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 = Δu - \nabla \cdot (u f(|\nabla v|^2 )\nabla v) + g(u), & \[2mm] 0= Δv -m(t)+ u , \quad \int_Ωv ,dx=0, & \[2mm] u(x,0)= u_0(x), & \end{cases} \end{equation*} in $\mathrm{\Omega }\times (0,\mathrm{\infty })$, with $\mathrm{\Omega }$ a ball in ${\mathbb{R}}^N$, $N\ge 3$, under homogeneous Neumann boundary conditions, where $g(u)=\lambda u-\mu u^k$ , $\lambda >0,\mu >0$, and $k>1$, $f(\mathrm{\mid }\mathrm{\nabla }v{\mathrm{\mid }}^2)=k_f(1+\mathrm{\mid }\mathrm{\nabla }v{\mathrm{\mid }}^2{)}^{-\alpha }$, $\alpha >0$, 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. $m(t):=\frac{1}{\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }}{\int }_{\mathrm{\Omega }}u(x,t)dx$. Under smallness conditions on $\alpha$ and $k$, we prove that the solution $u(x,t)$ blows up in $L^{\mathrm{\infty }}$-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.