $ L^p $-norm. On the other hand, if $ \chi\alpha-\xi\gamma0 $ and $ k $ is large, we prove that the solution is global in time.