Behavior in time of solutions of a Keller–Segel system with flux limitation and source term

In this paper we consider radially symmetric solutions of the following parabolic--elliptic cross-diffusion system \begin{equation*} \begin{cases} u_t = \Delta u - \nabla \cdot (u f(|\nabla v|^2 )\nabla v) + g(u), & \\[2mm] 0= \Delta v -m(t)+ u , \quad \int_{\Omega}v \,dx=0, & \\[2mm] u(x,0)= u_0(x), & \end{cases} \end{equation*} in $\Omega \times (0,\infty)$, with $\Omega$ a ball in $\mathbb{R}^N$, $N\geq 3$, under homogeneous Neumann boundary conditions, where $g(u)= \lambda u - \mu u^k$ , $\lambda >0, \ \mu >0$, and $ k >1$, $f(|\nabla v|^2 )= k_f(1+ |\nabla v|^2)^{-\alpha}$, $\alpha>0$, which describes gradient-dependent limitation of cross diffusion fluxes. The function $m(t)$ is the time dependent spatial mean of $u(x,t)$ i.e. $m(t) := \frac 1 {|\Omega|} \int_{\Omega} u(x,t) \,dx$. Under smallness conditions on $\alpha$ and $k$, we prove that the solution $u(x,t)$ blows up in $L^{\infty}$-norm at finite time $T_{max}$ and for some $p>1$ it blows up also in $L^p$-norm. In addition a lower bound of blow-up time is derived. Finally, under largeness conditions on $\alpha$ or $k$, we prove that the solution is global and bounded in time