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 {u(t)=Delta u-del(uf(|del v|(2))del v), 0=Delta v-mu(t)+g(u), mu(t)=1/|Omega|integral(Omega)g(u(& sdot;,t))dx u(x,0)=u(0)(x), in Omega x(0,infinity), with Omega a ball in R-N, N >= 1 under homogeneous Neumann boundary conditions, g(u) a regular function with the prototype g(u)=u(k), u >= 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 >= 3, under conditions on f, g and initial data, we prove that if the solution u(x,t)blows up in L-infinity-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.