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 $$\begin{aligned} {\left\{ \begin{array}{ll} u_t = \Delta u - \nabla \cdot (u f(|\nabla v|^2 )\nabla v) + g(u), &{} \\ 0= \Delta v -m(t)+ u, \quad \int _{\Omega }v \,dx=0, &{} \\ u(x,0)= u_0(x), &{} \end{array}\right. } \end{aligned}$$u t = Δ u - ∇ · ( u f ( | ∇ v | 2 ) ∇ v ) + g ( u ) , 0 = Δ v - m ( t ) + u , ∫ Ω v d x = 0 , u ( x , 0 ) = u 0 ( x ) , in $$\Omega \times (0,\infty )$$Ω × ( 0 , ∞ ) , with $$\Omega $$Ω a ball in $${\mathbb {R}}^N$$R N , $$N\ge 3$$N ≥ 3 , under homogeneous Neumann boundary conditions, where $$g(u)= \lambda u - \mu u^k$$g ( u ) = λ u - μ u k , $$\lambda>0, \ \mu >0$$λ > 0 , μ > 0 , and $$ k >1$$k > 1 , $$f(|\nabla v|^2 )= k_f(1+ |\nabla v|^2)^{-\alpha }$$f ( | ∇ v | 2 ) = k f ( 1 + | ∇ v | 2 ) - α , $$\alpha >0$$α > 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. $$m(t):= \frac{1}{|\Omega |} \int _{\Omega } u(x,t) \,dx$$m ( t ) : = 1 | Ω | ∫ Ω u ( x , t ) d x . Under smallness conditions on $$\alpha $$α and k, we prove that the solution u(x, t) blows up in $$L^{\infty }$$L ∞ -norm at finite time $$T_{max}$$T max and for some $$p>1$$p > 1 it blows up also in $$L^p$$L p -norm. In addition a lower bound of blow-up time is derived. Finally, under largeness conditions on $$\alpha $$α or k, we prove that the solution is global and bounded in time.