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

$$ \left \{ \textstyle\begin{array}{l} \begin{aligned} &u_{t} = \Delta u - \nabla (u f(|\nabla v|^{2} )\nabla v), \\ &0= \Delta v -\mu (t)+ g(u), \quad \mu (t)= \frac{1}{|\Omega |} \int _{\Omega } g(u(\cdot , t))dx \\ &u(x,0)= u_{0}(x), \end{aligned} \end{array}\displaystyle \right . $$

in \(\Omega \times (0,\infty )\), with \(\Omega \) a ball in \(\mathbb{R}^{N}\), \(N\geq 1\) under homogeneous Neumann boundary conditions, \(g(u)\) a regular function with the prototype \(g(u)= u^{k}\), \(u\geq 0\), \(k>0\). The function \(f(\xi ) = k_{f} (1+ \xi )^{-\alpha }\), \(k_{f} >0\), describes gradient-dependent limitation of cross diffusion fluxes. Under suitable conditions on the data, we prove that the solution is global in time. If \(N\geq 3\), under conditions on \(f\), \(g\) and initial data, we prove that if the solution \(u(x,t)\) blows up in \(L^{\infty }\)-norm at finite time \(T_{max}\) then for some \(p>1\) it blows up also in \(L^{p}\)-norm. Moreover a lower bound of blow-up time is derived.