Qualitative Behavior of Solutions of a Chemotaxis System with Flux Limitation and Nonlinear Signal Production

In this paper we consider radially symmetric solutions of the following parabolic-elliptic cross-diffusion system $\{\begin{array}{l}\begin{array}{rl}& u_t=\mathrm{\Delta }u-\mathrm{\nabla }(uf(|\mathrm{\nabla }v{|}^2)\mathrm{\nabla }v),\\ & 0=\mathrm{\Delta }v-\mu (t)+g(u),\mu (t)=\frac{1}{|\mathrm{\Omega }|}{\int }_{\mathrm{\Omega }}g(u(\cdot ,t))dx\\ & u(x,0)=u_0(x),\end{array}\end{array}$ \( \left \{ \textstyle\begin{array}{l} \begin{aligned} &u_{t} = \Delta u - \nabla (u f(|\nabla v|^{2} )\nabla v), \\ &0= \Delta v -\mu (t)+ g(u), \quad \mu (t)= \frac{1}{|\Omega |} \int _{\Omega } g(u(\cdot , t))dx \\ &u(x,0)= u_{0}(x), \end{aligned} \end{array}\displaystyle \right . \) in $\mathrm{\Omega }\times (0,\mathrm{\infty })$ $\Omega \times (0,\infty )$ , with $\mathrm{\Omega }$ $\Omega $ a ball in ${\mathbb{R}}^N$ $\mathbb{R}^{N}$ , $N\ge 1$ $N\geq 1$ under homogeneous Neumann boundary conditions, $g(u)$ $g(u)$ a regular function with the prototype $g(u)=u^k$ $g(u)= u^{k}$ , $u\ge 0$ $u\geq 0$ , $k>0$ $k>0$ . The function $f(\xi )=k_f{(1+\xi )}^{-\alpha }$ $f(\xi ) = k_{f} (1+ \xi )^{-\alpha }$ , $k_f>0$ $k_{f} >0$ , describes gradient-dependent limitation of cross diffusion fluxes. Under suitable conditions on the data, we prove that the solution is global in time. If $N\ge 3$ $N\geq 3$ , under conditions on $f$ $f$ , $g$ $g$ and initial data, we prove that if the solution $u(x,t)$ $u(x,t)$ blows up in $L^{\mathrm{\infty }}$ $L^{\infty }$ -norm at finite time $T_{max}$ $T_{max}$ then for some $p>1$ $p>1$ it blows up also in $L^p$ $L^{p}$ -norm. Moreover a lower bound of blow-up time is derived.