Blow-up and boundedness in a chemotaxis system with flux-limited diffusion and logistic source

In this paper we consider radially symmetric solutions of the parabolic-elliptic cross-diffusion system with flux limitation term, { u(t)=del & sdot;(u del u/ root u(2)+|del u|(2))-chi del & sdot;(u del v)+lambda u-mu u(k),x is an element of Omega,t>0, 0=Delta v-m(t)+u,m(t)=1/ |Omega|integral(Omega)u(x,t)dx,x is an element of Omega,t>0, u(x,0)=u(0)(x),x is an element of Omega under no-flux boundary conditions, where Omega=B-R(0)subset of R-N (N >= 1) is a ball, chi, lambda, mu are positive constants and k>1 . Under suitable conditions on the data, we prove that the solution is global in time. If N >= 3, under conditions on the data, we prove that the solution u(x,t) blows up in L-infinity-norm at finite time T-max. Moreover for some p>1 we prove that the solution blows up also in L-p-norm and a lower bound of the blow-up time is derived.