Blow-up and boundedness in a chemotaxis system with flux-limited diffusion and logistic source

In this paper we consider radially symmetric solutions of the parabolic–elliptic cross-diffusion system with flux limitation term, $\left(\begin{array}{cc}{u}_t=\nabla \cdot \left(\frac{u\nabla u}{\sqrt{u^2+{|\nabla u|}^2}}\right)-\chi \nabla \cdot (u\nabla v)+\lambda u-\mu u^k,& x\in \Omega ,t>0,\\ 0=\Delta v-m\left(t\right)+u,m\left(t\right)=\frac{1}{\left|\Omega \right|}{\int }_{\Omega }u(x,t)dx,& x\in \Omega ,t>0,\\ u(x,0)=u_0\left(x\right),& x\in \Omega \end{array}\right)$under no-flux boundary conditions, where $\Omega =B_R\left(0\right)\subset R^N$ ($N\ge 1$) is a ball, $\chi$, $\lambda$, $\mu$ are positive constants and $k>1$ . Under suitable conditions on the data, we prove that the solution is global in time. If $N\ge 3$, under conditions on the data, we prove that the solution $u(x,t)$ blows up in $L^{\infty }$-norm at finite time $T_{max}$. Moreover for some $p>1$ we prove that the solution blows up also in $L^p$-norm and a lower bound of the blow-up time is derived.